10. Hypothesis Testing for Two Samples

For ${\bar{x}}_1=40,{\sigma }_1=8.3,n_1=40,{\bar{x}}_2=39,{\sigma }_2=5.8$, & $n_2=50$, perform a hypothesis test to test the claim that ${\mu }_1>{\mu }_2$ for $\alpha =0.05$.

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1=μ2; α=0.1

Population statistics:σ1=3.4 and σ2=1.5

Sample Statistics: x̅1=16, n1=29, x̅2=14, n2=28

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1>μ2; α=0.10

Population statistics:σ1=40 and σ2=15

Sample Statistics: x̅1=500, n1=100, x̅2=495, n2=75

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Bed-in-a-Box To compare customer satisfaction with mattresses that are delivered compressed in a box and traditional mattresses, a researcher randomly selects 30 ratings of mattresses in boxes and 30 ratings of traditional mattresses. The mean rating of mattresses in boxes is 68.7 out of 100. Assume the population standard deviation is 6.6. The mean rating of traditional mattresses is 70.9 out of 100. Assume the population standard deviation is 5.6. At α=0.01, can the researcher support the claim that the mean rating of traditional mattresses is greater than the mean rating of mattresses in a box? (Adapted from Consumer Reports)

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Repair Costs: Washing Machines You want to buy a washing machine, and a salesperson tells you that the mean repair costs for Model A and Model B are equal. You research the repair costs. The mean repair cost of 24 Model A washing machines is \$208. Assume the population standard deviation is \$18. The mean repair cost of 26 Model B washing machines is \$221. Assume the population standard deviation is \$22. At α=0.01, can you reject the salesperson’s claim?

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Braking Distances To compare the dry braking distances from 60 to 0 miles per hour for two makes of automobiles, a safety engineer conducts braking tests for 16 compact SUVs and 11 midsize SUVs. The mean braking distance for the compact SUVs is 131.8 feet. Assume the population standard deviation is 5.5 feet. The mean braking distance for the midsize SUVs is 132.8 feet. Assume the population standard deviation is 6.7 feet. At α=0.10 , can the engineer support the claim that the mean braking distances are different for the two categories of SUVs? (Adapted from Consumer Reports)

Testing a Difference Other Than Zero Sometimes a researcher is interested in testing a difference in means other than zero. In Exercises 27 and 28, you will test the difference between two means using a null hypothesis of Ho: μ1-μ2=k, Ho: μ1-μ2>=k or Ho: μ1-μ2<=k . The standardized test statistic is still

Architect Salaries Is the difference between the mean annual salaries of entry level architects in Denver, Colorado, and Lincoln, Nebraska, equal to \$9000? To decide, you select a random sample of entry level architects from each city. The results of each survey are shown in the figure. Assume the population standard deviations are σ1=\$6560 and σ2=\$6100 . At α=0.01 what should you conclude? (Adapted from Salary.com)

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

a. Identify the claim and state Ho and Ha

The mean score on a reading assessment test for 49 randomly selected male high school students was 279. Assume the population standard deviation is 41. The mean score on the same test for 50 randomly selected female high school students was 292. Assume the population standard deviation is 39. (Adapted from National Center for Education Statistics)

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

b. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.

The mean score on a reading assessment test for 49 randomly selected male high school students was 279. Assume the population standard deviation is 41. The mean score on the same test for 50 randomly selected female high school students was 292. Assume the population standard deviation is 39. At α=0.05, can you support the claim that the mean score on the reading assessment test for male high school students is less than the mean score for female high school students? (Adapted from National Center for Education Statistics)

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

c. Find the critical value(s) and identify the rejection region(s).

The mean score on a reading assessment test for 49 randomly selected male high school students was 279. Assume the population standard deviation is 41. The mean score on the same test for 50 randomly selected female high school students was 292. Assume the population standard deviation is 39. At α=0.05, can you support the claim that the mean score on the reading assessment test for male high school students is less than the mean score for female high school students? (Adapted from National Center for Education Statistics)

Take this quiz as you would take a quiz in class. After you are done, check your work against the answers given in the back of the book.For each exercise, perform the steps below.

f. Interpret the decision in the context of the original claim.

The mean score on a reading assessment test for 49 randomly selected male high school students was 279. Assume the population standard deviation is 41. The mean score on the same test for 50 randomly selected female high school students was 292. Assume the population standard deviation is 39. At α=0.05, can you support the claim that the mean score on the reading assessment test for male high school students is less than the mean score for female high school students? (Adapted from National Center for Education Statistics)

In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1=μ2; α=0.01

Population statistics: σ1= 52 and σ2= 68

Sample statistics: x̅1 = 5595, n1 = 156, and x̅2= 5575, n2= 216

In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2; α=0.10

Population statistics: σ1= 0.11 and σ2= 0.10

Sample statistics: x̅1 = 0.28, n1 = 41, and x̅2= 0.33, n2= 34

Designing a Study Stock fund managers are investment professionals who decide which stocks should be part of a portfolio. In an article in the Wall Street Journal (“Not a Stock-Picker’s Market,” WSJ, January 25, 2014), the performance of stock fund managers was considered based on dispersion in the market. In the stock market, risk is measured by the standard deviation rate of return of stock (dispersion). When dispersion is low, then the rate of return of the stocks that make up the market are not as spread out. That is, the return on Company X is close to that of Y is close to that of Z, and so on. When dispersion is high, then the rate of return of stocks is more spread out; meaning some stocks outperform others by a substantial amount. Since 1991, the dispersion of stocks has been about 7.1%. In some years, the dispersion is higher (such as 2001 when dispersion was 10%), and in some years it is lower (such as 2013 when dispersion was 5%). So, in 2001, stock fund managers would argue, one needed to have more investment advice in order to identify the stock market winners, whereas in 2013, since dispersion was low, virtually all stocks ended up with returns near the mean, so investment advice was not as valuable.

e. Suppose this study was conducted and the data yielded a P-value of 0.083. Explain what this result suggests.

In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1≠μ2; α=0.05

Population statistics: σ1= 14 and σ2= 15

Sample statistics: x̅1 = 87, n1 = 410, and x̅2= 85, n2= 340