Stat - Chapter 10

In a clinical​ trial, 31 out of 900 patients taking a prescription drug complained of flulike symptoms. Suppose that it is known that 2.3​% of patients taking competing drugs complain of flulike symptoms. Is there sufficient evidence to conclude that more than 2.3​% of this​ drug's users experience flulike symptoms as a side effect at the α=0.05 level of​ significance? (a) What are the null and alternative​ hypotheses? (b) What is the P-value?

(a) H0: p=0.023 versus H1: p>0.023 (b) 0.014; Since ​P-value<α​, reject the null hypothesis and conclude that there is sufficient evidence that more than 2.3​% of the users experience flulike symptoms.

Five years​ ago, 10.5​% of high school students had tried marijuana for the first time before the age of 13. A school resource officer​ (SRO) thinks that the proportion of high school students who have tried marijuana for the first time before the age of 13 has increased since then. ​(a) Determine the null and alternative hypotheses. Which of the following is​ correct? (b) Suppose sample data indicate that the null hypothesis should be rejected. State the conclusion of the researcher. (c) Suppose, in​ fact, that the proportion of high school students who have tried marijuana before the age of 13 was 10.5​%. Was a type I or type II error​ committed?

(a) H0: p=0.105; H1:p>0.105 (b) There is sufficient evidence to conclude that the proportion of high school students has increased. (c) The SRO committed a type II error because he rejected the null hypothesis when, in fact, it is true.

In a previous​ poll, 49​% of adults with children under the age of 18 reported that their family ate dinner together 7 nights a week. Suppose​ that, in a more recent​ poll, 554 of 1150 adults with children under the age of 18 reported that their family ate dinner together 7 nights a week. Is there sufficient evidence that the proportion of families with children under the age of 18 who eat dinner together 7 nights a week has decreased at the α=0.05 significance​ level? (a) What are the null and alternative​ hypotheses? (b) What is the P-value? (c)

(a) H0: p=0.36 versus H1: p<0.36 (b) P-value=0.064; ​Yes, there is sufficient evidence because the​ P-value is less than the level of significance.​Therefore, reject the null hypothesis.

In a recent​ survey, 37​% of employed U.S. adults reported that basic mathematical skills were critical or very important to their job. The supervisor of the job placement office at a​ 4-year college thinks this percentage has increased due to increased use of technology in the workplace. She takes a random sample of 800 employed adults and finds that 321 of them feel that basic mathematical skills are critical or very important to their job. Is there sufficient evidence to conclude that the percentage of employed adults who feel basic mathematical skills are critical or very important to their job has increased at the α=0.01 level of​ significance? (a) What are the null and alternative hypotheses? (b) Determine the test statistic? (c)

(a) H0: p=0.37 versus H1: p>0.37 (b)

​Previously, 35​% of parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 120 of 300 parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. Do parents feel differently​ today? Use the α=0.05 level of significance. (a) What are the null and alternative​ hypotheses? (b) Determine the test statistic? (c) What is the P-value? (d) Interpret the P-value. (e) Determine the conclusion for this hypothesis test.

(a) H0: p=0.48 versus H1: p≠0.48 (b) Z0=1.27 (c) P-value=0.204 (d) The​ P-value is the probability of observing a sample statistic as extreme or more extreme than the one obtained if the population proportion equals 0.48. (e) Since P-value>α​, do not reject the null hypothesis and conclude that there is not sufficient evidence that parents feel differently today.

In a previous​ year, 65​% of females aged 15 years of age and older lived alone. A sociologist tests whether this percentage is different today by conducting a random sample of 700 females aged 15 years of age and older and finds that 462 are living alone. Is there sufficient evidence at the α=0.05 level of significance to conclude the proportion has​ changed? (a) Identify the null and alternative hypotheses for this test. (b) Find the test statistic for this hypothesis test. (c) Determine the P-value for this hypothesis test. (d) State the conclusion for this hypothesis test.

(a) H0: p=0.63; H1: p≠0.63 (b) 1.52 (c) 0.129 (d) 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.

(a) Determine the null and alternative​ hypotheses, (b) explain what it would mean to make a type I​ error, and​ (c) explain what it would mean to make a type II error. Three years​ ago, the mean price of a​ single-family home was ​$243,782. A real estate broker believes that the mean price has increased since then. (a) Which of the following is the hypothesis test to be conducted? (b) Which of the following is a type I error? (c) Which of the following is a type II error?

(a) H0: μ=$243,782; H1: μ>$243,782. (b) The broker rejects the hypothesis that the mean price is $243,782, when it is the true mean cost. (c) The broker fails to reject the hypothesis that the mean price is $243,782, when the true mean price is greater than $243,782.

Calcium is essential to tree growth because it promotes the formation of wood and maintains cell walls. In​ 1990, the concentration of calcium in precipitation in a certain area was 0.12 milligrams per liter​ (mg/L). A random sample of 10 precipitation dates in 2007 results in the following data table. Complete parts​ (a) through​ (c) below. (a) State the hypotheses for determining if the mean concentration of calcium precipitation has changed since 1990. (b) Construct a 95​% confidence interval about the sample mean concentration of calcium precipitation. (c) Does the sample evidence suggest that calcium concentrations have changed since​ 1990?

(a) H0: μ=0.12 mg/L H1: μ≠0.12 mg/L (b) The lower bound is 0.0931. The upper bound is 0.2183. (c) No, because the confidence interval contains 0.12 mg/L.

The average daily volume of a computer stock in 2011 was μ=35.1 million​ shares, according to a reliable source. A stock analyst believes that the stock volume in 2014 is different from the 2011 level. Based on a random sample of 30 trading days in​ 2014, he finds the sample mean to be 29.8 million​ shares, with a standard deviation of s=14.4 million shares. Test the hypotheses by constructing a 95​% confidence interval. Complete parts​ (a) through​ (c) below. (a) State the hypotheses for the test. (b) Construct a 95​% confidence interval about the sample mean of stocks traded in 2014. (c) Will the researcher reject the null​ hypothesis?

(a) H0: μ=35.1 million shares. H1: μ≠35.1 million shares (b) The lower bound is 24.423 million shares. The upper bound is 35.177 million shares. (c) Do not reject the null hypothesis because μ=35.1 million shares falls in the confidence interval.

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 64.9 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 0.16. Explain what this value represents. ​(c) Write a conclusion for this hypothesis test assuming an α=0.10 level of significance.

(a) H0: μ=63.7 in. versus H1: μ>63.7 in. (b) There is a 0.16 probability of obtaining a sample mean height of 64.9 inches or taller from a population whose mean height is 63.7 inches. (c) 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.

It has long been stated that the mean temperature of humans is 98.6°F. However, two researchers currently involved in the subject thought that the mean temperature of humans is less than 98.6°F. They measured the temperatures of 56 healthy adults 1 to 4 times daily for 3​ days, obtaining 250 measurements. The sample data resulted in a sample mean of 98.2°F and a sample standard deviation of 11°F. Use the​ P-value approach to conduct a hypothesis test to judge whether the mean temperature of humans is less than 98.6°F at the α=0.01 level of significance. (a) State the hypotheses. (b) Find the test statistic. (c) Find the P-value. (d) What can be concluded?

(a) H0: μ=98.6°F H1: μ<98.6°F (b) t=-6.32 (c) P-value=0.000 (d) Reject H0 since the P-value is less than the significance level.

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=110 H1​: σ<110

(a) Left-tailed test (b) Population mean

Real estate taxes are used to fund local education​ institutions, police​ protection, libraries, and other local public services. A​ household's real estate tax bill is based on the value of the house. In a recent​ year, the mean real estate tax bill per​ $1,000 assessed value for all households in a certain country was $8.43. A random sample of 50 rural households results in a sample mean real estate bill per​ $1,000 assessed value of $6.57 with a standard deviation of $5.98. Use the level of significance α=0.05. Complete parts​ (a) and​ (b). ​(a) Based on the histogram and boxplot​ shown, why is a large sample necessary to conduct a hypothesis test about the​ mean? (b) Conduct a hypothesis test using the​ P-value approach and a level of significance of α=0.05 to determine if the evidence suggests the mean real estate bill for rural households per​ $1,000 assessed value differs from that of all households. First determine the appropriate hypothesis. Find the test statistic. Find the P-value. Use the α=0.05 level of significance. What can be concluded from the hypothesis test?

(a) The real estate tax bill distribution is skewed right with outliers. (b) H0: μ=8.43 H1: μ≠8.43 t=-2.20 P-value=0.033 The evidence suggests that rural households have a different mean real estate tax bill.

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 data table. Complete parts​ (a) and​ (b) below. (a) Because the sample size is​ small, the engineer must verify that the diameter is normally distributed and the sample does not contain any outliers. The normal probability plot is shown below and the sample correlation coefficient is known to be r=0.952. Are the conditions for testing the hypothesis​ satisfied? (b) Do the golf balls conform to the​ standards? Conduct a hypothesis test using the​ P-value approach and a level of significance of α=0.05. First determine the appropriate hypothesis. Find the test statistic. Find the P-value. Use the α=0.05 level of significance. What can be concluded from the hypothesis test?

(a) Yes, the conditions are satisfied. The normal probability plot is linear enough, since the correlation coefficient is greater than the critical value. (b) H0: μ=1.68 H1: μ≠1.68 t=0.75 P-value=0.468 There is not sufficient evidence to conclude that the golf balls do not conform to the association's standards.

The mean waiting time at the​ drive-through of a​ fast-food restaurant from the time an order is placed to the time the order is received is 86.1 seconds. A manager devises a new​ drive-through system that she believes will decrease wait time. As a​ test, she initiates the new system at her restaurant and measures the wait time for 10 randomly selected orders. The wait times are provided in the table to the right. Complete parts​ (a) and​ (b) below. (a) Because the sample size is​ small, the manager must verify that the wait time is normally distributed and the sample does not contain any outliers. The normal probability plot is shown below and the sample correlation coefficient is known to be r=0.987. Are the conditions for testing the hypothesis​ satisfied? (b) Is the new system​ effective? Conduct a hypothesis test using the​ P-value approach and a level of significance of α=0.05. First determine the appropriate hypothesis. Find the test statistic. Find the P-value. Use the α=0.05 level of significance. What can be concluded from the hypothesis test?

(a) Yes, the conditions are satisfied. The normal probability plot is linear enough, since the correlation coefficient is greater than the critical value. In addition, a box plot does not show any outliers. (b) H0: μ=86.1 H1: μ<86.1 t=-1.58 P-value=0.074 The P-value is greater than the level of significance so there is not sufficient evidence to conclude the new system is effective.

Test the hypothesis using the​ P-value approach. Be sure to verify the requirements of the test. H0​: p=0.8 versus H1​: p>0.8 n=125; x=115; α=0.05 (a) Is np0 (1-p0)≥​10? (b) What is the P-value?

(a) Yes; 60≥10 (b) P-value=0.000 Reject the null hypothesis, because the P-value is less than α.

To test H0: μ=20 versus H1: μ<20, a simple random sample of size n = 16 is obtained from a population that is known to be normally distributed. Answers parts (a)-(d) (a) If x overbar = 18.3 and s = 4.4, compute the test statistic. (b) Draw a​ t-distribution with the area that represents the​ P-value shaded. Which of the following graphs shows the correct shaded​ region? (c) Approximate the​ P-value. Choose the correct range for the​ P-value below. (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) t=-1.55 (b) Graph shown. (c) 0.05 < P-value < 0.10 (d) The researcher will not reject the null hypothesis since the P-value is not less than α.

To test H0: μ=80 versus H1: μ≠80, a simple random sample of size n = 26 is obtained from a population that is known to be normally distributed. Complete parts (a) through (c) below. (a) If x overbar = 83.1 and s = 14.3, compute the test statistic. (b) Suppose a researcher wants to test this hypothesis at a level of significance of α=0.02. Approximate the​ P-value using the​ t-distribution table.

(a) t=1.105 (b) 0.20 ≤ P-value ≤ 0.30 (c) No​, the researcher will not reject the null hypothesis because the​ p-value is greatergreater than the level of significance.

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

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

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.

False