Q. 44 - The Practice of Statistics for AP

Question

The General Social Survey asked a random sample of adults, “Do you favour or oppose the death penalty for persons convicted of murder?” The following table gives the responses of people whose highest education was a high school degree and of people with a bachelor’s degree: 

We can test the hypothesis of “no difference” in support for the death penalty among people in these educational categories in two ways: with a two-sample z test or with a chi-square test. 

(a) Minitab output for a chi-square test is shown below. State appropriate hypotheses and interpret the P-value in context. What conclusion would you draw? Chi-Square Test: C1, C2 Expected counts are printed below-observed counts Chi-Square contributions are printed below expected counts 

(b) Minitab output for a two-sample z test is shown below. Explain how these results are consistent with the test in part (a). 

Step-by-Step Solution

Answer

(a) $H_0$ : There is no association between the variables

$H_a$: There is an association between the variables

If there is no association between the variables, then we have a probability of $0.0\%$ of obtaining a similar or more extreme sample.

(b) We can deduce from this P-value that there is a difference between the population proportions, which is the case if the variables are related.

As a result, the two-sample z-test and the chi-square test are equivalent, as seen by the similar P-values.

Part (a) Step 1: Given information

The given data is 

Part (a) Step 2: Explanation

The null hypothesis states that there is no association between the variables:

$H_0$ : There is no association between the variables

(type of degree and favoring death penalty)

The alternative hypothesis states that there is an association between the variables:

$H_a$ : There is an association between the variables

(type of degree and favoring death penalty)

The p-value is given as

$P=0.000=0.0\%$

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme if the null hypothesis is true.

If there is no association between the variables, then we have a probability of $0.0\%$obtaining a similar or more extreme sample.

$H_0$ : There is no association between the variables

$H_a$: There is an association between the variables.

Part (b) Step 1: Given information

The given data is

Part (b) Step 2: Explanation

Find the hypothesis

$H_0:p_1-p_2=0$

$H_0:p_1-p_2\ne 0$

The P- value is

$P=0.000=0.0\%$

We can deduce from this P-value that there is a difference between the population proportions, which is the case if the variables are related.

As a result, the two-sample z-test and the chi-square test are equivalent, as seen by the similar P-values.