Test 5: Chapter 9&10

The trade magazine QSR routinely checks the​ drive-through service times of​ fast-food restaurants. A 95​% confidence interval that results from examining 519 customers in Taco​ Bell's drive-through has a lower bound of 171.5 seconds and an upper bound of 177.9 seconds. Complete parts​ (a) through​ (c). ​(a) What is the mean service time from the 519 ​customers? The mean service time from the 519 customers is ____ seconds ​(Type an integer or a decimal. Do not​ round.) ​(b) What is the margin of error for the confidence​ interval? The margin of error is ___ seconds. ​(Type an integer or a decimal. Do not​ round.) ​(c) Interpret the confidence interval. Select the correct choice below and fill in the answer boxes to complete your choice. ​(Type integers or decimals. Do not​ round.) A. One can be nothing​% confident that the mean​ drive-through service time of Taco Bell is nothing seconds. B. The mean​ drive-through service time of Taco Bell is nothing seconds nothing​% of the time. C. There is a nothing​% probability that the mean​ drive-through service time of Taco Bell is between nothing seconds and nothing seconds. D. One can be 9595​% confident that the mean​ drive-through service time of Taco Bell is between 171.5171.5 seconds and 177.9177.9 seconds.

(a)174.7 ​(b)3.2 ​(c)D. One can be 9595​% confident that the mean​ drive-through service time of Taco Bell is between 171.5171.5 seconds and 177.9177.9 seconds.

A simple random sample of size n is drawn from a population that is known to be normally distributed. The sample​ variance, s2​, is determined to be 13.2. Complete parts​ (a) through​ (c). ​(a) Construct a​ 90% confidence interval for σ2 if the sample​ size, n, is 20. The lower bound is _____. ​(Round to two decimal places as​ needed.) The upper bound is ____. ​(Round to two decimal places as​ needed.) ​(b) Construct a​ 90% confidence interval for σ2 if the sample​ size, n, is 30. The lower bound is ____. ​(Round to two decimal places as​ needed.) The upper bound is ____. ​(Round to two decimal places as​ needed.) How does increasing the sample size affect the width of the​ interval? The width decreases The width increases The width does not change ​(c) Construct a​ 98% confidence interval for σ2 if the sample​ size, n, is 20. The lower bound is ____ ​(Round to two decimal places as​ needed.) The upper bound is ____. ​(Round to two decimal places as​ needed.) Compare the results with those obtained in part​ (a). How does increasing the level of confidence affect the confidence​ interval? The width decreases The width does not change The width increases

(a)The lower bound is 8.32 The upper bound is 24.79 ​(b) The lower bound is 9. The upper bound is 21.62 The width decreases ​(c) The lower bound is 6.93 The upper bound is 32.86​ The width increases

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 alternative is true. C. Fail to reject the null hypothesis and the null is true. D. Reject the null hypothesis and the null is true.

A. Fail to reject the null hypothesis and the alternative is true.

Test the hypothesis using the​ P-value approach. Be sure to verify the requirements of the test. H0: p=0.84 versus H1: p≠0.84 n=500, x=410, 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. ​Yes, because np01−p0=___ B. ​No, because np01−p0=___ Now find p. p=____ ​(Type an integer or a decimal. Do not​ round.) Find the test statistic z0. z0=−1.22 ​(Round to two decimal places as​ needed.) Find the​ P-value. ​P-value=0.2230.223 ​(Round to three decimal places as​ needed.) State the conclusion of the hypothesis test. ________, because the​ P-value is ____ than α.

A. ​Yes, because np01−p0=67.267.2. p=0.82 z0=−1.22 P-value=0.223 Do not reject the null hypothesis; greater

Explain what a​ P-value is. What is the criterion for rejecting the null hypothesis using the​ P-value approach? Explain what a​ P-value is. Choose the correct answer below. A. A​ P-value is the value used to designate the area α in either the​ left- or​ right-tail of the normal curve. 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 number of standard deviations that the observed proportion is from the proportion stated in the null hypothesis. 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<α​, reject the null hypothesis. C. 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.

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. If ​P-value<α​, reject the null hypothesis.

True or​ false: The​ chi-square distribution is symmetric. Choose the best answer below. skewed neither left nor right A. False. The​ chi-square distribution is skewed to the left. B. False. The​ chi-square distribution is skewed to the right C. True. Since the​ chi-square distribution is skewed to the left and​ right, it is symmetric. D. True. Since the​ chi-square distribution is skewed neither left nor​ right, it is symmetric.

B. False. The​ chi-square distribution is skewed to the right

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 ​$243,798. A real estate broker believes that the mean price has increased since then. Which of the following is the correct​ conclusion? A. 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. C. There is not sufficient evidence to conclude that the mean price of a​ single-family home has not changed. D. 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.

Explain why the​ t-distribution has less spread as the number of degrees of freedom increases. Choose the correct answer below. A. The​ t-distribution has less spread as the degrees of freedom increase​ because, for large values of​ n, n≥​30, the​ t-distribution and the normal distribution are the same. B. The​ t-distribution has less spread as the degrees of freedom increase​ because, as n​ increases, s becomes closer to σ by the law of large numbers. C. The​ t-distribution has less spread as the degrees of freedom increase because the variability introduced into the​ t-statistic becomes greater as n increases. D. The​ t-distribution has less spread as the degrees of freedom increase​ because, as n​ increases, less information is known about σ by the law of large numbers.

B. The​ t-distribution has less spread as the degrees of freedom increase​ because, as n​ increases, s becomes closer to σ by the law of large numbers.

In 1945​, an organization asked 1539 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 785 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. What are the null and alternative​ hypotheses? H0​:___ ___ ____ versus H1​: ___ ___ ___ ​(Type integers or​ decimals.) Determine the test​ statistic, z0. z0=___ ​(Round to two decimal places as​ needed.) Use technology to determine the​ P-value for the test statistic. The​ P-value is ____ ​(Round to three decimal places as​ needed.) 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 ___ sufficient evidence to conclude that the majority of the citizens feel the country could develop a way to protect itself from atomic bombs.

H0​: p=0.5 versus H1​: p>0.5 0.79 0.215 greater;do not reject; is not

A credit score is used by credit agencies​ (such as mortgage companies and​ banks) to assess the creditworthiness of individuals. Values range from 300 to​ 850, with a credit score over 700 considered to be a quality credit risk. According to a​ survey, the mean credit score is 703.6. A credit analyst wondered whether​ high-income individuals​ (incomes in excess of​ $100,000 per​ year) had higher credit scores. He obtained a random sample of 41 ​high-income individuals and found the sample mean credit score to be 721.7 with a standard deviation of 84.1. Conduct the appropriate test to determine if​ high-income individuals have higher credit scores at the α=0.05 level of significance. State the null and alternative hypotheses. H0​:μ___ ___ H1​:μ__ ___ ​(Type integers or decimals. Do not​ round.) Identify the​ t-statistic. t0=____ ​(Round to two decimal places as​ needed.) Identify the​ P-value. ​P-value=____ ​(Round to three decimal places as​ needed.) Make a conclusion regarding the hypothesis. ____ the null hypothesis. There ____ sufficient evidence to claim that the mean credit score of​ high-income individuals is_____ _____.

H0​:μ=703.6 H1​:μ>703.6 1.38 0.088 Fail to reject; is not; greater than; 703.6

Construct a 95​% confidence interval of the population proportion using the given information. x=125, n=250 LOADING... Click here to view the table of critical values. The lower bound is ____ The upper bound is ____ ​(Round to three decimal places as​ needed.)

The lower bound is 0.438 The upper bound is 0.562

Fill in the blanks to complete the statement. The​ _______ _______ is a statement of no change, no effect, or no difference.

null hypothesis

To test H0​: p=0.55 versus H1​: p<0.55​, a simple random sample of n=450 individuals is obtained and x=234 successes are observed. If the true population proportion is 0.52 and the level of significance of α=0.05 is​ used, β=0.6422 and the power of the test is 0.3578. What is the β and power of the test if redone at α=0.01 level of​ significance? β=____ Power=_____ ​(Round to four decimal places as​ needed.) 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 is 0. C. When the level of significance is​ lowered, the power of the test decreases. D. When the level of significance is​ lowered, the power of the test increases.

β=0.8515 Power=0.1485 C. When the level of significance is​ lowered, the power of the test decreases.

To test H0: σ=4.4 versus H1: σ≠4.4​, a random sample of size n=19 is obtained from a population that is known to be normally distributed. ​(a) If the sample standard deviation is determined to be s=6.2​, 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 ___ than the level of​ significance, the researcher ___ reject the null hypothesis H0: σ=4.4.

​(a)35.74 ​(b)0.015 ​(c)less; will

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=28.4 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​:___ ___ 26 H1​:___ ___ 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 less than the α=0.01 level of significance. B. Do not reject H0 because the​ P-value is greater 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. Reject H0 because the​ P-value is greater than the α=0.01 level of significance. ​(d)There ____ sufficient evidence at the α=0.01 level of significance to conclude that the population mean is different from 26.

​(a)H0​:μ=26 H1​:μ≠26 ​(b)0.162 ​(c)B. Do not reject H0 because the​ P-value is greater than the α=0.01 level of significance. ​(d) is not

To test H0​: μ=20 versus H1​: μ<20​, a simple random sample of size n=20 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=3.9​, 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 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 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 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. ​(c) Approximate the​ P-value. Choose the correct range for the​ P-value below. A. 0.10<P-value<0.15 B. 0.20<P-value<0.25 C. 0.05<P-value<0.10 D. 0.15<P-value<0.20 ​(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 not reject the null hypothesis since the P-value is not less than α. C. The researcher will 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)t=−1.72 ​(b)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. ​(c)C. 0.05<P-value<0.10 ​(d)B. The researcher will not reject the null hypothesis since the P-value is not less than α.