Question

For each of the following hypothesis testing scenarios, indicate whether or not the appropriate hypothesis test would be for a difference in population means. If not, explain why not. Scenario 1: A researcher at the Medical College of Virginia conducted a study of 60 randomly selected male soccer players and concluded that players who frequently "head" the ball in soccer have a lower mean IQ (USA Today, August 14,1995 ). The soccer players were divided into two samples, based on whether they averaged 10 or more headers per game, and IQ was measured for each player. You would like to determine if the data support the researcher's conclusion. Scenario 2: A credit bureau analysis of undergraduate students" credit records found that the mean number of credit cards in an undergraduate's wallet was 4.09 ("Undergraduate Students and Credit Cards in \(2004,{ }^{n}\) Nellie Mae, May 2005 ). It was also reported that in a random sample of 132 undergraduates, the mean number of credit cards that the students said they carried was 2.6. You would like to determine if there is convincing evidence that the mean number of credit cards that undergraduates report carrying is less than the credit bureau's figure of \(4.09 .\) Scenario 3: Some commercial airplanes recirculate approximately \(50 \%\) of the cabin air in order to increase fuel efficiency. The authors of the paper "Aircraft Cabin Air Recirculation and Symptoms of the Common Cold" (Journal of the American Medical Association \([2002]: 483-486)\) studied 1,100 airline passengers who flew from San Francisco to Denver. Some passengers traveled on airplanes that recirculated air, and others traveled on planes that did not. Of the 517 passengers who flew on planes that did not recirculate air,

Short answer

Scenario 1 requires a test for a difference in population means. Scenario 2 does not require a test for a difference in population means because it's a test comparing a sample mean to a known population mean. Scenario 3 also does not require a test for a difference in population means because it's about proportions rather than means.

Step-by-step solution

Analyzing Scenario 1

In Scenario 1, a researcher conducted a study on 60 randomly selected male soccer players divided into two groups based on whether they averaged 10 or more headers per game. IQ was measured for each player. It's clear that the hypothesis test will be for the difference in population means because there are two independent samples (players who 'head' the ball frequently vs players who don't) and the measurement is a continuous variable (IQ).

Analyzing Scenario 2

In Scenario 2, a credit bureau analysis of undergraduate students' credit records found that the mean number of credit cards in an undergraduate's wallet was 4.09. In a random sample of 132 undergraduates, it was also found that the mean number of credit cards that the students reported was 2.6. This is still a question about a difference in means. The question is whether the true mean number of credit cards carried by students is less than 4.09. Since it's a test comparing a sample mean to a known population mean, it's a one-sample t-test, not a test for a difference in population means.

Analyzing Scenario 3

In Scenario 3, the authors of a research paper studied 1,100 airline passengers who flew from San Francisco to Denver. Some passengers traveled on airplanes that recirculated air, and others traveled on planes that did not. The study is supposed to show whether passengers in one group are more likely to suffer from common cold symptoms compared to the other group. The critical information is not about the mean, but rather the proportions or percentages of passengers with cold symptoms in each group. Therefore, this scenario is likely best analyzed with a chi-squared test or a two-proportion z-test, not a test for difference in means.