Statistics Chapter 10 Homework

10.1 The standard deviation in the pressure required to open a certain valve is known to be σ=1.2 psi. Due to changes in the manufacturing​ process, the​ quality-control manager feels that the pressure variability has increased. ​(a) State the null and alternative hypotheses in words. ​(b) State the null and alternative hypotheses symbolically. ​(c) Explain what it would mean to make a Type I error. ​(d) Explain what it would mean to make a Type II error. ​(a) State the null hypothesis in words. Choose the correct answer below. A. The standard deviation in the pressure required to open any valve is 1.2 psi. B. The standard deviation in the pressure required to open a certain valve is different from 1.2 psi. C. The standard deviation in the pressure required to open a certain valve is 1.2 psi. D. The standard deviation in the pressure required to open a certain valve is greater than 1.2 psi. State the alternative hypothesis in words. Choose the correct answer below. A. The standard deviation in the pressure required to open a certain valve is different from 1.2 psi. B. The standard deviation in the pressure required to open a certain valve is greater than 1.2 psi. C. The standard deviation in the pressure required to open a certain valve is 1.2 psi. D. The standard deviation in the pressure required to open any valve is 1.2 psi. (b) State the hypotheses symbolically. H0​: ___ __ ___ H1​: ___ ___ ___ ​(Type integers or decimals. Do not​ round.) ​(c) What would it mean to make a Type I​ error? The manager _____ the hypothesis that the pressure variability is ___ ____ ​psi, when the true pressure variability is ____ ____ psi. ​(Type integers or decimals. Do not​ round.) ​(d) What would it mean to make a Type II​ error? The manager ____ the hypothesis that the pressure variability is ____ ____ ​psi, when the true pressure variability is ___ ____ psi. ​(Type integers or decimals. Do not​ round.)

a) C. The standard deviation in the pressure required to open a certain valve is 1.2 psi. B. The standard deviation in the pressure required to open a certain valve is greater than 1.2 psi. b)H0​:σ=1.2psi H1​:σ>1.2psi c)rejects; equal to; 1.2; equal to; 1.2 d)fails to reject; equal to; 1.2; greater than; 1.2

10.2 Explain what a​ P-value is. What is the criterion for rejecting the null hypothesis using the​ P-value approach? a) Explain what a​ P-value is. Choose the correct answer below. A. A​ P-value is the number of standard deviations that the observed proportion is from the proportion stated in the null hypothesis. B. A​ P-value is the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption that the statement in the null hypothesis is true. C. A​ P-value is the value used to designate the area α in either the​ left- or​ right-tail of the normal curve. b) What is the criterion for rejecting the null hypothesis using the​ P-value approach? Choose the correct answer below. A. If ​P-value<α​, reject the null hypothesis. B. If P-value<−zα for a​ left-tailed test, or if ​P-value>zα for a​ right-tailed test, or if P-value<−zα/2 or P-value>zα/2 for a​ two-tailed test, then reject the null hypothesis. C. If ​P-value>α​, reject the null hypothesis.

a) b. A​ P-value is the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption that the statement in the null hypothesis is true. b)A. If ​P-value < α​, reject the null hypothesis. Note: The criterion is if ​P-value<α​, reject the null hypothesis.

10.2 Test the hypothesis using the​ P-value approach. Be sure to verify the requirements of the test. H0: p=0.91 versus H1: p≠0.91 n=500, x=440, α=0.1 Is np01−p0≥10​? Select the correct choice below and fill in the answer box to complete your choice. ​(Type an integer or a decimal. Do not​ round.) A. ​No, because np01−p0=___ ​b. Yes, because np01−p0=_____. Now find p. p=____ ​(Type an integer or a decimal. Do not​ round.) Find the test statistic z0. z0=_____ ​(Round to two decimal places as​ needed.) Find the​ P-value. ​P-value=____ ​(Round to three decimal places as​ needed.) State the conclusion of the hypothesis test. _______ because the​ P-value is ______ than α.

​b. Yes, because np01−p0= 40.95 0.88 -2.34 0.019 Reject null Hyp; less

10.1 According to the Centers for Disease Control and​ Prevention, 10.6​% of high school students currently use electronic cigarettes. A high school counselor is concerned the use of​ e-cigs at her school is higher. Complete parts​ (a) through​ (c) below.

(a) Determine the null and alternative hypotheses. Hypothesis testing is a​ procedure, based on sample evidence and​ probability, used to test statements regarding a characteristic of one or more populations. The parameter being tested is the proportion of high school students at the​ counselor's high school that use electronic cigarettes. The null​ hypothesis, H0​, is a statement to be​ tested; it is a statement of equality. The null hypothesis is assumed to be true until evidence indicates otherwise. It is a statement regarding the value of a population parameter. The alternative​ hypothesis, H1​, is a statement to be​ tested; it is a statement of inequality. We are trying to find evidence for the alternative hypothesis. It is a statement regarding the value of the population parameter. Convert the percent 10.6​% to a decimal. 10.6​%=0.106 The hypotheses are stated below. H0​:p=0.106 H1​:p>0.106 ​(b) If the sample data indicate that the null hypothesis should not be​ rejected, state the conclusion of the high school counselor. When the null hypothesis is not​ rejected, the alternative hypothesis is not supported. There is not sufficient evidence to conclude that the proportion of high school students exceeds 0.106 at this​ counselor's high school. ​(c) ​Suppose, in​ fact, that the proportion of students at the​ counselor's high school who use electronic cigarettes is 0.293. Was a type I or type II error​ committed? A Type I error is committed when the null hypothesis is rejected​ when, in​ fact, it is true. A Type II error is committed when the null hypothesis is not rejected​ when, in​ fact, the alternative hypothesis is true. ​Thus, a Type II error was committed because the sample evidence led the counselor to conclude the proportion of​ e-cigs users at her school was 0.106​, ​when, in​ fact, the proportion is higher.

10.5 A simple random sample of size n=15 is drawn from a population that is normally distributed. The sample mean is found to be x=21.6 and the sample standard deviation is found to be s=6.5. Determine if the population mean is different from 28 at the α=0.01 level of significance. Complete parts ​(a) through ​(d) below.

(a) Determine the null and alternative hypotheses. The first step is to determine which parameter is being tested. This problem is testing the population​ mean, μ​, to see if it has changed from a previous value. Next set up the hypothesis test. The test is​ two-tailed. The null and alternative hypotheses are shown below. H0​: μ=28 H1​: μ≠28 ​(b) Calculate the​ P-value. Since the parameter is the mean and the standard deviation of the population is not known use the​ Student's t-distribution. The sample size needs to be greater than 30 or the sample must come from a population that is normally distributed. Since the sample​ size, mean, and standard deviation are​ known, technology can be used to perform a​ one-sample t-test. Use technology to determine the exact​ P-value, rounding to three decimal places. Keep in mind that this is a​ two-tailed test. P=0.002 ​(c) State the conclusion for the test. If the​ P-value is less than the α ​value, reject H0. If the​ P-value is greater than the α ​value, do not reject H0. ​(d) State the conclusion in context of the problem. If the​ P-value is less than the α ​value, reject H0 and conclude that there is sufficient evidence at the α=0.01 level of significance that the population mean is different than 28. If the​ P-value is greater than the α ​value, do not reject and conclude there is not sufficient evidence that the population mean is different than 28.

10.6 To test H0​: p=0.65 versus H1​: p<0.65​, a simple random sample of n=200 individuals is obtained and x=122 successes are observed. ​(a) What does it mean to make a Type II error for this​ test? ​(b) If the researcher decides to test this hypothesis at the α=0.05 level of​ significance, compute the probability of making a Type II​ error, β​, if the true population proportion is 0.61. What is the power of the​ test? ​(c) Redo part​ (b) if the true population proportion is 0.55. ​

(a) If the null hypothesis is​ false, but the null hypothesis is not​ rejected, a Type II error has been made. For a Type II error to occur in this​ experiment, H0 would not be rejected and the true population proportion is less than 0.65. ​(b) If the researcher decides to test this hypothesis at the α=0.05 level of​ significance, compute the probability of making a Type II​ error, β​, if the true population proportion is 0.61. What is the power of the​ test? The critical value for a α=0.05 level of significance is 1.645. This means that any value of the sample​ statistic, p​, that satisfies the following inequality will lead to rejecting H0. Note that p0 is the assumed true population proportion. p<p0−1.645•p01−p0n Substitute the appropriate values into the formula and determine the maximum sample proportion that leads to rejecting H0​, rounding to four decimal places. p0−1.645•p01−p0n =0.65−1.645•0.65(1−0.65)200 =0.5945 Determine the probability of not rejecting H0​, assuming the true population proportion is 0.61. Rewrite the inequality using the​ z-score, rounding to four decimal places. β = Pp>0.5945 = Pz>0.5945−0.610.61(1−0.61)200 = P(z>−0.4494) Use technology to find the area under the standard normal curve to the right of the previously calculated​ z-score, −0.4494​, rounding to four decimal places. β=P(z>−0.4494) β=0.6734 The probability of not rejecting H0​: p=0.65 ​when, in​ fact, H1​: p<0.65 is​ true, if the true population proportion is 0.61​, is 0.6734. The power of the test is 1−β​, where β is the probability computed in the previous step. Power=1−0.6734 Power=0.3266. ​(c) Redo part​ (b) if the true population proportion is 0.55. The values of the sample​ proportion, p0​, that lead to rejecting H0 are the same as in part​ (b), because they only depend on the null and alternative​ hypotheses, and do not depend on the true population proportion. Determine the probability of not rejecting H0​, assuming the true population proportion is 0.55. Rewrite the inequality using the​ z-score, rounding to four decimal places. β = Pp>0.5945 = Pz>0.5945−0.550.55(1−0.55)200 = P(z>1.2650) Use technology to find the area under the standard normal curve to the right of the previously calculated​ z-score, 1.2650​, rounding to four decimal places. β=P(z>1.2650) β=0.1029 The probability of not rejecting H0​: p=0.65 ​when, in​ fact, H1​: p<0.65 is​ true, if the true population proportion is 0.55​, is 0.1029. The power of the test is 1−β​, where β is the probability computed in the previous step. Power=1−0.1029 Power=0.8971

10.3 To test H0​: μ=20 versus H1​: μ<20​, a simple random sample of size n=19 is obtained from a population that is known to be normally distributed. Answer parts​ (a)-(d). LOADING... Click here to view the​ t-Distribution Area in Right Tail.

(a) If x=18.5 and s=4.1​, compute the test statistic. Use the formula t0= x−μ0 / s/ sqrt n to calculate the test statistic because the objective is to test a statement about a population mean with unknown standard deviation. Substitute the values given in the problem statement into the formula and evaluate. t0= 18.5−20 / 4.1/sqrt 19 ≈−1.59 ​Thus, the test statistic is −1.59. ​(b) Draw a​ t-distribution with the area that represents the​ P-value shaded. Determine whether to use a​ two-tailed, a​ left-tailed, or a​ right-tailed test. The table below shows which test to use depending on the null and alternative hypotheses. ​ Since the null and alternative hypotheses are H0​: μ=20 and H1​: μ<20​, use a​ left-tailed test. The graph below shows the area that represents the​ P-value shaded. A symmetric bell-shaped curve is plotted over a horizontal axis. On the left side of the graph, a vertical line runs from the axis to the curve. The area under the curve to the left of the vertical line is shaded. ​(c) Approximate the​ P-value. Use a​ calculator, statistical​ software, or a​ t-distribution table to approximate the​ P-value using the test statistic from part​ (a). The​ P-value is equal to the area in the left tail. Recall that P(t<−1.59)=P(t>1.59). Using the​ t-distribution table, we find that the area with test statistic 1.59 and 18 degrees of freedom is between 0.05 and 0.10. ​Thus, 0.05<P-value<0.10. ​(d) If the researcher decides to test this hypothesis at the α=0.05 level of​ significance, will the researcher reject the null​ hypothesis? Reject the null hypothesis only if the​ P-value is less than α. Since the​ P-value is not less than α​, the researcher will not reject the null hypothesis.

10.6 Explain what it means to make a Type II error. Choose the correct answer below. A. Fail to reject the null hypothesis and the alternative is true. B. Reject the null hypothesis and the null is true. C. Fail to reject the null hypothesis and the null is true. D. Reject the null hypothesis and the alternative is true.

A. Fail to reject the null hypothesis and the alternative is true. Note: A Type II error is made when we fail to reject the null hypothesis and the alternative hypothesis is true.

10.3 Explain the difference between statistical significance and practical significance. Choose the correct answer below. A. Statistical significance means that the sample statistic is not likely to come from the population whose parameter is stated in the null hypothesis. Practical significance refers to whether the difference between the sample statistic and the parameter stated in the null hypothesis is large enough to be considered important in an application. B. Statistical significance refers to how an unusual event is unlikely to actually appear in a real world​ application, such as every entry in a sample of size 50 having the same value. Practical significance refers to how an unusual event is likely to actually appear in a real world​ application, such as a rejection of a null hypothesis using data that looks feasible. C. Statistical significance refers to the type of hypothesis test needed to analyze a​ population, with some tests being more important than Z tests. Practical significance refers to how difficult a desired hypothesis test is to perform in an​ application, with some tests being easier to perform than others. D. Statistical significance means that the hypothesis test being performed is useful for building theoretical foundations for other statistical work. Practical significance means that the particular application of the hypothesis test is of great importance to the real world.

A. Statistical significance means that the sample statistic is not likely to come from the population whose parameter is stated in the null hypothesis. Practical significance refers to whether the difference between the sample statistic and the parameter stated in the null hypothesis is large enough to be considered important in an application. A statistically significant result may be of no practical significance. Note: While different hypothesis tests are used in different​ situations, it is not correct to say that some are​ "more important" than others. A statistically significant result may be of no practical significance.

10.1 Suppose the null hypothesis is rejected. State the conclusion based on the results of the test. Three years​ ago, the mean price of a​ single-family home was ​$243,770. A real estate broker believes that the mean price has increased since then. Which of the following is the correct​ conclusion? A. There is not sufficient evidence to conclude that the mean price of a​ single-family home has increased. B. There is sufficient evidence to conclude that the mean price of a​ single-family home has increased. C. There is not sufficient evidence to conclude that the mean price of a​ single-family home has not changed. D. There is sufficient evidence to conclude that the mean price of a​ single-family home has not changed.

B. There is sufficient evidence to conclude that the mean price of a​ single-family home has increased.

10.1 If the consequences of making a Type I error are​ severe, would you choose the level of​ significance, α​, to equal​ 0.01, 0.05, or​ 0.10? Choose the correct answer below. A. 0.10 B. 0.05 C. 0.01

C. 0.01 Note: Choose a small α to make it difficult to reject H0.

10.1 State the conclusion based on the results of the test. According to the Federal Housing Finance​ Board, the mean price of a​ single-family home two years ago was ​$299,600. A real estate broker believes that because of the recent credit​ crunch, the mean price has increased since then. The null hypothesis is rejected. Choose the correct answer below. A. There is sufficient evidence to conclude that the mean price of a​ single-family home has decreased from its level two years ago of ​$299,600. B. There is not sufficient evidence to conclude that the mean price of a​ single-family home has decreased from its level two years ago of ​$299,600. C. There is sufficient evidence to conclude that the mean price of a​ single-family home has increased from its level two years ago of ​$299,600. D. There is not sufficient evidence to conclude that the mean price of a​ single-family home has increased from its level two years ago of ​$299,600.

C. There is sufficient evidence to conclude that the mean price of a​ single-family home has increased from its level two years ago of ​$299,600.

10.5 A psychologist obtains a random sample of 20 mothers in the first trimester of their pregnancy. The mothers are asked to play Mozart in the house at least 30 minutes each day until they give birth. After 5​ years, the child is administered an IQ test. It is known that IQs are normally distributed with a mean of 100. If the IQs of the 20 children in the study result in a sample mean of 104.7 and sample standard deviation of 16.1​, is there evidence that the children have higher​ IQs? Use the α=0.05 level of significance.

Determine the appropriate type of hypothesis test to conduct for this research. The first step is to determine which parameter is being tested. The parameter being tested is the mean​ IQ, μ​, of the children at age 5. Use this to determine the appropriate type of hypothesis test. The appropriate hypothesis test is a hypothesis test on a population mean. ​Next, determine the null and alternative hypotheses. The null​ hypothesis, denoted H0​, is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The alternative​ hypothesis, denoted H1​, is a statement that we are trying to find evidence to support. Determine the null and alternative hypotheses. H0: μ=100 H1: μ>100 ​Next, check the conditions for testing hypotheses regarding a population mean. ​First, the sample must be obtained using simple random sampling or from a randomized experiment. It is given that the sample is a simple random sample of mothers and their​ infants, so this condition is met. ​Next, the sample must have no outliers and either the population from which the sample is drawn is normally​ distributed, or the sample​ size, n, is large ​(n≥​30). It is given that IQs are normally​ distributed, so the population from which the sample is drawn is normally distributed. It is reasonable to assume that the sample has no​ outliers, as the sample standard deviation is not large when compared to the sample mean. ​Finally, the sampled values should be independent of each other. That​ is, the sample size is less than​ 5% of the population size. The sample size is less than​ 5% of the population​ size, because 20 mothers is less than​ 5% of all mothers. ​Thus, all of the conditions are met and the hypothesis test can proceed. The test statistic for the hypothesis test on a population mean can be calculated using technology or the following​ formula, where x is the sample​ mean, μ0 is the assumed value of the population​ mean, s is the sample standard​ deviation, and n is the sample size. t0=x−μ0sn For this​ exercise, use technology to calculate the test statistic. Identify the values of x​, μ0​, ​s, and n to be entered into technology. x=104.7 μ0=100 s=16.1 n=20 Enter the values x=104.7​, μ0=100​, s=16.1​, and n=20 into technology to determine the value of the test statistic t0​, rounding to two decimal places. t0=1.31 The​ P-value is the probability of observing a sample statistic as extreme or more extreme than one observed under the assumption that the statement in the null hypothesis is true. For the hypothesis test on a population mean with the population standard deviation​ unknown, the test statistic follows​ Student's t-distribution with n−1 degrees of freedom. While technology or a​ t-distribution Table can be used to find the​ P-value, in this​ problem, technology will be used. Use the same technology output that was used to find the test statistic to find the​ P-value, rounding to three decimal places. P-value=0.104 State the conclusion for the test. If the​ P-value is less than the level of significance α​, reject H0. ​Otherwise, do not reject H0. Compare the​ P-value to the level of significance. Use this information to draw an appropriate conclusion about the​ P-value.

10.1 Determine whether the following statement is true or false. Sample evidence can prove that a null hypothesis is true. Choose the correct answer below. True False

False Note: because although sample data is used to test the null​ hypothesis, it cannot be stated with​ 100% certainty that the null hypothesis is true. It can only be determined whether the sample data supports or does not support the null hypothesis.

10.3 A golf association requires that golf balls have a diameter that is 1.678 inches. To determine if golf balls conform to the​ standard, a random sample of golf balls was selected. Their diameters are shown in the accompanying data table. Do the golf balls conform to the​ standards? Use the α=0.10 level of significance. LOADING... Click the icon to view the data table. Golf Ball Diameter​ (inches) 1.681 1.676 1.683 1.680 1.678 1.682 1.684 1.683 1.672 1.684 1.684 1.673

First determine the null and alternative hypotheses. The null​ hypothesis, denoted H0​, is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The alternative​ hypothesis, denoted H1​, is a statement that the test is trying to find evidence to support. Identify the null and alternative hypotheses for this test. H0​: μ=1.678 H1: μ≠1.678 Next select a level of​ significance, α. As​ given, use the α=0.10 level of significance. Now find the test statistic t0. This can be found using technology or the formula​ below, where x is the sample​ mean, μ0 is the hypothesized​ mean, s is the sample standard​ deviation, n is the sample​ size, and the test statistic follows​ Student's t-distribution with n−1 degrees of freedom. t0=x−μ0sn For the purposes of this​ explanation, use technology to perform the test. Note that this is a​ two-tailed test because the alternative hypothesis uses the ≠ symbol. Use technology to find the test​ statistic, rounding to two decimal places. Enter the given data into technology to test H0​: μ=1.678 vs. H1: μ≠1.678. t0=1.61 Find the​ P-value for this hypothesis test. Identify the​ P-value from your technology​ output, rounding to three decimal places. ​P-value=0.136 If ​P-value<α​, reject the null hypothesis.​ Otherwise, do not reject the null hypothesis. Then state the conclusion.

10.4 Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than 7​%. A​ mutual-fund rating agency randomly selects 31 months and determines the rate of return for a certain fund. The standard deviation of the rate of return is computed to be 5.12​%. Is there sufficient evidence to conclude that the fund has moderate risk at the α=0.05 level of​ significance? A normal probability plot indicates that the monthly rates of return are normally distributed.

First state the null and alternative hypotheses for this test. The population parameter that is being tested is the population standard​ deviation, σ. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The null hypothesis is always a statement of equality. The hypothesis H0​: σ=0.07 states that the population standard deviation is equal to the assumed value. The alternative hypothesis is a statement you are trying to find evidence to support. The alternative hypothesis is always a statement of inequality. The hypothesis H1​: σ<0.07 states the values for the population standard deviation needed to show that the mutual fund qualifies as having moderate risk. To test the hypotheses H0​: σ=0.07 versus H1​: σ<0.07​, first calculate the test statistic χ20​, where n is the sample​ size, s is the sample standard​ deviation, and σ is the population standard deviation. χ20=(n−1)s2σ20 Assume σ=0.07. Identify the value of​ n, the sample size. n=31 Identify the value of​ s, the sample​ deviation, and convert it to a decimal. s=5.12​%=0.0512 Now substitute 31 for​ n, 0.0512 for​ s, and 0.07 for σ. Then simplify and round to two decimal places. χ20 = (n−1)s2σ20 = (31−1)(0.0512)2(0.07)2 Substitute. = 16.05 Simplify. Next find the​ P-value for the test statistic. Since this is a​ left-tailed test, the​ P-value is the probability of getting a value of χ2 less than the test statistic χ20. Note that ​P(χ2<χ20​), the area to the left of the test​ statistic, is 1−​P(χ2>χ20​), the area to the right. Use technology to find the area to the right of χ20=16.05 under the curve of the χ2 distribution with 31−1=30 degrees of​ freedom, rounding to three decimal places. ​P(χ2>16.05​)=0.982 Now find the​ P-value. 1−0.982=0.018 If the​ P-value is less than the level of significance α​, reject the null hypothesis.​ Otherwise, do not reject the null hypothesis. Compare the​ P-value, 0.018​, to the level of​ significance, α=0.05. The​ P-value is less than α. Use this information to determine the correct conclusion and interpret the conclusion in the context of the problem.

10.1 If we do not reject the null hypothesis when the statement in the alternative hypothesis is​ true, we have made a Type​ _______ error.

II Note: A Type I error occurs if the null hypothesis is rejected​ when, in​ fact, the null hypothesis is true. A Type II error occurs if the null hypothesis is not rejected​ when, in​ fact, the alternative hypothesis is true.

10.6 To test H0​: p=0.55 versus H1​: p<0.55​, a simple random sample of n=375 individuals is obtained and x=195 successes are observed. If the true population proportion is 0.52 and the level of significance of α=0.05 is​ used, β=0.6832 and the power of the test is 0.3168.

If the true population proportion is 0.52 and the level of significance of α=0.05 is​ used, β=0.6832 and the power of the test is 0.3168. Now repeat the test with a significance level of α=0.01. The critical value for a α=0.01 level of significance is 2.326. This means that any value of the sample​ statistic, p​, that satisfies the following inequality will lead to rejecting H0. p<p0−2.326•p01−p0n Determine the maximum sample​ proportion, rounding to four decimal​ places, that leads to rejecting H0. p = 0.55−2.326•0.55(1−0.55)375 = 0.4902 Determine the probability of not rejecting H0​, assuming that the true population proportion is 0.52. Rewrite the inequality using the​ z-score, rounding to four decimal places. β = Pp>0.4902 = Pz>0.4902−0.520.52(1−0.52)375 = P(z>−1.1551) Use technology to find the area under the standard normal curve to the right of the previously calculated​ z-score, −1.1551​, rounding to four decimal places. β = P(z>−1.1551) = 0.8760 The power of the test is 1−β​, where β is the probability computed in the previous​ step, rounding to four decimal places. Power=1−0.8760 Power=0.1240 When the level of significance was lowered from α=0.05 to α=​0.01, the power of the test went from 0.3168 to 0.1240. As the probability of rejecting a true H0 is​ lowered, the probability of not rejecting a false H0 ​increases, thus lowering the power.

10.1 The null and alternative hypotheses are given. Determine whether the hypothesis test is​ left-tailed, right-tailed, or​ two-tailed. What parameter is being​ tested? H0​:μ=120 H1​:μ<120 Choose the correct answer below. Two-tailed Left-tailed Right-tailed What parameter is being​ tested? p σ μ

Left-tailed; μ

10.5 A simple random sample of size n=250 drivers with a valid​ driver's license is asked if they drive an​ American-made automobile. Of the 250 drivers​ surveyed, 133 responded that they drive an​ American-made automobile. Determine if a majority of those with a valid​ driver's license drive an​ American-made automobile at the α=0.05 level of significance.

Qualitative, or​ categorical, variables allow for classification of individuals based on some attribute or characteristic. Quantitative variables provide numerical measures of individuals. The values of a quantitative variable can be added or subtracted and provide meaningful results. A discrete variable is a quantitative variable that has either a finite number of possible values or a countable number of possible values. A continuous variable is a quantitative variable that has an infinite number of possible values that are not countable. A continuous variable may take on every possible value between any two values. The variable​ "drive an​ American-made automobile, or​ not" is​ qualitative, with two possible outcomes of driving an​ American-made automobile and not driving an​ American-made automobile. Determine the appropriate type of hypothesis test to conduct for this research. The first step is to determine which parameter is being tested. This problem is testing the​ proportion, p, of drivers that drive an​ American-made automobile. Use this to determine the appropriate type of hypothesis test. The appropriate hypothesis test is a hypothesis test on a population proportion. ​Next, determine the null and alternative hypotheses. The null​ hypothesis, denoted H0​, is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The alternative​ hypothesis, denoted H1​, is a statement that we are trying to find evidence to support. Determine the null and alternative hypotheses. H0​: p=0.5 H1​: p>0.5 The sampling distribution of the sample proportion p is approximately​ normal, provided that the sample is a simple random​ sample, that np01−p0 is greater than or equal to 10 where n is the sample size and p0 is the hypothesized population​ proportion, and that the sampled values are independent of each other​ (or that the sample size is less than​ 5% of the population​ size). Check these requirements. It is given that the sample is a simple random sample of all drivers with valid​ driver's licenses. To check that np01−p0 is at least​ 10, first find the values of n and p0. n = 250 p0 = 0.5 Find the value of np01−p0. np01−p0 = 250​(0.5​)(1−0.5​) = 62.5 ​Therefore, the value of np01−p0 is greater than or equal to 10. Also notice that the sample size is less than​ 5% of the population​ size, because 250 drivers is less than​ 5% of all drivers. Since the sample is a simple random​ sample, np01−p0=62.5 is greater than​ 10, and 250 drivers is less than​ 5% of all​ drivers, the normal distribution can be used. The next step is to calculate the test statistic. The test statistic for the hypothesis test on a population proportion can be calculated using technology or the formula​ below, where p=xn is the sample​ proportion, x is the number of individuals in the sample with the specified​ characteristic, n is the sample​ size, and p0 is the hypothesized population proportion. z0=p−p0p01−p0n For this​ exercise, use technology to calculate the test statistic.​ First, identify the value of x. x=133 Enter the values n=250​, x=133​, and p=0.5 into technology to determine the value of the test statistic z0​, rounding to two decimal places. z0=1.01 The​ P-value is the probability of observing a sample statistic as extreme or more extreme than one observed under the assumption that the statement in the null hypothesis is true. For the hypothesis test on a population​ proportion, the test statistic follows the standard normal distribution. While technology or a standard normal distribution table can be used to find the​ P-value, in this​ problem, technology will be used. Use the same technology output that was used to find the test statistic to find the​ P-value, rounding to three decimal places. ​P-value=0.156 State the conclusion of the hypothesis test. If the​ P-value is less than the level of significance α=0.05​, reject H0. ​Otherwise, do not reject H0. Compare the​ P-value to the level of significance. Use the comparison to draw an appropriate conclusion about the​ P-value.

10.1 The null and alternative hypotheses are given. Determine whether the hypothesis test is​ left-tailed, right-tailed, or​ two-tailed. What parameter is being​ tested?. H0​:μ=120 H1​:μ>120 Is the hypothesis test​ left-tailed, right-tailed, or​ two-tailed? Two​-tailed Right​-tailed Left​-tailed What parameter is being​ tested?. Population standard deviation Population proportion Population mean

Right​-tailed; Population mean

10.3 The life expectancy of a male during the course of the past 100 years is approximately 27,682 days. Use the table to the right to conduct a test using α=0.10 to determine whether the evidence suggests that chief justices live longer than the general population of males. Suggest a reason why the conclusion drawn may be flawed. Chief Justice Life Span​ (Days) A 31,126 I 30,843 B 31,819 J 30,539 C 24,613 K 32,877 D 30,525 L 32,142 E 28,496 M 29,039 F 29,223 N 26,014 G 28,686 O 33,749 H 31,338 P 27,204

State the appropriate null and alternative hypotheses to assess whether chief justices live longer than the general population of males. The hypotheses can be structured in one of three ways. Two-tailedLeft-tailedRight-tailedH0: μ=μ0H0: μ=μ0H0: μ=μ0H1: μ≠μ0H1: μ<μ0H1: μ>μ0 Identify the value of μ0​, the assumed value of the population mean. μ0=27,682 Determine which type of test should be used. A​ right-tailed test should be used. Now determine the hypotheses that should be used. H0​: μ=27,682 versus H1​: μ>27,682 Use the​ P-value approach at the α=0.10 level of significance to test the hypotheses. The​ P-value is the probability of observing a sample statistic as extreme or more extreme than one observed under the assumption that the statement in the null hypothesis is true. The test statistic for this test follows​ Student's t-distribution with n−1 degrees of freedom. First find​ n, the size of the sample . n=16 Now use technology to find x​, the sample average. Round to the nearest integer. x≈29,890 Use technology to find​ s, the sample standard deviation. Round to the nearest integer. s≈2485 Use technology to find the​ P-value by substituting 16 for​ n, 29,890 for x​, and 2485 for s. Recall that this hypothesis test is​ right-tailed and the alternative hypothesis is in the form of H1: μ>μ0. Round the result to three decimal places. ​P-value≈0.001 If the​ P-value is less than the level of​ significance, α​, reject H0. ​Otherwise, do not reject H0. Write a conclusion based on the results. Reject H0 if the​ P-value is less than the level of​ significance, α=0.10​, and conclude that there is sufficient evidence to support the statement in H1. ​Otherwise, do not reject H0 and conclude that there is not sufficient evidence to support the statement in H1. Suggest a reason why the conclusion drawn may be flawed. To test hypotheses regarding the population​ mean, the sample must be obtained using simple random sampling or from a randomized experiment. The sample must not have any​ outliers, and the population from which the sample is drawn must be normally distributed or the sample​ size, n, must be large ​(n≥​30). The sampled values must also be independent of each other. Recall that​ n, the size of the​ sample, is 16. Use this information to determine why the conclusion may be flawed.

10.3 In a​ study, researchers wanted to measure the effect of alcohol on the hippocampal​ region, the portion of the brain responsible for​ long-term memory​ storage, in adolescents. The researchers randomly selected 16 adolescents with alcohol use disorders to determine whether the hippocampal volumes in the alcoholic adolescents were less than the normal volume of 9.02 cm3. An analysis of the sample data revealed that the hippocampal volume is approximately normal with no outliers and x=8.36 cm3 and s=0.7 cm3. Conduct the appropriate test at the α=0.05 level of significance.

State the null and alternative hypotheses. The null​ hypothesis, denoted H0​, is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The alternative​ hypothesis, denoted H1​, is a statement such that evidence is gathered in an attempt to support. Identify the null hypothesis. H0​: μ=9.02 Identify the alternative hypothesis. H1​: μ<9.02 Identify the​ t-statistic. The formula below can compute the test​ statistic, where x is the sample​ mean, μ0 is the hypothesized population​ mean, s is the sample standard​ deviation, and n is the sample size. t0=x−μ0sn While either the formula or technology can be used to compute the test​ statistic, for the purposes of this​ exercise, use technology. Use technology to compute the test​ statistic, rounding to two decimal places. t0=−3.77 Review the technology output from the previous step to determine the​ P-value, rounding to three decimal places. ​P-value=0.001 Make a conclusion regarding the hypothesis. The null hypothesis is rejected if the​ P-value is less than the level of significance.​ Otherwise, fail to reject the null hypothesis. Use this information and the hypotheses from a previous step to make a conclusion for the hypothesis test.

10.5 In 1945​, an organization asked 1600 randomly sampled American​ citizens, "Do you think we can develop a way to protect ourselves from atomic bombs in case others tried to use them against​ us?" with 808 responding yes. Did a majority of the citizens feel the country could develop a way to protect itself from atomic bombs in 1945​? Use the α=0.1 level of significance.

The first step is to determine which parameter is being tested. The parameter being tested is​ proportion, p, because the exercise is testing a​ majority, which is a proportion. To test the​ hypothesis, first verify the​ requirements; that​ is, the sample must be a simple random​ sample, np01−p0≥​10, and the sample size cannot be more than​ 5% of the population size. Determine the value of p0 by interpreting the given statement. The statement is that the majority of the citizens feel the country could develop a way to protect itself from atomic bombs. A portion is said to have the majority if it is greater than​ 50%. The value of p0 is 0.50. Compute np01−p0 to verify that np01−p0≥10. np01−p0 = ​(1600​)(0.50​)(1−0.50​) = 400 ​Thus, np01−p0=400≥10 Determine if the sample size is less than​ 5% of the population size. Divide the sample size by​ 5% (0.05). 16000.05=32,000 We can assume that the population of American citizens is greater than 32,000. It was also stated that the sample is a simple random sample.​ Thus, all of the requirements are satisfied. Now that all the requirements have been​ verified, the next step is to determine the null and alternative hypotheses. Determine the null and alternative hypotheses. H0​: p=0.50 H1​: p>0.50 Now find p using the formula p=xn​, where x is the number of individuals in the sample with the specified characteristic and n is the sample size. p = xn = 808/1600 = 0.505 Calculate the test statistic z0. Substitute 0.505 for p​, 0.50 for p0​, and 1600 for n into the formula below. z0 = p−p0p01−p0n = 0.505−0.500.50(1−0.50)1600 = 0.4 While technology or a standard normal distribution table can be used to find the​ area, in this​ problem, technology will be used. Find the​ P-value associated with z0=0.4​, rounding to three decimal places. ​P-value=0.345 Compare the​ P-value to the level of significance. If the​ P-value is less than the α ​value, reject H0. If the​ P-value is greater than the α ​value, do not reject H0.

10.1 The null and alternative hypotheses are given. Determine whether the hypothesis test is​ left-tailed, right-tailed, or​ two-tailed. What parameter is being​ tested? H0​: σ=7.8 H1​: σ≠7.8

The null​ hypothesis, denoted H0​, is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference. The null hypothesis is assumed true until evidence indicates otherwise. It will be a statement regarding the value of a population parameter. The alternative​ hypothesis, denoted H1​, is assumed to be true if evidence shows the null hypothesis to be false. It will be a statement regarding the value of a population parameter. The three different types of hypothesis tests are described below. 1. Equal hypothesis versus not equal hypothesis​ (two-tailed test), if H0 parameter=x and H1 parameter≠x. 2. Equal versus less than​ (left-tailed test), if H0 parameter=x and H1 parameter<x. 3. Equal versus greater than​ (right-tailed test), if H0 parameter=x and H1 parameter>x. Notice that in the given hypothesis​ test, the null hypothesis uses ​"=​" and the alternative hypothesis uses ​"≠​". ​Therefore, the hypothesis test is​ two-tailed. What parameter is being​ tested? Look closely at the problem statement. Since both the null and alternative hypotheses given in the problem statement contain the parameter σ​, it is the parameter that is being tested.

10.2 Test the hypothesis using the​ P-value approach. Be sure to verify the requirements of the test. H0​: p=0.4 versus H1​: p>0.4 n=300​; x=145​, α=0.1

To test the​ hypothesis, first verify the​ requirements; that​ is, the sample must be a simple random​ sample, np0(1−p0)≥​10, and the sample size cannot be more than​ 5% of the population size. Compute np0(1−p0) to verify that np0(1−p0)≥10. Since no other information is​ given, assume the other requirements are satisfied. np0(1−p0)=​(300​)(0.4​)(1−0.4​)=72 ​Thus, np0(1−p0)=72≥10 and all of the requirements are satisfied. The next step in this approach is to find the​ P-value. Use technology to find the​ P-value, rounding to three decimal places. ​P-value=0.002 ​Finally, compare the​ P-value with α. If ​P-value<α​, reject the null hypothesis.

10.2 In a clinical​ trial, 20 out of 875 patients taking a prescription drug daily complained of flulike symptoms. Suppose that it is known that 1.5​% of patients taking competing drugs complain of flulike symptoms. Is there sufficient evidence to conclude that more than 1.5​% of this​ drug's users experience flulike symptoms as a side effect at the α=0.05 level of​ significance?

To test the​ hypothesis, first verify the​ requirements; that​ is, the sample must be a simple random​ sample; np01−p0≥​10, where n is the sample size and p0 is the assumed population​ proportion; and the sample size cannot be more than​ 5% of the population size. It is not explicitly stated that the sample is random.​ However, since this is a clinical​ trial, it is reasonable to assume that the sample is a simple random sample. It is reasonable to assume that 875 is less than​ 5% of the population size. Compute np01−p0 to verify that np01−p0≥​10, rounding to one decimal place. np01−p0 =(875​)(0.015​)(1−0.015​) =12.9 ​Thus, np01−p0=12.9≥10 and all of the requirements are satisfied. Now that all the requirements have been​ verified, the next step is to determine the null and alternative hypotheses. The null​ hypothesis, denoted H0​, is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The alternative​ hypothesis, denoted H1​, is a statement that one is trying to find evidence to support. The statement is that more than 1.5​% of this​ drug's users experience flulike symptoms as a side effect. The null and alternative hypotheses are shown below. H0​:p=0.015 versus H1​: p>0.015 Notice that this is a ​right-tailed test because the hypotheses are of the form H0​: p=p0 versus H1​: p>p0. The formula for the test​ statistic, z0​, is shown​ below, where p=xn is the sample​ proportion, p0 is the assumed population​ proportion, and n is the sample size. z0=p−p0 / sqrt p0(1−p0)/n While either technology or formulas and tables can be used to conduct this hypothesis​ test, in this​ problem, use technology.​ First, identify the sample size. n=875 ​Next, identify the number of successes in the sample. Let a success be someone complaining of flulike symptoms. x=20 Determine the test statistic for a​ right-tailed test with a sample of size n=875 with x=20 ​successes, rounding to two decimal places. z0=1.91 Now continue with the​ P-value approach. Use the technology output generated when finding the test statistic to determine the​ P-value, rounding to three decimal places. ​P-value=0.028 Compare the​ P-value with the level of significance α. If the​ P-value is less than α​, reject the null hypothesis.​ Otherwise, do not reject the null hypothesis.

10.2 In a previous​ poll, 49​% of adults with children under the age of 18 reported that their family ate dinner together seven nights a week. Suppose​ that, in a more recent​ poll, 1207 adults with children under the age of 18 were selected at​ random, and 575 of those 1207 adults reported that their family ate dinner together seven nights a week. Is there sufficient evidence that the proportion of families with children under the age of 18 who eat dinner together seven nights a week has​ decreased? Use the α=0.1 significance level.

To test the​ hypothesis, first verify the​ requirements; that​ is, the sample must be a simple random​ sample; np01−p0≥​10, where n is the sample size and p0 is the assumed population​ proportion; and the sample size cannot be more than​ 5% of the population size. Notice that the adults in the survey were selected at random.​ Therefore, the requirement of a random sample is satisfied. It is reasonable to assume that 1207 is less than 5 percent of the number of adults with children under the age of 18 in the population. Since no other information is​ given, assume that the sample is a random sample. Compute np01−p0 to verify that np01−p0≥​10, rounding to one decimal place. np01−p0 =(1207​)(0.49​)(1−0.49​) =301.6 ​Thus, np01−p0=301.6≥10 and all of the requirements are satisfied. Now that all the requirements have been​ verified, the next step is to determine the null and alternative hypotheses. The null​ hypothesis, denoted H0​, is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The alternative​ hypothesis, denoted H1​, is a statement that one is trying to find evidence to support. The statement is that the proportion of families with children under the age of 18 who eat dinner together seven nights a week has decreased. The null and alternative hypotheses are shown below. H0​: p=0.49 versus H1​: p<0.49 Notice that this is a​ left-tailed test because the hypotheses are of the form H0​: p=p0 versus H1​: p<p0. The formula for the test​ statistic, z0​, is shown​ below, where p=xn is the sample​ proportion, p0 is the assumed population​ proportion, and n is the sample size. z0=p−p0p01−p0n While either technology or formulas and tables can be used to conduct this hypothesis​ test, in this​ problem, use technology.​ First, identify the sample size. n=1207 ​Next, identify the number of successes in the sample. Let a success be a family eating dinner together seven nights a week x=575 Determine the test statistic for a​ left-tailed test with a sample of size n=1207 with x=575 ​successes, rounding to two decimal places. z0=−0.95 Now continue with the​ P-value approach. Use the technology output generated when finding the test statistic to determine the​ P-value, rounding to three decimal places. ​P-value=0.172

10.4 To test H0: σ=20 versus H1: σ<20​, a random sample of size n=28 is obtained from a population that is known to be normally distributed. ​(a) If the sample standard deviation is determined to be s=14.2​, compute the test statistic. ​(b) If the researcher decides to test this hypothesis at the α=0.05 level of​ significance, use technology to determine the​ P-value. ​(c) Will the researcher reject the null​ hypothesis? ​(a) The test statistic is χ20=___ ​(Round to two decimal places as​ needed.) ​(b) The​ P-value is ____ ​(Round to three decimal places as​ needed.) ​(c) Since the​ P-value is _____ the level of​ significance, the researcher ___ reject the null hypothesis H0: σ=20.

a) 13.61 b)0.015 Note: Since this is a​ left-tailed test, use technology to determine the probability of getting a value of χ2 less than the test statistic χ20 with n−1 degrees of freedom. c) less;will Note: If the​ P-value is less than the level of significance α​, reject the null hypothesis.​ Otherwise, do not reject the null hypothesis.

10.4 To test H0: σ=70 versus H1: σ<70​, a random sample of size n=21 is obtained from a population that is known to be normally distributed. ​(a) If the sample standard deviation is determined to be s=63.4​, compute the test statistic. ​(b) If the researcher decides to test this hypothesis at the α=0.05 level of​ significance, use technology to determine the​ P-value. ​(c) Will the researcher reject the null​ hypothesis? ​(a) The test statistic is χ20=___ ​(Round to two decimal places as​ needed.) ​(b) The​ P-value is ____ ​(Round to three decimal places as​ needed.) ​(c) Since the​ P-value is _____ the level of​ significance, the researcher ___ reject the null hypothesis H0: σ=70.

a) 16.41 b)0.309 c) greater;will not

10.2 In a previous​ year, 54​% of females aged 15 and older lived alone. A sociologist tests whether this percentage is different today by conducting a random sample of 650 females aged 15 and older and finds that 345 are living alone. Is there sufficient evidence at the α=0.05 level of significance to conclude the proportion has​ changed? a) Because np0(1−p0)=___ ___10, the sample size is ____ ​5% of the population​ size, and the sample _______ the requirements for testing the hypothesis ______ satisfied. ​(Round to one decimal place as​ needed.) b) What are the null and alternative​ hypotheses? H0​: ____ ____ ____ versus H1​:___ ___ ____ ​(Type integers or decimals. Do not​ round.) c) Find the test statistic z0. z0=_____ ​(Round to two decimal places as​ needed.) d)Find the​ P-value. ​P-value=____ ​(Round to three decimal places as​ needed.) e) State the conclusion for this hypothesis test. A. Reject H0. There is not sufficient evidence at the α=0.05 level of significance to conclude that the proportion of females who are living alone has changed. B. Do not reject H0. There is not sufficient evidence at the α=0.05 level of significance to conclude that the proportion of females who are living alone has changed. C. Reject H0. There is sufficient evidence at the α=0.05 level of significance to conclude that the proportion of females who are living alone has changed. D. Do not reject H0. There is sufficient evidence at the α=0.05 level of significance to conclude that the proportion of females who are living alone has changed.

a) 161.5​; >; less than; is given to be random; are b)H0​:p=0.54 versus H1​: p≠0.54 c) -0.47 d)0.637 e) B. Do not reject H0. There is not sufficient evidence at the α=0.05 level of significance to conclude that the proportion of females who are living alone has changed.

10.2 In a previous​ poll, 41​% of adults with children under the age of 18 reported that their family ate dinner together seven nights a week. Suppose​ that, in a more recent​ poll, 1130 adults with children under the age of 18 were selected at​ random, and 442 of those 1130 adults reported that their family ate dinner together seven nights a week. Is there sufficient evidence that the proportion of families with children under the age of 18 who eat dinner together seven nights a week has​ decreased? Use the α=0.05 significance level. a) Because np0(1−p0)=___ ___10, the sample size is ____ ​5% of the population​ size, and the sample _______ the requirements for testing the hypothesis ______ satisfied. ​(Round to one decimal place as​ needed.) b) What are the null and alternative​ hypotheses? H0​: ____ ____ ____ versus H1​:___ ___ ____ ​(Type integers or decimals. Do not​ round.) c) Find the test statistic z0. z0=_____ ​(Round to two decimal places as​ needed.) d)Find the​ P-value. ​P-value=____ ​(Round to three decimal places as​ needed.) e)Interpret the results. Since the​ P-value is____ than α​, ______ the null hypothesis. There ____ sufficient evidence at the α=nothing level of significance to conclude that the proportion of families with children under the age of 18 who eat dinner together seven nights a week is ____ _____. ​(Type integers or decimals. Do not​ round.)

a) 273.3 greater than; less than; were; are b)H0 p=0.41 vs H1 p<0.41 c) -1.29 d) 0.099 e) greater; do not reject; is not; 0.05;less than 0.41

10.4 To test H0: σ=4.8 versus H1: σ≠4.8​, a random sample of size n=15 is obtained from a population that is known to be normally distributed. ​(a) If the sample standard deviation is determined to be s=7.1​, compute the test statistic. ​(b) If the researcher decides to test this hypothesis at the α=0.10 level of​ significance, use technology to determine the​ P-value. ​(c) Will the researcher reject the null​ hypothesis? ​(a) The test statistic is χ20=___ ​(Round to two decimal places as​ needed.) ​(b) The​ P-value is ____ ​(Round to three decimal places as​ needed.) ​(c) Since the​ P-value is _____ the level of​ significance, the researcher ___ reject the null hypothesis H0: σ=4.8

a) 30.63 b)0.013 c) less ;will

10.4 To test H0: σ=1.6 versus H1: σ>1.6​, a random sample of size n=23 is obtained from a population that is known to be normally distributed. ​(a) If the sample standard deviation is determined to be s=2.2​, compute the test statistic. ​(b) If the researcher decides to test this hypothesis at the α=0.10 level of​ significance, use technology to determine the​ P-value. ​(c) Will the researcher reject the null​ hypothesis? ​(a) The test statistic is χ20=___ ​(Round to two decimal places as​ needed.) ​(b) The​ P-value is ____ ​(Round to three decimal places as​ needed.) ​(c) Since the​ P-value is _____ the level of​ significance, the researcher ___ reject the null hypothesis H0: σ=1.6

a) 41.59 b)0.007 c) less ;will

10.1 According to a​ report, the standard deviation of monthly cell phone bills was ​$5.05 in 2017. A researcher suspects that the standard deviation of monthly cell phone bills is higher today. ​(a) State the null and alternative hypotheses in words. ​(b) State the null and alternative hypotheses symbolically. ​(c) Explain what it would mean to make a Type I error. ​(d) Explain what it would mean to make a Type II error.

a) A hypothesis is a statement regarding a characteristic of one or more populations. Hypothesis testing is a​ procedure, based on sample evidence and​ probability, used to test statements regarding a characteristic of one or more populations. The parameter being tested is the standard deviation of monthly cell phone bills. The null​ hypothesis, denoted H0 ​(read "H-naught"), is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. In other words the null hypothesis is a statement of status quo or no difference and always contains a statement of equality. It is a statement regarding the value of a population parameter. The alternative​ hypothesis, denoted H1 ​(read "H-one"), is a statement that we are trying to find evidence to support. The null and alternative hypotheses in words are as follows. H0​: The standard deviation of monthly cell phone bills is ​$5.05. H1​: The standard deviation of monthly cell phone bills is greater than ​$5.05. ​(b) The hypotheses in symbols are as follows. H0​:σ=​$5.05 H1​:σ>​$5.05 ​(c) A type I error is committed when the null hypothesis is rejected​ when, in​ fact, it is true. The sample evidence led the researcher to believe the standard deviation of monthly cell phone bills is higher than ​$5.05 ​when, in​ fact, the standard deviation of the bill is ​$5.05. ​(d) A type II error is committed when the null hypothesis is not rejected when the alternative hypothesis is true. For this​ problem, a type II error occurs when the sample evidence did not lead the researcher to believe the standard deviation of monthly cell phone bills is higher than ​$5.05 ​when, in​ fact, the standard deviation of bills is higher than ​$5.05.

10.1 For students who first enrolled in​ two-year public institutions in a recent​ semester, the proportion who earned a​ bachelor's degree within six years was 0.395. The president of a certain junior college believes that the proportion of students who enroll in her institution have a lower completion rate. ​(a) State the null and alternative hypotheses in words. ​(b) State the null and alternative hypotheses symbolically. ​(c) Explain what it would mean to make a Type I error. ​(d) Explain what it would mean to make a Type II error. ​(a) State the null hypothesis in words. Choose the correct answer below. A. Among students who enroll at the certain junior​ college, the completion rate is greater than 0.395. B. Among students who enroll at the certain junior​ college, the completion rate is 0.395. C. Among students who enroll at the certain junior​ college, the completion rate is less than 0.395. D. Among students who first enroll in​ two-year public​ institutions, the completion rate is 0.395 State the alternative hypothesis in words. Choose the correct answer below. A. Among students who first enroll in​ two-year public​ institutions, the completion rate is 0.395. B. Among students who enroll at the certain junior​ college, the completion rate is greater than 0.395. C. Among students who enroll at the certain junior​ college, the completion rate is less than 0.395. D. Among students who enroll at the certain junior​ college, the completion rate is 0.395. ​(b) State the hypotheses symbolically. H0​: ___ __ ___ H1​: ___ ___ ___ ​(Type integers or decimals. Do not​ round.) (c) What would it mean to make a Type I​ error? The president _________ the hypothesis that the proportion of students who earn a​ bachelor's degree within six years is _____ _____​when, in​ fact, the proportion is _____ ______ ​(Type integers or decimals. Do not​ round.) (d) What would it mean to make a Type II​ error? The president ______ the hypothesis that the proportion of students who earn a​ bachelor's degree within six years is ____ ____when, in​ fact, the proportion is ____ _____. ​(Type integers or decimals. Do not​ round.)

a) B. Among students who enroll at the certain junior​ college, the completion rate is 0.395. C. Among students who enroll at the certain junior​ college, the completion rate is less than 0.395. b)​H0​:p=0.395 H1​: p< 0.395 Note: The null​ hypothesis, denoted H0 (read "H-naught"), is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The alternative​ hypothesis, denoted H1 (read "H-one"), is a statement that we are trying to find evidence to support. Be sure to type integers or decimals and not to round. c)rejects; equal to; 0.395​; equal to; 0.395 d)fails to reject; equal to; 0.395; less than; 0.395

10.6 To test H0​: p=0.45 versus H1​: p<0.45​, a simple random sample of n=300 individuals is obtained and x=126 successes are observed. ​(a) What does it mean to make a Type II error for this​ test? ​(b) If the researcher decides to test this hypothesis at the α=0.05 level of​ significance, compute the probability of making a Type II​ error, β​, if the true population proportion is 0.42. What is the power of the​ test? ​(c) Redo part​ (b) if the true population proportion is 0.38. ​(a) What does it mean to make a Type II error for this​ test? Choose the correct answer below. A. H0 is rejected and the true population proportion is greater than 0.45. B. H0 is not rejected and the true population proportion is less than 0.45. C. H0 is not rejected and the true population proportion is equal to 0.45. D. H0 is rejected and the true population proportion is less than 0.45. (b) If the researcher decides to test this hypothesis at the α=0.05 level of​ significance, compute the probability of making a Type II​ error, β​, if the true population proportion is 0.42. What is the power of the​ test? β=____ Power=____ ​(Type integers or decimals rounded to four decimal places as​ needed.) c) Redo part​ (b) if the true population proportion is 0.38. β=____ Power=____ ​(Type integers or decimals rounded to four decimal places as​ needed.)

a) B. H0 is not rejected and the true population proportion is less than 0.45. b) 0.7274; 0.2725 c) 0.2084; 0.7916

10.3 Several years​ ago, the mean height of women 20 years of age or older was 63.7 inches. Suppose that a random sample of 45 women who are 20 years of age or older today results in a mean height of 64.3 inches. ​(a) State the appropriate null and alternative hypotheses to assess whether women are taller today. ​(b) Suppose the​ P-value for this test is 0.04. Explain what this value represents. ​(c) Write a conclusion for this hypothesis test assuming an α=0.05 level of significance. ​(a) State the appropriate null and alternative hypotheses to assess whether women are taller today. A. H0​: μ=64.3 in. versus H1​: μ≠64.3 in. B. H0​: μ=63.7 in. versus H1​: μ>63.7 in C. H0​: μ=63.7 in. versus H1​: μ≠63.7 in. D. H0​: μ=64.3 in. versus H1​: μ>64.3 in. E. H0​: μ=63.7 in. versus H1​: μ<63.7 in. F. H0​: μ=64.3 in. versus H1​: μ<64.3 in. ​(b) Suppose the​ P-value for this test is 0.04. Explain what this value represents. A. There is a 0.04 probability of obtaining a sample mean height of 64.3 inches or taller from a population whose mean height is 63.7 inches. B. There is a 0.04 probability of obtaining a sample mean height of 63.7 inches or taller from a population whose mean height is 64.3 inches. C. There is a 0.04 probability of obtaining a sample mean height of exactly 64.3 inches from a population whose mean height is 63.7 inches. D. There is a 0.04 probability of obtaining a sample mean height of 64.3 inches or shorter from a population whose mean height is 63.7 inches. ​(c) Write a conclusion for this hypothesis test assuming an α=0.05 level of significance. A. Do not reject the null hypothesis. There is not sufficient evidence to conclude that the mean height of women 20 years of age or older is greater today. B. Reject the null hypothesis. There is not sufficient evidence to conclude that the mean height of women 20 years of age or older is greater today. C. Do not reject the null hypothesis. There is sufficient evidence to conclude that the mean height of women 20 years of age or older is greater today. D. Reject the null hypothesis. There is sufficient evidence to conclude that the mean height of women 20 years of age or older is greater today.

a) B. H0​: μ=64.6 in. versus H1​: μ<64.6 in. Note: The hypotheses can be structured in one of three​ ways: Two-tailedLeft-tailedRight-tailedH0: μ=μ0H0: μ=μ0H0: μ=μ0H1: μ≠μ0H1: μ<μ0H1: μ>μ0 ​Note: μ0 is the assumed value of the population mean. b)A. There is a 0.04 probability of obtaining a sample mean height of 64.3 inches or taller from a population whose mean height is 63.7 inches. Note: For a​ two-tailed test, the​ P-value is the probability of obtaining a sample mean that is more than ​|z0​| standard deviations away from the mean stated in the null​ hypothesis, μ0. For a​ left-tailed test, the​ P-value is the probability of obtaining a sample mean of x or smaller under the assumption that H0 is true. In other​ words, it is the probability of obtaining a sample mean that is more than ​|z0​| standard deviations to the left of μ0. For a​ right-tailed test, the​ P-value is the probability of obtaining a sample mean of x or larger under the assumption that H0 is true. In other​ words, it is the probability of obtaining a sample mean that is more than z0 standard deviations to the right of μ0. c)D. Reject the null hypothesis. There is sufficient evidence to conclude that the mean height of women 20 years of age or older is greater today. Note: Reject the null hypothesis if the​ P-value is less than the level of​ significance, α. There is sufficient evidence to conclude that H1 is true. Do not reject the null hypothesis if the​ P-value is greater than the level of​ significance, α. There is not sufficient evidence to conclude that H1 is true.

10.3 To test H0: μ=105 versus H1: μ≠105 a simple random sample of size n=35 is obtained. Complete parts a through e below. LOADING... Click here to view the​ t-Distribution Area in Right Tail. ​(a) Does the population have to be normally distributed to test this​ hypothesis? Why? A. ​No, because the test is​ two-tailed. B. ​No, because n≥30. C. ​Yes, because the sample is random. D. ​Yes, because n≥30. ​(b) If x=102.0 and s=5.8​, compute the test statistic. The test statistic is t0=____. ​(Round to two decimal places as​ needed.) ​(c) Draw a​ t-distribution with the area that represents the​ P-value shaded. Choose the correct graph below. A. A symmetric bell-shaped curve is plotted over a horizontal axis. On the left side of the graph, a vertical line runs from the axis to the curve. The area under the curve to the left of the vertical line is shaded. B. A symmetric bell-shaped curve is plotted over a horizontal axis. Two vertical lines, equidistant from the curve's peak at the center, extend from the axis to the curve on the left and right sides of the graph. The area under the curve between the vertical lines is shaded. C. A symmetric bell-shaped curve is plotted over a horizontal axis. Two vertical lines, equidistant from the curve's peak at the center, extend from the axis to the curve on the left and right sides of the graph. The areas under the curve to the left of the left vertical line and to the right of the right vertical line are shaded. ​(d) Approximate the​ P-value. Choose the correct answer below. A. 0.002<​P-value<0.005 B. 0.001<​P-value<0.002 C. 0.005<​P-value<0.01 D. 0.01<​P-value<0.02 Interpret the​ P-value. Choose the correct answer below. A. If 1000 random samples of size n=35 are​ obtained, about 4 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=105. B. If 1000 random samples of size n=35 are​ obtained, about 10 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=105. C. If 100 random samples of size n=35 are​ obtained, about 4 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=105. D. If 1000 random samples of size n=35 are​ obtained, about 4 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=102.0. (e) If the researcher decides to test this hypothesis at the α=0.05 level of​ significance, will the researcher reject the null​ hypothesis? Yes No

a) B. ​No, because n≥30. b)-3.06 c)C. A symmetric bell-shaped curve is plotted over a horizontal axis. Two vertical lines, equidistant from the curve's peak at the center, extend from the axis to the curve on the left and right sides of the graph. The areas under the curve to the left of the left vertical line and to the right of the right vertical line are shaded. d)A. 0.002<​P-value<0.005 A. If 1000 random samples of size n=35 are​ obtained, about 4 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=105. e) Yes

10.1 For students who first enrolled in​ two-year public institutions in a recent​ semester, the proportion who earned a​ bachelor's degree within six years was 0.392. The president of a certain junior college believes that the proportion of students who enroll in her institution have a lower completion rate. ​(a) State the null and alternative hypotheses in words. ​(b) State the null and alternative hypotheses symbolically. ​(c) Explain what it would mean to make a Type I error. ​(d) Explain what it would mean to make a Type II error. ​(a) State the null hypothesis in words. Choose the correct answer below. A. Among students who enroll at the certain junior​ college, the completion rate is less than 0.392. B. Among students who first enroll in​ two-year public​ institutions, the completion rate is 0.392. C. Among students who enroll at the certain junior​ college, the completion rate is 0.392. D. Among students who enroll at the certain junior​ college, the completion rate is greater than 0.392. State the alternative hypothesis in words. Choose the correct answer below. A. Among students who first enroll in​ two-year public​ institutions, the completion rate is 0.392. B. Among students who enroll at the certain junior​ college, the completion rate is greater than 0.392. C. Among students who enroll at the certain junior​ college, the completion rate is 0.392. D. Among students who enroll at the certain junior​ college, the completion rate is less than 0.392. ​(b) State the hypotheses symbolically. H0​: ___ __ ___ H1​: ___ ___ ___ ​(Type integers or decimals. Do not​ round.) ​(c) What would it mean to make a Type I​ error? The president _________ the hypothesis that the proportion of students who earn a​ bachelor's degree within six years is _____ _____​when, in​ fact, the proportion is _____ ______ ​(Type integers or decimals. Do not​ round.) (d) What would it mean to make a Type II​ error? The president ______ the hypothesis that the proportion of students who earn a​ bachelor's degree within six years is ____ ____when, in​ fact, the proportion is ____ _____. ​(Type integers or decimals. Do not​ round.)

a) C. Among students who enroll at the certain junior​ college, the completion rate is 0.392. Note: The null​ hypothesis, denoted H0 ​(read "H-naught"), is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. D. Among students who enroll at the certain junior​ college, the completion rate is less than 0.392. b)h0: p=0.392 h2: p<0.392 Note: The null​ hypothesis, denoted H0 (read "H-naught"), is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The alternative​ hypothesis, denoted H1 (read "H-one"), is a statement that we are trying to find evidence to support. Be sure to type integers or decimals and not to round. c)rejects; equal to; 0.392​; equal to; 0.392 Note: A Type I error is committed when the null hypothesis is rejected​ when, in​ fact, it is true. Be sure to type integers or decimals and not to round. d)fails to reject; equal to; 0.392; less than; 0.392 Note: A Type II error is committed when the null hypothesis is not rejected​ when, in​ fact, the alternative hypothesis is true. Be sure to type integers or decimals and not to round.

10.3 To test H0: μ=105 versus H1: μ≠105 a simple random sample of size n=35 is obtained. Complete parts a through e below. LOADING... Click here to view the​ t-Distribution Area in Right Tail.

a) Does the population have to be normally distributed to test this​ hypothesis? Why? If the sample is obtained using simple random​ sampling, the sample has no​ outliers, and the population from which the sample is drawn is normally distributed or the sample​ size, n, is large ​(n≥​30), use the​ t-distribution to test hypotheses regarding the population mean with unknown σ. Since n=35≥30 and the sample is obtained using simple random​ sampling, the population does not have to be normally distributed to test this hypothesis. ​(b) If x=102.0 and s=5.6​, compute the test statistic. The test statistic is given by the formula below. t0=x−μ0sn Compute the test statistic. t0=x−μ0sn t0=102.0−1055.635 t0≈−3.17 ​(c) Draw a​ t-distribution with the area that represents the​ P-value shaded. Since the null and alternative hypotheses are H0: μ=105 and H1: μ≠105​, the test is a​ two-tailed test. Draw a​ t-distribution with the area that represents the​ P-value shaded. The correct graph is shown below. A symmetric bell-shaped curve is plotted over a horizontal axis. Two vertical lines, equidistant from the curve's peak at the center, extend from the axis to the curve on the left and right sides of the graph. The areas under the curve to the left of the left vertical line and to the right of the right vertical line are shaded. ​(d) Approximate and interpret the​ P-value. Since the test is a​ two-tailed test, ​P-value=​P(t0<−3.17 or t0>3.17​). ​P-value=​P(t0<−3.17 or t0>3.17​) ​P-value=​P(t0<−3.17​)+​P( t0>3.17​) ​P-value=​2P(t0>3.17​) Use the table of the​ t-distribution. Note that df=n−1=34. Since 3.17 is between 3.002 and​ 3.348, ​P(t0>3.17​) is between 0.001 and 0.0025. ​Thus, since ​P-value=​2P(t0>3.17​), the​ P-value is between 0.002 and 0.005. A​ P-value is the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption that the null hypothesis is true. Put another​ way, the​ P-value is the likelihood or probability that a sample will result in a sample mean such as the one obtained if the null hypothesis is true. Interpret the​ P-value. If 1000 random samples of size n=35 are​ obtained, about 3 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=105. ​(e) If the researcher decides to test this hypothesis at the α=0.01 level of​ significance, will the researcher reject the null​ hypothesis? To state the appropriate​ conclusion, compare the​ P-value to the level of​ significance, α. The null hypothesis is rejected if ​P-value<α. Since the​ P-value is less than the α=0.01 level of​ significance, there is sufficient evidence to reject the null hypothesis.

10.5 A simple random sample of size n=40 is drawn from a population. The sample mean is found to be 106.6​, and the sample standard deviation is found to be 23.4. Is the population mean greater than 100 at the α=0.10 level of​ significance? a) Determine the null and alternative hypotheses. H0​:____ H1​:____ b)Compute the test statistic. __ = ___​(Round to two decimal places as​ needed.) c)Determine the​ P-value. The​ P-value is ____. ​(Round to three decimal places as​ needed.) d) What is the result of the hypothesis​ test? ____ the null hypothesis because the​ P-value is _______ the level of significance. At the α=0.10 level of​ significance, the population mean ___ ___

a) H0​: μ=100 H1​: μ>100 b)t0=1.78 c)0.041 d) Reject ; less than; is; greater than

10.1 According to the Centers for Disease Control and​ Prevention, 10.2​% of high school students currently use electronic cigarettes. A high school counselor is concerned the use of​ e-cigs at her school is higher. Complete parts​ (a) through​ (c) below. ​(a) Determine the null and alternative hypotheses. H0​: ____ ____ ____ H1​:___ ___ ____ ​(Type integers or decimals. Do not​ round.) ​(b) If the sample data indicate that the null hypothesis should not be​ rejected, state the conclusion of the high school counselor. A. There is not sufficient evidence to conclude that the proportion of high school students exceeds 0.102 at this​ counselor's high school. B. There is not sufficient evidence to conclude that the proportion of high school students stayed 0.102 at this​ counselor's high school. C. There is sufficient evidence to conclude that the proportion of high school students stayed 0.102 at this​ counselor's high school. D. There is sufficient evidence to conclude that the proportion of high school students exceeds 0.102 at this​ counselor's high school. (c) ​Suppose, in​ fact, that the proportion of students at the​ counselor's high school who use electronic cigarettes is 0.222. Was a type I or type II error​ committed? A. A Type II error was committed because the sample evidence led the counselor to conclude the proportion of​ e-cig users was 0.222​, ​when, in​ fact, the proportion is lower. B. A Type II error was committed because the sample evidence led the counselor to conclude the proportion of​ e-cig users was 0.102​, ​when, in​ fact, the proportion is higher. C. A Type I error was committed because the sample evidence led the counselor to conclude the proportion of​ e-cig users was 0.222​, ​when, in​ fact, the proportion is lower. D. A Type I error was committed because the sample evidence led the counselor to conclude the proportion of​ e-cig users was 0.102​, ​when, in​ fact, the proportion is higher.

a) H0​:p=0.102 H1​:p>0.102 b)A. There is not sufficient evidence to conclude that the proportion of high school students exceeds 0.102 at this​ counselor's high school Note: When the null hypothesis is​ rejected, we say that there is sufficient evidence to support the statement. When the null hypothesis is not​ rejected, we say that there is not sufficient evidence to support the statement. We never say that the null hypothesis is true. c)B. A Type II error was committed because the sample evidence led the counselor to conclude the proportion of​ e-cig users was 0.102​, ​when, in​ fact, the proportion is higher.

10.5 In 1945​, an organization asked 1409 randomly sampled American​ citizens, "Do you think we can develop a way to protect ourselves from atomic bombs in case others tried to use them against​ us?" with 728 responding yes. Did a majority of the citizens feel the country could develop a way to protect itself from atomic bombs in 1945​? Use the α=0.01 level of significance. a) What are the null and alternative​ hypotheses? H0​: ___ ___ ___ H1: ___ __ ___ ​(Type integers or​ decimals.) b)Determine the test​ statistic, z0. z0=_____ ​(Round to two decimal places as​ needed.) c)Use technology to determine the​ P-value for the test statistic. The​ P-value is ____ ​(Round to three decimal places as​ needed.) d)What is the correct conclusion at the α=0.01 level of​ significance? Since the​ P-value is _____ than the level of​ significance, _____the null hypothesis. There ___ not sufficient evidence to conclude that the majority of the citizens feel the country could develop a way to protect itself from atomic bombs.

a) H0​:p=0.50 H1​:p>0.50 b)1.25 c)0.105 d)greater; do not reject; is not

10.5 In 1945​, an organization asked 1437 randomly sampled American​ citizens, "Do you think we can develop a way to protect ourselves from atomic bombs in case others tried to use them against​ us?" with 744 responding yes. Did a majority of the citizens feel the country could develop a way to protect itself from atomic bombs in 1945​? Use the α=0.01 level of significance. a) What are the null and alternative​ hypotheses? H0​: ___ ___ ___ H1: ___ __ ___ ​(Type integers or​ decimals.) b)Determine the test​ statistic, z0. z0=_____ ​(Round to two decimal places as​ needed.) c)Use technology to determine the​ P-value for the test statistic. The​ P-value is ____ ​(Round to three decimal places as​ needed.) d)What is the correct conclusion at the α=0.01 level of​ significance? Since the​ P-value is _____ than the level of​ significance, _____the null hypothesis. There ___ not sufficient evidence to conclude that the majority of the citizens feel the country could develop a way to protect itself from atomic bombs.

a) H0​:p=0.50 H1​:p>0.50 b)1.35 c)0.089 d)greater; do not reject; is not

10.3 A golf association requires that golf balls have a diameter that is 1.68 inches. To determine if golf balls conform to the​ standard, a random sample of golf balls was selected. Their diameters are shown in the accompanying data table. Do the golf balls conform to the​ standards? Use the α=0.01 level of significance. LOADING... Click the icon to view the data table. Golf Ball Diameter​ (inches) 1.682 1.676 1.681 1.686 1.677 1.686 1.684 1.685 1.673 1.685 1.682 1.675 a)First determine the appropriate hypotheses.. H0​:μ __ ___ H1​: μ ___ ___ ​(Type integers or decimals. Do not​ round.) b)Find the test statistic. ___ ​(Round to two decimal places as​ needed.) c)Find the​ P-value. ___ ​(Round to three decimal places as​ needed.) d)What can be concluded from the hypothesis​ test? A. Do not reject H0. There is not sufficient evidence to conclude that the golf balls do not conform to the​ association's standards at the α=0.01 level of significance. B. Reject H0. There is not sufficient evidence to conclude that the golf balls do not conform to the​ association's standards at the α=0.01 level of significance. C. Reject H0. There is sufficient evidence to conclude that the golf balls do not conform to the​ association's standards at the α=0.01 level of significance. D. Do not reject H0. There is sufficient evidence to conclude that the golf balls do not conform to the​ association's standards at the α=0.01 level of significance.

a) H0​:μ=1.68 H1: μ≠1.68 b) 0.75 c)0.468 d)A. Do not reject H0. There is not sufficient evidence to conclude that the golf balls do not conform to the​ association's standards at the α=0.01 level of significance.

10.4 Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than 3​%. A​ mutual-fund rating agency randomly selects 28 months and determines the rate of return for a certain fund. The standard deviation of the rate of return is computed to be 2.05​%. Is there sufficient evidence to conclude that the fund has moderate risk at the α=0.10 level of​ significance? A normal probability plot indicates that the monthly rates of return are normally distributed. a)What are the correct hypotheses for this​ test? The null hypothesis is H0​: ___ ___ ___ The alternative hypothesis is H1​: ___ ___ ___ b)Calculate the value of the test statistic. χ20=__ ​(Round to two decimal places as​ needed.) c)Use technology to determine the​ P-value for the test statistic. The​ P-value is ___ ​(Round to three decimal places as​ needed.) d) What is the correct conclusion at the α=0.10 level of​ significance? Since the​ P-value is ____ than the level of​ significance, ______ the null hypothesis. There _____ sufficient evidence to conclude that the fund has moderate risk at the 0.10 level of significance.

a) H0​:σ=0.03 H1​: σ<0.03. b) 12.61 c)0.009 d) less, reject, is Note: If the​ P-value is less than the level of significance α​, reject the null hypothesis.​ Otherwise, do not reject the null hypothesis.

10.3 A college entrance exam company determined that a score of 21 on the mathematics portion of the exam suggests that a student is ready for​ college-level mathematics. To achieve this​ goal, the company recommends that students take a core curriculum of math courses in high school. Suppose a random sample of 150 students who completed this core set of courses results in a mean math score of 21.4 on the college entrance exam with a standard deviation of 4.2. Do these results suggest that students who complete the core curriculum are ready for​ college-level mathematics? That​ is, are they scoring above 21 on the mathematics portion of the​ exam? Complete parts​ a) through​ d) below.

a) State the appropriate null and alternative hypotheses. The null​ hypothesis, denoted H0​, is a statement of no​ change, no​ effect, or no difference that is tested in an experiment. The alternative​ hypothesis, denoted H1​, is a statement that is contrary to the null hypothesis. The hypotheses can be structured in one of three ways. Two-tailedLeft-tailedRight-tailedH0: μ=μ0H0: μ=μ0H0: μ=μ0H1: μ≠μ0H1: μ<μ0H1: μ>μ0 Identify the value of μ0​, which is the assumed population mean. μ0=21 Note that the goal of the test is to determine if students are scoring above a certain score. The key word​ "above" means that a​ right-tail test is the appropriate test. Now determine the hypotheses that should be used. The hypotheses below can be used to represent this problem. H0​: μ=21 versus H1​: μ>21 ​b) Verify that the requirements to perform the test using the​ t-distribution are satisfied. To test hypotheses regarding the population​ mean, the sample must be obtained using simple random sampling or from a randomized experiment. The sample must not have any​ outliers, and the population from which the sample is drawn must be normally distributed or the sample​ size, n, must be large ​(n≥​30). The sampled values must also be independent of each other. Identify the value of​ n, the sample size. n=150 Use the value of n and the information in the problem statement to determine if the requirements to perform the test using the​ t-distribution are satisfied. ​c) Use the​ P-value approach at the α=0.05 level of significance to test the hypotheses in part​ (a). The formula displayed below is used to find the test statistic. Either the formula or technology can be used. For the purpose of this​ problem, use technology. t0=x−μ0sn Identify the value of x​, the sample mean. x=21.4 Identify the value of​ s, the sample standard deviation. s=4.2 Use technology to compute the test​ statistic, rounding to two decimal places. t0 = 1.17 Review the technology output from the previous step to determine the​ P-value, rounding to three decimal places. ​P-value=0.123 ​d) Write a conclusion based on the results. Reject H0 if the test statistic is more extreme than the critical​ value, and conclude that there is sufficient evidence that the population mean is greater than 21. ​Otherwise, do not reject H0 and conclude there is not sufficient evidence that the population mean is greater than 21.

10.2 Test the hypothesis using the​ P-value approach. Be sure to verify the requirements of the test. H0​: p=0.9 versus H1​: p>0.9 n=250​; x=230​, α=0.05 a) Is np0(1−p0)≥​10? No Yes b) Use technology to find the​ P-value. ​P-value=_____ ​(Round to three decimal places as​ needed.) c)______ the null​hypothesis, because the​P-value is ______ α.

a) yes b)0.146 c) Do not reject; greater than

10.5 A simple random sample of size n=15 is drawn from a population that is normally distributed. The sample mean is found to be x=27.4 and the sample standard deviation is found to be s=6.3. Determine if the population mean is different from 24 at the α=0.01 level of significance. Complete parts ​(a) through ​(d) below. ​(a) Determine the null and alternative hypotheses. H0​:μ __ 24 H1​: μ ___ 24 (b) Calculate the​ P-value. ​P-value=_____ ​(Round to three decimal places as​ needed.) (c) State the conclusion for the test. A. Reject H0 because the​ P-value is greater than the α=0.01 level of significance. B. Reject H0 because the​ P-value is less than the α=0.01 level of significance. C. Do not reject H0 because the​ P-value is less than the α=0.01 level of significance. D. Do not reject H0 because the​ P-value is greater than the α=0.01 level of significance. (d) State the conclusion in context of the problem. There ___sufficient evidence at the α=0.01 level of significance to conclude that the population mean is different from 24.

a) μ=24 μ≠24 b) 0.055 c)D. Do not reject H0 because the​ P-value is greater than the α=0.01 level of significance. d) is not Note: The correct answer is is not because the​P-value is greater than the level of significance.

10.5 A simple random sample of size n=15 is drawn from a population that is normally distributed. The sample mean is found to be x=28.1 and the sample standard deviation is found to be s=6.3. Determine if the population mean is different from 26 at the α=0.01 level of significance. Complete parts ​(a) through ​(d) below. ​(a) Determine the null and alternative hypotheses. H0​:μ __ 26 H1​: μ ___ 26 (b) Calculate the​ P-value. ​P-value=_____ ​(Round to three decimal places as​ needed.) (c) State the conclusion for the test. A. Reject H0 because the​ P-value is greater than the α=0.01 level of significance. B. Reject H0 because the​ P-value is less than the α=0.01 level of significance. C. Do not reject H0 because the​ P-value is less than the α=0.01 level of significance. D. Do not reject H0 because the​ P-value is greater than the α=0.01 level of significance. (d) State the conclusion in context of the problem. There ___sufficient evidence at the α=0.01 level of significance to conclude that the population mean is different from 26.

a) μ=26 μ≠26 b) 0.218 c)D. Do not reject H0 because the​ P-value is greater than the α=0.01 level of significance. d) is not

10.3 To test H0​: μ=20 versus H1​: μ<20​, a simple random sample of size n=17 is obtained from a population that is known to be normally distributed. Answer parts​ (a)-(d). LOADING... Click here to view the​ t-Distribution Area in Right Tail. ​(a) If x=18.2 and s=4.2​, compute the test statistic. t=____ ​(Round to two decimal places as​ needed.) ​(b) Draw a​ t-distribution with the area that represents the​ P-value shaded. Which of the following graphs shows the correct shaded​ region? A. A symmetric bell-shaped curve is plotted over a horizontal axis. On the right side of the graph, a vertical line runs from the axis to the curve. The area under the curve to the right of the vertical line is shaded. B. A symmetric bell-shaped curve is plotted over a horizontal axis. Two vertical lines, equidistant from the curve's peak at the center, extend from the axis to the curve on the left and right sides of the graph. The areas under the curve to the left of the left vertical line and to the right of the right vertical line are shaded. C. A symmetric bell-shaped curve is plotted over a horizontal axis. On the left side of the graph, a vertical line runs from the axis to the curve. The area under the curve to the left of the vertical line is shaded. ​(c)Approximate the​ P-value. Choose the correct range for the​ P-value below. A. 0.025<P-value<0.05 B. 0.10<P-value<0.15 C. 0.15<P-value<0.20 D. 0.05<P-value<0.10 ​(d) If the researcher decides to test this hypothesis at the α=0.05 level of​ significance, will the researcher reject the null​ hypothesis? A. The researcher will not reject the null hypothesis since the​ P-value is less than α. B. The researcher will reject the null hypothesis since the​ P-value is not less than α. C. The researcher will not reject the null hypothesis since the P-value is not less than α. D. The researcher will reject the null hypothesis since the P-value is less than α.

a)-1.77 b) C. A symmetric bell-shaped curve is plotted over a horizontal axis. On the left side of the graph, a vertical line runs from the axis to the curve. The area under the curve to the left of the vertical line is shaded. c)A. 0.025<P-value<0.05 d) D. The researcher will reject the null hypothesis since the P-value is less than α. Note: Reject the null hypothesis only if the​ P-value is less than α.

10.6 To test H0​: p=0.55 versus H1​: p<0.55​, a simple random sample of n=100 individuals is obtained and x=52 successes are observed. If the true population proportion is 0.52 and the level of significance of α=0.05 is​ used, β=0.8499 and the power of the test is 0.1501. a)What is the β and power of the test if redone at α=0.01 level of​ significance? β=____ Power=____ ​(Type integers or decimals rounded to four decimal places as​ needed.) b)What effect does lowering the level of significance have on the power of the​ test? A. When the level of significance is​ lowered, the power of the test stays constant. B. When the level of significance is​ lowered, the power of the test decreases. C. When the level of significance is​ lowered, the power of the test is 0. D. When the level of significance is​ lowered, the power of the test increases.

a)0.9569 0.0431 b)B. When the level of significance is​ lowered, the power of the test decreases.

10.4 Determine the critical values for these tests of a population standard deviation. ​(a) A​ right-tailed test with 12 degrees of freedom at the α=0.05 level of significance ​(b) A​ left-tailed test for a sample of size n=27 at the α=0.1 level of significance ​(c) A​ two-tailed test for a sample of size n=26 at the α=0.01 level of significance LOADING... Click the icon to view a table a critical values for the​ Chi-Square Distribution. ​(a) The critical value for this​ right-tailed test is ____. ​(Round to three decimal places as​ needed.) (b) The critical value for this​ left-tailed test is ____. ​(Round to three decimal places as​ needed.) ​(c) The critical values for this​ two-tailed test are ___. ​(Round to three decimal places as needed. Use a comma to separate answers as​ needed.)

a)21.026 Note: To use the table to identify the critical​ value, locate the number of degrees of freedom along the left side and the value of α along the top. Then identify the value in that row and column of the table. b)17.292 Note:For a sample of size​ n, the number of degrees of freedom is n−1. To use the table to identify the critical​ value, locate the number of degrees of freedom along the left side and the value of 1−α along the top. Then identify the value in that row and column of the table. c)10.52, 46.928 Note: For a sample of size​ n, the number of degrees of freedom is n−1. To use the table to identify the critical​ values, locate the number of degrees of freedom along the left side and the values of 1−α/2 and α/2 along the top. Then identify the values in that row and those columns of the table.

10.2 In a previous​ year, 53​% of females aged 15 and older lived alone. A sociologist tests whether this percentage is different today by conducting a random sample of 400 females aged 15 and older and finds that 217 are living alone. Is there sufficient evidence at the α=0.05 level of significance to conclude the proportion has​ changed? a) Because np0(1−p0)=___ ___10, the sample size is ____ ​5% of the population​ size, and the sample _______ the requirements for testing the hypothesis ______ satisfied. ​(Round to one decimal place as​ needed.) b) What are the null and alternative​ hypotheses? H0​: ____ ____ ____ versus H1​:___ ___ ____ ​(Type integers or decimals. Do not​ round.) c) Find the test statistic z0. z0=_____ ​(Round to two decimal places as​ needed.) d)Find the​ P-value. ​P-value=____ ​(Round to three decimal places as​ needed.) e) State the conclusion for this hypothesis test. A. Reject H0. There is not sufficient evidence at the α=0.05 level of significance to conclude that the proportion of females who are living alone has changed. B. Do not reject H0. There is not sufficient evidence at the α=0.05 level of significance to conclude that the proportion of females who are living alone has changed. C. Do not reject H0. There is sufficient evidence at the α=0.05 level of significance to conclude that the proportion of females who are living alone has changed. D. Reject H0. There is sufficient evidence at the α=0.05 level of significance to conclude that the proportion of females who are living alone has changed.

a)99.6; >; less than; is given to be random; are b)H0​:p=0.53 versus H1​: p≠0.53 c) 0.5 d)0.616 e) B. Do not reject H0. There is not sufficient evidence at the α=0.05 level of significance to conclude that the proportion of females who are living alone has changed.

10.5 A psychologist obtains a random sample of 20 mothers in the first trimester of their pregnancy. The mothers are asked to play Mozart in the house at least 30 minutes each day until they give birth. After 5​ years, the child is administered an IQ test. It is known that IQs are normally distributed with a mean of 100. If the IQs of the 20 children in the study result in a sample mean of 104.5 and sample standard deviation of 15.8​, is there evidence that the children have higher​ IQs? Use the α=0.10 level of significance. a) What type of hypothesis test is appropriate to conduct for this​ research? A. Hypothesis test on a population mean B. Hypothesis test on a population standard deviation C. Hypothesis test on a population proportion b)Determine the null and alternative hypotheses. H0: __ __ __ H1: __ ___ __ ​(Type integers or decimals. Do not​ round.) c)Which distribution should be used for this hypothesis​ test? A. The​ chi-square distribution because the parameter is σ or σ2​, and the model conditions are satisfied. B. The​ Student's t-distribution because the parameter is the​ mean, σ is not​ known, and the model conditions are satisfied. C. The normal distribution because the parameter is the​ mean, σ is​ known, and the model conditions are satisfied. D. The normal distribution because the parameter is a​ proportion, p, and the model conditions are satisfied. d)Calculate the test statistic. Test statistic=____ ​(Round to two decimal places as​ needed.) e)Calculate the​ P-value. ​P-value=____ ​(Round to three decimal places as​ needed.) f) State the conclusion for the test. Choose the correct answer below. A. Do not reject H0. There is not sufficient evidence at the α=0.10 level of significance to conclude that mothers who listen to Mozart have children with higher IQs. B. Reject H0. There is sufficient evidence at the α=0.10 level of significance to conclude that mothers who listen to Mozart have children with higher IQs. C. Do not reject H0. There is sufficient evidence at the α=0.10 level of significance to conclude that mothers who listen to Mozart have children with lower IQs. D. Reject H0. There is not sufficient evidence at the α=0.10 level of significance to conclude that mothers who listen to Mozart have children with lower IQs.

a)A. Hypothesis test on a population mean b)H0: μ=100 H1: μ>100 Note: The null​ hypothesis, denoted H0​, is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The alternative​ hypothesis, denoted H1​, is a statement that we are trying to find evidence to support. Determine which parameter is being tested. Next determine if the test is​ two-tailed, left-tailed, or​ right-tailed. Reread the problem statement and determine if the parameter is greater​ than, less​ than, or not equal to a certain number. Remember to type integers or decimals and to not round. c)B. The​ Student's t-distribution because the parameter is the​ mean, σ is not​ known, and the model conditions are satisfied. d)1.27 Note: Calculate the test statistic using technology or the following​ formula, where x is the sample​ mean, μ0 is the assumed value of the population​ mean, s is the sample standard​ deviation, and n is the sample size. e)0.109 f) A. Do not reject H0. There is not sufficient evidence at the α=0.10 level of significance to conclude that mothers who listen to Mozart have children with higher IQs.

10.5 A simple random sample of size n=200 drivers with a valid​ driver's license is asked if they drive an​ American-made automobile. Of the 200 drivers​ surveyed, 116 responded that they drive an​ American-made automobile. Determine if a majority of those with a valid​ driver's license drive an​ American-made automobile at the α=0.05 level of significance. a) What type of variable is​ "drive an​ American-made automobile, or​ not"? A. Qualitative with two possible outcomes B. Qualitative with more than two possible outcomes C. Continuous D. Discrete b) What type of hypothesis test is appropriate to conduct for this​ research? A. Hypothesis test on a population standard deviation B. Hypothesis test on a population proportion C. Hypothesis test on a population mean c) Determine the null and alternative hypotheses. H0​:μ __ ___ H1​: μ ___ ___ ​(Type integers or decimals. Do not​ round.) d)Which distribution should be used for this hypothesis​ test? A. The normal distribution because the parameter is the​ mean, σ is​ known, and the model conditions are satisfied. B. The normal distribution because the parameter is a​ proportion, p, and the model conditions are satisfied. C. The​ Student's t-distribution because the parameter is the​ mean, σ is not​ known, and the model conditions are satisfied. D. The​ chi-square distribution because the parameter is σ or σ2​, and the model conditions are satisfied. e)Calculate the test statistic. Test statistic=__ ​(Round to two decimal places as​ needed.) f)Calculate the​ P-value. ​P-value=___ ​(Round to three decimal places as​ needed.) g)State the conclusion for the test. _____H0. There ____ sufficient evidence at the α=0.05 level of significance to conclude that _______ of all drivers with a valid​ driver's license drive an​ American-made automobile.

a)A. Qualitative with two possible outcomes Note: ​Qualitative, or​ categorical, variables allow for classification of individuals based on some attribute or characteristic. Quantitative variables provide numerical measures of individuals. The values of a quantitative variable can be added or subtracted and provide meaningful results. A discrete variable is a quantitative variable that has either a finite number of possible values or a countable number of possible values. A continuous variable is a quantitative variable that has an infinite number of possible values that are not countable. A continuous variable may take on every possible value between any two values. b) Hypothesis test on a population proportion Note: the parameter being tested is the proportion of individuals with a valid​ driver's license that drive an​ American-made automobile. c)p=0.5​ p>0.5 Note: The null​ hypothesis, denoted H0​, is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The alternative​ hypothesis, denoted H1​, is a statement that we are trying to find evidence to support. Determine which parameter is being tested. Next determine if the test is​ two-tailed, left-tailed, or​ right-tailed. Reread the problem statement and determine if the parameter is greater​ than, less​ than, or not equal to a certain number. Remember to type an integer or a decimal and to not round. d) B. The normal distribution because the parameter is a​ proportion, p, and the model conditions are satisfied. e)2.26 f)0.012 Note: The​ P-value is the probability of observing a sample statistic as extreme or more extreme than one observed under the assumption that the statement in the null hypothesis is true. For the hypothesis test on a population​ proportion, the test statistic follows the standard normal distribution. Use technology or the standard normal distribution table to find the​ P-value. g)Reject, is ,more than half Note: Compare the​ P-value to the level of signifance α. If the​ P-value is less than the level of significance α​, reject H0. ​Otherwise, do not reject H0.

10.5 A simple random sample of size n=200 drivers with a valid​ driver's license is asked if they drive an​ American-made automobile. Of the 200 drivers​ surveyed, 108 responded that they drive an​ American-made automobile. Determine if a majority of those with a valid​ driver's license drive an​ American-made automobile at the α=0.05 level of significance. a) What type of variable is​ "drive an​ American-made automobile, or​ not"? A. Qualitative with two possible outcomes B. Qualitative with more than two possible outcomes C. Continuous D. Discrete b) What type of hypothesis test is appropriate to conduct for this​ research? A. Hypothesis test on a population standard deviation B. Hypothesis test on a population proportion C. Hypothesis test on a population mean c) Determine the null and alternative hypotheses. H0​:μ __ ___ H1​: μ ___ ___ ​(Type integers or decimals. Do not​ round.) d)Which distribution should be used for this hypothesis​ test? A. The normal distribution because the parameter is the​ mean, σ is​ known, and the model conditions are satisfied. B. The normal distribution because the parameter is a​ proportion, p, and the model conditions are satisfied. C. The​ Student's t-distribution because the parameter is the​ mean, σ is not​ known, and the model conditions are satisfied. D. The​ chi-square distribution because the parameter is σ or σ2​, and the model conditions are satisfied. e)Calculate the test statistic. Test statistic=__ ​(Round to two decimal places as​ needed.) f)Calculate the​ P-value. ​P-value=___ ​(Round to three decimal places as​ needed.) g)State the conclusion for the test. _____H0. There ____ sufficient evidence at the α=0.05 level of significance to conclude that _______ of all drivers with a valid​ driver's license drive an​ American-made automobile.

a)A. Qualitative with two possible outcomes b) Hypothesis test on a population proportion c)p=0.5​ p>0.5 d) B. The normal distribution because the parameter is a​ proportion, p, and the model conditions are satisfied. e)1.13 f)0.012 g)Do not Reject, is not ,more than half

10.3 The life expectancy of a male during the course of the past 100 years is approximately 27,759 days. Use the table to the right to conduct a test using α=0.05 to determine whether the evidence suggests that chief justices live longer than the general population of males. Suggest a reason why the conclusion drawn may be flawed. Chief Justice Life Span​ (Days) A 31,645 I 30,322 B 31,028 J 28,357 C 32,483 K 26,631 D 30,694 L 32,340 E 32,654 M 30,492 F 29,035 N 29,410 G 24,782 O 31,712 H 32,055 P 30,768 a) State the appropriate null and alternative hypotheses. A. H0​: μ=30,276 versus H1​: μ<30,276 B. H0​: μ=27,759 versus H1​: μ>27,759 C. H0​: μ=30,276 versus H1​: μ≠30,276 D. H0​: μ=27,759 versus H1​: μ≠27,759 E. H0​: μ=27,759 versus H1​: μ<27,759 F. H0​: μ=30,276 versus H1​: μ>30,276 b)Use the​ P-value approach at the α=0.05 level of significance to test the hypotheses. ​P-value=____ ​(Round to three decimal places as​ needed.) c) State the conclusion for the test. Choose the correct answer below. A. Reject the null hypothesis. There is sufficient evidence to conclude that the mean life span of males is longer than 27,759 days.​ Thus, there is sufficient evidence to indicate that chief justices live longer than the general population of males. B. Reject the null hypothesis. There is not sufficient evidence to conclude that the mean life span of males is longer than 27,759 days.​ Thus, there is not sufficient evidence to indicate that chief justices live longer than the general population of males. C. Do not reject the null hypothesis. There is not sufficient evidence to conclude that the mean life span of males is longer than 27,759 days.​ Thus, there is not sufficient evidence to indicate that chief justices live longer than the general population of males. D. Do not reject the null hypothesis. There is sufficient evidence to conclude that the mean life span of males is longer than 30,276 days.​ Thus, there is sufficient evidence to indicate that chief justices live longer than the general population of males. d) Suggest a reason why the conclusion drawn may be flawed. Choose the correct answer below. A. The sample is not obtained using simple random sampling or from a randomized experiment. B. The sample size is very large and the data are normally distributed. C. The sampled values are not independent of each other. D. The sample consists of outliers.

a)B. H0​: μ=27,759 versus H1​: μ>27,759 b)0 c)A. Reject the null hypothesis. There is sufficient evidence to conclude that the mean life span of males is longer than 27,759 days.​ Thus, there is sufficient evidence to indicate that chief justices live longer than the general population of males. Note: Reject H0 if the​ P-value is less than the level of​ significance, α​, and conclude that there is sufficient evidence to support the statement in H1. ​Otherwise, do not reject H0 and conclude that there is not sufficient evidence to support the statement in H1. d)A. The sample is not obtained using simple random sampling or from a randomized experiment. Note: To test hypotheses regarding the population​ mean, the sample must be obtained using simple random sampling or from a randomized experiment. The sample must not have any​ outliers, and the population from which the sample is drawn must be normally distributed or the sample​ size, n, must be large ​(n≥​30). The sampled values must also be independent of each other. Recall that​ n, the size of the​ sample, is 16. Use this information to determine why the conclusion may be flawed.

10.5 A psychologist obtains a random sample of 20 mothers in the first trimester of their pregnancy. The mothers are asked to play Mozart in the house at least 30 minutes each day until they give birth. After 5​ years, the child is administered an IQ test. It is known that IQs are normally distributed with a mean of 100. If the IQs of the 20 children in the study result in a sample mean of 104.3 and sample standard deviation of 14.6​, is there evidence that the children have higher​ IQs? Use the α=0.05 level of significance. a) What type of hypothesis test is appropriate to conduct for this​ research? A. Hypothesis test on a population proportion B. Hypothesis test on a population mean C. Hypothesis test on a population standard deviation b)Determine the null and alternative hypotheses. H0: __ __ __ H1: __ ___ __ ​(Type integers or decimals. Do not​ round.) c)Which distribution should be used for this hypothesis​ test? A. The​ chi-square distribution because the parameter is σ or σ2​, and the model conditions are satisfied. B. The​ Student's t-distribution because the parameter is the​ mean, σ is not​ known, and the model conditions are satisfied. C. The normal distribution because the parameter is the​ mean, σ is​ known, and the model conditions are satisfied. D. The normal distribution because the parameter is a​ proportion, p, and the model conditions are satisfied. d)Calculate the test statistic. Test statistic=____ ​(Round to two decimal places as​ needed.) e)Calculate the​ P-value. ​P-value=____ ​(Round to three decimal places as​ needed.) f) State the conclusion for the test. Choose the correct answer below.

a)B. Hypothesis test on a population mean b)H0: μ=100 H1: μ>100 c)B. The​ Student's t-distribution because the parameter is the​ mean, σ is not​ known, and the model conditions are satisfied. d)1.32 e)0.102 f) A. Do not reject H0. There is not sufficient evidence at the α=0.10 level of significance to conclude that mothers who listen to Mozart have children with higher IQs.

10.1 According to a​ report, the standard deviation of monthly cell phone bills was ​$4.81 in 2017. A researcher suspects that the standard deviation of monthly cell phone bills is different today. ​(a) State the null and alternative hypotheses in words. ​(b) State the null and alternative hypotheses symbolically. ​(c) Explain what it would mean to make a Type I error. ​(d) Explain what it would mean to make a Type II error. ​(a) State the null hypothesis in words. Choose the correct answer below. A. The standard deviation of monthly cell phone bills is less than ​$4.81. B. The standard deviation of monthly cell phone bills is ​$4.81. C. The standard deviation of monthly cell phone bills is different from ​$4.81. D. The standard deviation of monthly cell phone bills is greater than ​$4.81. State the alternative hypothesis in words. Choose the correct answer below. A. The standard deviation of monthly cell phone bills is greater than ​$4.81. B. The standard deviation of monthly cell phone bills is different from ​$4.81. C. The standard deviation of monthly cell phone bills is less than ​$4.81. D. The standard deviation of monthly cell phone bills is ​$4.81. (b) State the hypotheses symbolically. H0​: ___ __ ___ H1​: ___ ___ ___ ​(Type integers or decimals. Do not​ round.) (c) What would it mean to make a Type I​ error? A. The sample evidence led the researcher to believe the standard deviation of monthly cell phone bills is higher than ​$4.81 ​when, in​ fact, the standard deviation of bills is $4.81. B. The sample evidence led the researcher to believe the standard deviation of monthly cell phone bills is different from ​$4.81 ​when, in​ fact, the standard deviation of bills is ​$4.81. C. The sample evidence did not lead the researcher to believe the standard deviation of monthly cell phone bills is higher than ​$4.81 ​when, in​ fact, the standard deviation of bills is higher than ​$4.81. D. The sample evidence did not lead the researcher to believe the standard deviation of monthly cell phone bills is different from ​$4.81 ​when, in​ fact, the standard deviation of bills is different from ​$4.81. (d) What would mean to make a Type II​ error? A. The sample evidence did not lead the researcher to believe the standard deviation of monthly cell phone bills is higher than ​$4.81 ​when, in​ fact, the standard deviation of bills is higher than ​$4.81. B. The sample evidence led the researcher to believe the standard deviation of monthly cell phone bills is different from ​$4.81 ​when, in​ fact, the standard deviation of bills is different from ​$4.81. C. The sample evidence did not lead the researcher to believe the standard deviation of monthly cell phone bills is different from ​$4.81 ​when, in​ fact, the standard deviation of bills is different from ​$4.81. D. The sample evidence led the researcher to believe the standard deviation of monthly cell phone bills is different from ​$4.81 ​when, in​ fact, the standard deviation of bills is ​$4.81.

a)B. The standard deviation of monthly cell phone bills is ​$4.81. B. The standard deviation of monthly cell phone bills is different from ​$4.81. b) b)H0​:σ=$4.81 H1​:σ≠$4.81 c) B. The sample evidence led the researcher to believe the standard deviation of monthly cell phone bills is different from ​$4.81 ​when, in​ fact, the standard deviation of bills is ​$4.81. d)C. The sample evidence did not lead the researcher to believe the standard deviation of monthly cell phone bills is different from ​$4.81 ​when, in​ fact, the standard deviation of bills is different from ​$4.81.

10.2 In a clinical​ trial, 21 out of 862 patients taking a prescription drug daily complained of flulike symptoms. Suppose that it is known that 2.1​% of patients taking competing drugs complain of flulike symptoms. Is there sufficient evidence to conclude that more than 2.1​% of this​ drug's users experience flulike symptoms as a side effect at the α=0.01 level of​ significance? a) Because np0(1−p0)=___ ___10, the sample size is ____ ​5% of the population​ size, and the sample _______ the requirements for testing the hypothesis ______ satisfied. ​(Round to one decimal place as​ needed.) b) What are the null and alternative​ hypotheses? H0​: ____ ____ ____ versus H1​:___ ___ ____ ​(Type integers or decimals. Do not​ round.) c) Find the test statistic z0. z0=_____ ​(Round to two decimal places as​ needed.) d)Find the​ P-value. ​P-value=____ ​(Round to three decimal places as​ needed.) e)Choose the correct conclusion below. A. Since ​P-value<α​, do not reject the null hypothesis and conclude that there is sufficient evidence that more than 2.1​% of the users experience flulike symptoms. B. Since ​P-value>α​, reject the null hypothesis and conclude that there is not sufficient evidence that more than 2.1​% of the users experience flulike symptoms. C. Since ​P-value>α​, do not reject the null hypothesis and conclude that there is not sufficient evidence that more than 2.1​% of the users experience flulike symptoms. D. Since ​P-value<α​, reject the null hypothesis and conclude that there is sufficient evidence that more than 2.1​% of the users experience flulike symptoms.

a)Because np01−p0= __17.7>__​10,the sample size is _less than__ 5% of the population​ size and the sample __can be reasonably assumed to be random,__ the requirements for testing the hypothesis __are___ satisfied. b) H0​:p=0.021 versus H1​: p>0.021 c) 0.69 d)0.246 e)C. Since ​P-value>α​, do not reject the null hypothesis and conclude that there is not sufficient evidence that more than 2.1​% of the users experience flulike symptoms. Note: Compare the​ P-value with α. If the​ P-value is less than α​, reject the null hypothesis.​ Otherwise, do not reject the null hypothesis

10.3 To test H0: μ=107 versus H1: μ≠107 a simple random sample of size n=35 is obtained. Complete parts a through e below. LOADING... Click here to view the​ t-Distribution Area in Right Tail. ​(a) Does the population have to be normally distributed to test this​ hypothesis? Why? A. ​Yes, because the sample is random. B. ​No, because the test is​ two-tailed. C. ​Yes, because n≥30. D. ​No, because n≥30. (b) If x=104.0 and s=5.7​, compute the test statistic. The test statistic is t0=_____. ​(Round to two decimal places as​ needed.) (c) Draw a​ t-distribution with the area that represents the​ P-value shaded. Choose the correct graph below. A. A symmetric bell-shaped curve is plotted over a horizontal axis. Two vertical lines, equidistant from the curve's peak at the center, extend from the axis to the curve on the left and right sides of the graph. The areas under the curve to the left of the left vertical line and to the right of the right vertical line are shaded. B. A symmetric bell-shaped curve is plotted over a horizontal axis. Two vertical lines, equidistant from the curve's peak at the center, extend from the axis to the curve on the left and right sides of the graph. The area under the curve between the vertical lines is shaded C. A symmetric bell-shaped curve is plotted over a horizontal axis. On the left side of the graph, a vertical line runs from the axis to the curve. The area under the curve to the left of the vertical line is shaded. ​(d)Approximate the​ P-value. Choose the correct answer below. A. 0.01<​P-value<0.02 B. 0.002<​P-value<0.005 C. 0.001<​P-value<0.002 D. 0.005<​P-value<0.01 Interpret the​ P-value. Choose the correct answer below. A. If 100 random samples of size n=35 are​ obtained, about 4 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=107. B. If 1000 random samples of size n=35 are​ obtained, about 4 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=104.0. C. If 1000 random samples of size n=35 are​ obtained, about 10 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=107. D. If 1000 random samples of size n=35 are​ obtained, about 4 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=107. (e) If the researcher decides to test this hypothesis at the α=0.05 level of​ significance, will the researcher reject the null​ hypothesis? Yes No

a)D. ​No, because n≥30. Note: If the sample is obtained using simple random​ sampling, the sample has no​ outliers, and the population from which the sample is drawn is normally distributed or the sample​ size, n, is large ​(n≥​30), use the​ t-distribution to test hypotheses regarding the population mean with unknown σ. b)-3.11 c)A. A symmetric bell-shaped curve is plotted over a horizontal axis. Two vertical lines, equidistant from the curve's peak at the center, extend from the axis to the curve on the left and right sides of the graph. The areas under the curve to the left of the left vertical line and to the right of the right vertical line are shaded. Note: Since the null and alternative hypotheses are H0: μ=107 and H1: μ≠107​, the test is a​ two-tailed test. d)B. 0.002<​P-value<0.005 D. If 1000 random samples of size n=35 are​ obtained, about 4 samples are expected to result in a mean as extreme or more extreme than the one observed if μ=107. Note: A​ P-value is the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption that the null hypothesis is true. Put another​ way, the​ P-value is the likelihood or probability that a sample will result in a sample mean such as the one obtained if the null hypothesis is true. e)Yes

10.3 A college entrance exam company determined that a score of 21 on the mathematics portion of the exam suggests that a student is ready for​ college-level mathematics. To achieve this​ goal, the company recommends that students take a core curriculum of math courses in high school. Suppose a random sample of 200 students who completed this core set of courses results in a mean math score of 21.6 on the college entrance exam with a standard deviation of 3.8. Do these results suggest that students who complete the core curriculum are ready for​ college-level mathematics? That​ is, are they scoring above 21 on the mathematics portion of the​ exam? Complete parts​ a) through​ d) below. ​a) State the appropriate null and alternative hypotheses. Fill in the correct answers below. The appropriate null and alternative hypotheses are H0​:__ ___ __versus H1​: __ ___ __ b) Verify that the requirements to perform the test using the​ t-distribution are satisfied. Check all that apply. A. The sample data come from a population that is approximately normal. B. The sample size is larger than 30. C. The students were randomly sampled. D. A boxplot of the sample data shows no outliers. E. The​ students' test scores were independent of one another F. None of the requirements are satisfied. ​c) Use the​ P-value approach at the α=0.10 level of significance to test the hypotheses in part​ (a). Identify the test statistic. t0=_____ ​(Round to two decimal places as​ needed.) P-value=_____ ​(Round to three decimal places as​ needed.) ​d) Write a conclusion based on the results. Choose the correct answer below. _______ the null hypothesis and claim that there ____ sufficient evidence to conclude that the population mean is ______than 21.

a)H0​:μ=21 versus H1​:μ>21. b)b,c,e B. The sample size is larger than 30. C. The students were randomly sampled. E. The​ students' test scores were independent of one another. c) 2.23 0.013 d)Reject; is; greater Note: Reject the null hypothesis if the test statistic is greater than the critical value—then there is sufficient evidence to conclude that H1 is true.​ Otherwise, do not reject H0 and conclude that there is not sufficient evidence to support the statement in H1.

10.3 In a​ study, researchers wanted to measure the effect of alcohol on the hippocampal​ region, the portion of the brain responsible for​ long-term memory​ storage, in adolescents. The researchers randomly selected 20 adolescents with alcohol use disorders to determine whether the hippocampal volumes in the alcoholic adolescents were less than the normal volume of 9.02 cm3. An analysis of the sample data revealed that the hippocampal volume is approximately normal with no outliers and x=8.19 cm3 and s=0.8 cm3. Conduct the appropriate test at the α=0.01 level of significance. a)State the null and alternative hypotheses. H0​:μ __ ___ H1​: μ ___ ___ ​(Type integers or decimals. Do not​ round.) b)Identify the​ t-statistic. t0=___ ​(Round to two decimal places as​ needed.) c)Identify the​ P-value. ​P-value=___ ​(Round to three decimal places as​ needed.) d)Make a conclusion regarding the hypothesis. _____ the null hypothesis. There _____ sufficient evidence to claim that the mean hippocampal volume is___ ____ cm3.

a)H0​:μ=9.02 H1​:μ<9.02 Note: The null​ hypothesis, denoted H0​, is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. The alternative​ hypothesis, denoted H1​, is a statement such that evidence is gathered in an attempt to support. b)-4.64 c) 0 d) Reject; is;less than 9.02

10.3 Several years​ ago, the mean height of women 20 years of age or older was 63.7 inches. Suppose that a random sample of 45 women who are 20 years of age or older today results in a mean height of 65.1 inches. ​(a) State the appropriate null and alternative hypotheses to assess whether women are taller today. ​(b) Suppose the​ P-value for this test is 0.08. Explain what this value represents. ​(c) Write a conclusion for this hypothesis test assuming an α=0.10 level of significance. ​(a) State the appropriate null and alternative hypotheses to assess whether women are taller today. A. H0​: μ=63.7 in. versus H1​: μ>63.7 in B. H0​: μ=65.1 in. versus H1​: μ<65.1 in. C. H0​: μ=65.1 in. versus H1​: μ>65.1 in. D. H0​: μ=63.7 in. versus H1​: μ≠63.7 in. E. H0​: μ=65.1 in. versus H1​: μ≠65.1 in. F. H0​: μ=63.7 in. versus H1​: μ<63.7 in. ​(b) Suppose the​ P-value for this test is 0.08. Explain what this value represents. A. There is a 0.08 probability of obtaining a sample mean height of 65.1 inches or taller from a population whose mean height is 63.7 inches B. There is a 0.08 probability of obtaining a sample mean height of exactly 65.1 inches from a population whose mean height is 63.7 inches. C. There is a 0.08 probability of obtaining a sample mean height of 63.7 inches or taller from a population whose mean height is 65.1 inches. D. There is a 0.08 probability of obtaining a sample mean height of 65.1 inches or shorter from a population whose mean height is 63.7 inches. ​(c) Write a conclusion for this hypothesis test assuming an α=0.10 level of significance. A. Do not reject the null hypothesis. There is sufficient evidence to conclude that the mean height of women 20 years of age or older is greater today. B. Reject the null hypothesis. There is not sufficient evidence to conclude that the mean height of women 20 years of age or older is greater today. C. Reject the null hypothesis. There is sufficient evidence to conclude that the mean height of women 20 years of age or older is greater today. D. Do not reject the null hypothesis. There is not sufficient evidence to conclude that the mean height of women 20 years of age or older is greater today.

a. A. H0​: μ=63.7 in. versus H1​: μ>63.7 in b. A. There is a 0.08 probability of obtaining a sample mean height of 65.1 inches or taller from a population whose mean height is 63.7 inches c. C. Reject the null hypothesis. There is sufficient evidence to conclude that the mean height of women 20 years of age or older is greater today.

10.1 The _____ _____ is a statement we are trying to find evidence to support.

alternative hypothesis

10.2 Determine if the following statement is true or false. When testing a hypothesis using the​ P-value Approach, if the​ P-value is​ large, reject the null hypothesis. This statement is ____

false Note: The null hypothesis is rejected when the​P-value is small. When the​P-value is​large, the null hypothesis is not rejected.

10.1 A ____ is a statement regarding a characteristic of one or more populations.

hypothesis Note: A hypothesis is a statement regarding a characteristic of one or more populations. The null hypothesis is a statement of no​ change, no​ effect, or no difference. The alternative hypothesis is a statement we are trying to find evidence to support.

10.1 The _____ _______ is a statement of no​ change, no​ effect, or no difference.

null hypothesis

10.1 The​ _______ _______ is a statement of no change, no effect, or no difference.

null hypothesis Note: In hypothesis​ testing, the null hypothesis is a statement of no​ change, no​ effect, or no difference and it is denoted H0.The alternative hypothesis is a statement we are trying to find evidence to support and it is denoted H1.

10.2 Test the hypothesis using the​ P-value approach. Be sure to verify the requirements of the test. H0: p=0.61 versus H1: p<0.61 n=150, x=81, α=0.01 Is np01−p0≥10​? Yes No Use technology to find the​ P-value. ​P-value=_____ ​(Round to three decimal places as​ needed.) _____ because the​ P-value is ____ than α.

yes 0.039 Do not reject the null hypothesis, because the​P-value is greater than α.

10.1 For students who first enrolled in​ two-year public institutions in a recent​ semester, the proportion who earned a​ bachelor's degree within six years was 0.379. The president of a certain junior college believes that the proportion of students who enroll in her institution have a lower completion rate. ​(a) State the null and alternative hypotheses in words. ​(b) State the null and alternative hypotheses symbolically. ​(c) Explain what it would mean to make a Type I error. ​(d) Explain what it would mean to make a Type II error.

​(a) A hypothesis is a statement regarding a characteristic of one or more populations. Hypothesis testing is a​ procedure, based on sample evidence and​ probability, used to test statements regarding a characteristic of one or more populations. The parameter being tested is the proportion of students who earned a​ bachelor's degree within six years at the institution. The null​ hypothesis, denoted H0 ​(read "H-naught"), is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. In other words the null hypothesis is a statement of status quo or no difference and always contains a statement of equality. It is a statement regarding the value of a population parameter. The alternative​ hypothesis, denoted H1 ​(read "H-one"), is a statement that we are trying to find evidence to support. The null and alternative hypotheses in words are as follows. H0​: Among students who enroll at the certain junior​ college, the completion rate is 0.379. H1​: Among students who enroll at the certain junior​ college, the completion rate is less than 0.379. ​(b) The hypotheses in symbols are as follows. H0​:p=0.379 H1​:p<0.379 ​(c) A Type I error is committed when the null hypothesis is rejected​ when, in​ fact, it is true. For​ example, a Type I error would occur if the sample evidence that leads the president to conclude the proportion of students who earn a​ bachelor's degree within six years is less than 0.379 ​when, in​ fact, the proportion is 0.379. ​(d) A Type II error is committed when the null hypothesis is not rejected​ when, in​ fact, the alternative hypothesis is true. For​ example, a Type II error would occur if the sample evidence that leads the president to conclude the proportion of students who earn a​ bachelor's degree within six years is equal to 0.379 ​when, in​ fact, the proportion is less than 0.379.

10.1 The standard deviation in the pressure required to open a certain valve is known to be σ=1.6 psi. Due to changes in the manufacturing​ process, the​ quality-control manager feels that the pressure variability has been reduced. ​(a) State the null and alternative hypotheses in words. ​(b) State the null and alternative hypotheses symbolically. ​(c) Explain what it would mean to make a Type I error. ​(d) Explain what it would mean to make a Type II error.

​(a) A hypothesis is a statement regarding a characteristic of one or more populations. Hypothesis testing is a​ procedure, based on sample evidence and​ probability, used to test statements regarding a characteristic of one or more populations. The parameter being tested is the standard deviation in pressure. The null​ hypothesis, denoted H0 ​(read "H-naught"), is a statement to be tested. The null hypothesis is a statement of no​ change, no​ effect, or no difference and is assumed true until evidence indicates otherwise. In other words the null hypothesis is a statement of status quo or no difference and always contains a statement of equality. It is a statement regarding the value of a population parameter. The alternative​ hypothesis, denoted H1 ​(read "H-one"), is a statement that we are trying to find evidence to support. The null and alternative hypotheses in words are as follows. H0​: The standard deviation in the pressure required to open a certain valve is 1.6 psi. H1​: The standard deviation in the pressure required to open a certain valve is less than 1.6 psi. ​(b) The hypotheses in symbols are as follows. H0​:σ=1.6psi H1​:σ<1.6psi ​(c) A Type I error is committed when the null hypothesis is rejected​ when, in​ fact, it is true. For​ example, a Type I error occurs if the​ quality-control manager rejects the hypothesis that the pressure variability is 1.6​, ​when, in​ fact, it is the true pressure variability. ​(d) A Type II error is committed when the null hypothesis is not rejected when the alternative hypothesis is true. For​ example, a Type II error occurs if the​ quality-control manager fails to reject the hypothesis that the pressure variability is 1.6​, ​when, in​ fact, the true pressure variability is less than 1.6.

10.4 To test H0: σ=25 versus H1: σ<25​, a random sample of size n=31 is obtained from a population that is known to be normally distributed. ​(a) If the sample standard deviation is determined to be s=23.3​, compute the test statistic. ​(b) If the researcher decides to test this hypothesis at the α=0.05 level of​ significance, use technology to determine the​ P-value. ​(c) Will the researcher reject the null​ hypothesis?

​(a) To test a hypothesis about a population standard​ deviation, the test statistic is χ20=(n−1)s^2 /σ^20. Use the given values of​ n, s, and σ to calculate the test statistic. Substitute 31 for​ n, 23.3 for​ s, and 25 for σ​, ​simplify, and round to two decimal places. χ20=(n−1)s^2 /σ^20 =(31−1)(23.3)^2/(25)^2 Substitute. =26.06 Simplify. ​(b) Since this is a​ left-tailed test, the​ P-value is the probability of getting a value of χ^2 less than the test statistic χ ^20. Note that ​P(χ^2<χ^20​), the area to the left of the test​ statistic, is 1−​P(χ^2>χ^20​), the area to the right. Use technology to find the area to the right of χ^20=26.06 under the curve of the χ^2 distribution with 31−1=30 degrees of​ freedom, rounding to three decimal places. ​P(χ^2>26.06​)=0.672 Now find the​ P-value. 1−0.672=0.328 ​(c) If the​ P-value is less than the level of significance α​, reject the null hypothesis.​ Otherwise, do not reject the null hypothesis. Compare the​ P-value, 0.328​, to the level of​ significance, α=0.05. The​ P-value is greater than α. Use this information to determine the correct conclusion