91Ó°ÊÓ

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,

Step by step solution

01

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).

02

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.

03

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Means

Population means are central to the study of statistics and hypothesis testing. They represent the average value of a particular characteristic within an entire population. For example, when examining the IQ scores of soccer players, the population mean would be the average IQ score of all soccer players meeting certain criteria, such as age and level of play. Understanding population means helps researchers make inferences about a group based on sample data.

When conducting hypothesis tests about population means, one aims to decide whether the data supports a given hypothesis or claim. There can be different scenarios in testing for population means, like comparing the means of two different independent samples or comparing a sample mean to a known population mean.

• Population means provide a baseline for what is considered 'normal' for a given group.
• Tests about population means involve calculations that check whether observed data could have occurred by random chance.

One-Sample t-Test

The one-sample t-test is a statistical method used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean. This test is applicable when the population standard deviation is unknown, which is often the case in real-world scenarios.

Let's look into how a one-sample t-test works using Scenario 2 from the exercise. Here, a known mean number of credit cards, 4.09, is compared to the sample mean of 2.6 from 132 undergraduates. The question is whether this sample provides evidence that students, in general, carry fewer credit cards than the mean reported by the credit bureau.

• The null hypothesis ( H_0 ) would be that the mean number of credit cards students have is equal or greater than 4.09.
• The alternative hypothesis ( H_a ) is that the mean number of credit cards is less than 4.09.
• The step involves calculating the t-statistic and comparing it to a critical value from the t-distribution with n-1 degrees of freedom.

The outcome helps to determine if the difference is statistically significant. If the calculated t-value is larger in absolute terms than the critical value, then the null hypothesis is rejected.

Two-Independent Samples

In statistical analysis, when comparing two groups, we use a hypothesis test for two independent samples. This test determines whether there is a significant difference between the means of two independent groups.

Scenario 1 from the exercise provides an example where you might use such a test. Here, two groups of soccer players are divided based on how often they head a soccer ball. This division results in two independent samples, each with its own mean IQ score. The analysis seeks to determine if the mean IQ score is different for those who head the ball frequently versus those who do not.

• A null hypothesis ( H_0 ) typically states there is no difference between the means of the two groups.
• The alternative hypothesis ( H_a ) would suggest a difference exists.
• Calculating the t-statistic involves considering the difference between the sample means, the standard deviations, and the sample sizes.

Understanding the difference between these groups can lead to insights about the impact of frequent heading on cognitive function, which was the focus of the research.