Q.52 - The Practice of Statistics for AP

Question

What is the most important reason that students buy from catalogs? The answer may differ for different groups of students. Here are results for separate random samples of American and Asian students at a large midwestern university 

(a) Should we use a chi-square test for homogeneity or a chi-square test of association/independence in this setting? Justify your answer. (b) State appropriate hypotheses for performing the type of test you chose in part (a). Minitab output from a chi-square test is shown below. 

(c) Check that the conditions for carrying out the test are met. 

(d) Interpret the P-value in context. What conclusion would you draw? 

Step-by-Step Solution

Answer

(a) The chi-square homogeneity test should be applied here.

(b) For the provided test, the proper null and alternative hypotheses may be stated as follows:

$H_0$ : There is no connection between the two variables.

$H_a$ A connection exists between two variables.

(c) Required conditions are fulfilled.

(d)  there is adequate evidence to reject the assertion that the two variables are related at the$5$ percent significance level.

Part (a)Step 1: Given information

Given in the question that,  Here are results for separate random samples of American and Asian students at a large midwestern university 

We need to verify that should we use a chi-square test for homogeneity or a chi-square test of association/independence in this setting.

Part(a) Step 2: Explanation

Let's consider the output:

For the one qualitative variable of two samples, a two-way table was generated. As a result, the chi-square homogeneity test should be used.

Part(b) Step 1: Given information

Given in the question that,  Here are results for separate random samples of American and Asian students at a large midwestern university. 

We need to state appropriate hypotheses for performing the type of test  chose in part (a) 

Part (b) Step 2: Explanation

For the provided test, the proper null and alternative hypotheses may be stated as follows:

$H_0$: There is no connection between the two variables.

$H_a$: A connection exists between two variables.

Part(c) Step 1: Given information

Given in the question that,  Here are results for separate random samples of American and Asian students at a large midwestern university 

We need to find that whether the conditions for carrying out the test are met.

Part(c) Step 2: Explanation

The following are the circumstances in this case:

1. A random sample is picked.

2. All of the projected counts are more than five.

3. The sample size is$10\%$lower than the population.

As a result, the required conditions might be considered to be met.

Part(d) Step 1: Given information

Given in the question that,  Here are results for separate random samples of American and Asian students at a large midwestern university 

 We need to interpret the P-value.

Part (d) Step 2: Explanation

If there is no association between the variables, the probability of having the same or extreme sample is $0.01$percent.

$0.05$ is the $p$-value. The null hypothesis is found to be false. Thus, there is adequate evidence to reject the assertion that the two variables are related at the$5$ percent significance level.