10. Hypothesis Testing for Two Samples

What is the requirement for the sample size of each sample when using the Wilcoxon rank sum test?

In a recent year, according to the Bureau of Labor Statistics, the median number of years that wage and salary employees had been with their current employer (called employee tenure) was 4.1 years. Information on employee tenure has been gathered since 1996 using the Current Population Survey (CPS), a monthly survey of about 60,000 households that provides information on employment, unemployment, earnings, demographics, and other characteristics of the U.S. population ages 16 and over. With respect to employee tenure, the questions measure how long employees have been with their current employers, not how long they plan to stay with their employers.

A congressional representative claims that the median tenure for employees from the representative’s district is less than the national median tenure of 4.1 years. The claim is based on the representative’s data, which is shown in the table at the right above. (Assume that the employees were randomly selected.)

a. Is it possible that the claim is true? What questions should you ask about how the data were collected?

In a recent year, according to the Bureau of Labor Statistics, the median number of years that wage and salary employees had been with their current employer (called employee tenure) was 4.1 years. Information on employee tenure has been gathered since 1996 using the Current Population Survey (CPS), a monthly survey of about 60,000 households that provides information on employment, unemployment, earnings, demographics, and other characteristics of the U.S. population ages 16 and over. With respect to employee tenure, the questions measure how long employees have been with their current employers, not how long they plan to stay with their employers.

A congressional representative claims that the median tenure for employees from the representative’s district is less than the national median tenure of 4.1 years. The claim is based on the representative’s data, which is shown in the table at the right above. (Assume that the employees were randomly selected.)

b. How would you test the representative’s claim? Can you use a parametric test, or do you need to use a nonparametric test?  

In a recent year, according to the Bureau of Labor Statistics, the median number of years that wage and salary employees had been with their current employer (called employee tenure) was 4.1 years. Information on employee tenure has been gathered since 1996 using the Current Population Survey (CPS), a monthly survey of about 60,000 households that provides information on employment, unemployment, earnings, demographics, and other characteristics of the U.S. population ages 16 and over. With respect to employee tenure, the questions measure how long employees have been with their current employers, not how long they plan to stay with their employers.

A congressional representative claims that the median tenure for employees from the representative’s district is less than the national median tenure of 4.1 years. The claim is based on the representative’s data, which is shown in the table at the right above. (Assume that the employees were randomly selected.)

c. State the null hypothesis and the alternative hypothesis.

In Exercises 11–16, 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. Assume (σ1)^2 ≠ (σ2)^2

Sample statistics: x̅1= 520, s1= 25, n1= 7 and x̅2= 500, s2= 55, n2= 6

In Exercises 1–5, (a) identify the claim and state H0 and Ha, (b) decide which nonparametric test to use, (c) find the critical value(s), (d) find the test statistic, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim.

An employment agency claims that there is a difference in the weekly earnings of employees who are union members and employees who are not union members. The table shows the weekly earnings (in dollars) for a random sample of nine union members and eight nonunion members. At , can you support the agency’s claim? (Adapted from U.S. Bureau of Labor Statistics)

In Exercises 11–16, 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. Assume (σ1)^2 ≠ (σ2)^2

Sample statistics: x̅1=0.015, s1=0.011, n1= 8 and x̅2=0.019, s2=0.004, n2= 6

A pediatrician claims that the mean birth weight of a single-birth baby is greater than the mean birth weight of a baby that has a twin. The mean birth weight of a random sample of 85 single-birth babies is 3086 grams. Assume the population standard deviation is 563 grams. The mean birth weight of a random sample of 68 babies that have a twin is 2263 grams. Assume the population standard deviation is 624 grams. At α=0.10, can you support the pediatrician’s claim? Interpret the decision in the context of the original claim.

Blue Crabs A marine researcher claims that the stomachs of blue crabs from one location contain more fish than the stomachs of blue crabs from another location. The stomach contents of a sample of 25 blue crabs from Location A contain a mean of 320 milligrams of fish and a standard deviation of 60 milligrams. The stomach contents of a sample of 15 blue crabs from Location B contain a mean of 280 milligrams of fish and a standard deviation of 80 milligrams. At , α= 0.01can you support the marine researcher’s claim? Assume the population variances are equal.

Annual Income

A politician claims that the mean household income in a recent year is greater in York County, South Carolina, than it is in Elmore County, Alabama. In York County, a sample of 23 residents has a mean household income of \$64,900 and a standard deviation of \$16,000. In Elmore County, a sample of 19 residents has a mean household income of \$59,500 and a standard deviation of \$23,600. At , α= 0.05can you support the politician’s claim? Assume the population variances are not equal. (Adapted from U.S. Census Bureau)

Independent and Dependent Samples In Exercises 5–8, classify the two samples as independent or dependent and justify your answer.

Sample 1: The IQ scores of 60 females

Sample 2: The IQ scores of 60 males

In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.

Sample 1: The fuel efficiencies of 12 cars

Sample 2: The fuel efficiencies of the same 12 cars using an alternative fuel

In Exercises 1–4, classify the two samples as independent or dependent and justify your answer.

Sample 1: The weights of 45 oranges

Sample 2: The weights of 40 grapefruits

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

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.

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

d. Find the appropriate standardized test statistic.

e. Decide whether to reject or fail to reject the null hypothesis.

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

A music teacher claims that the mean scores on a music assessment test for eighth grade students in public and private schools are equal. The mean score for 13 randomly selected public school students is 146 with a standard deviation of 49, and the mean score for 15 randomly selected private school students is 160 with a standard deviation of 42. At α=0.1, can you reject the teacher’s claim? Assume the populations are normally distributed and the population variances are equal. (Adapted from National Center for Education Statistics)

Take this test as you would take a test in class.For each exercise, perform the steps below.

a. Identify the claim and state and

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.

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

d. Find the appropriate standardized test statistic.

e. Decide whether to reject or fail to reject the null hypothesis.

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

A demographics researcher claims that the mean household income in a recent year is different for native-born households and foreign-born households. A sample of 18 native-born households has a mean household income of \$69,474 and a standard deviation of \$21,249. A sample of 21 foreign-born households has a mean household income of \$64,900 and a standard deviation of \$17,896. At α=0.01, can you support the demographics researcher’s claim? Assume the populations are normally distributed and the population variances are not equal. (Adapted from U.S. Census Bureau)