Question

Opinions about the death penaltyThe General Social Survey (GSS) asked separate random samples of people with only a high school degree and people with a bachelor’s degree, “Do you favor or oppose the death penalty for persons convicted of murder?” Of the $1379$people with only a high school degree, $1010$ favored the death penalty, while $319$of the $504$people with a bachelor’s degree favored the death penalty. 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. State appropriate hypotheses for a chi-square test.

b. Here is Minitab output for a chi-square test. Interpret the P-value. What conclusion would you draw?

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

Short answer

a. $H_0$: The rates of improvement in the two therapy groups are identical.

$H_a$: The two therapy groups progress at different speeds.

b. There is no convincing evidence that there is a difference in the improvement rate of the two treatment groups.

c. The p-value of chi-square test and z-test is same so, conclusion based on two tests are same.

Step-by-step solution

Part (a) Step 1 : Given Information

We have to determine the state the null and alternative hypotheses.

Part (a) Step 2 : Simplification

The following are the null and alternate hypotheses:

$H_0$:The rates of improvement in the two therapy groups are identical.
$H_a$: The two therapy groups progress at different speeds.

Part (b) Step 1 : Given Information

We have to state the conclusion.

Part (b) Step 2 : Simplification

The p-value for this study is $0.570$.

As a result, we may claim that there is a $57$ percent chance of getting the sample outcomes, or even more severe, when the improvement rates of the two treatment groups are identical.

Decision: If the P-value is greater than$0.05$, $H_0$is not rejected.

Conclusion: There is no persuasive evidence that the two therapy groups have different improvement rates.

Part (c) Step 1 : Given Information

We have to compare chi-square test and z-test.

Part (c) Step 2 : Simplification

The p-value for this study is $0.570.$

Thechi-squaretestandthez-testbothhavethesamep-value.

Asaresult,theconclusionsbasedonthetwotestsarethesame.

Decision: If the P-value is greater than $0.05$, H0 is not rejected.

In addition, the chi-square test statistic is $0.322$, and the z-test statistic is$-0.57$.

The property that the square of z becomes the chi-square is well-known.

As a result,

$z^2={(-0.57)}^2=0.322={\chi }^2$

This property has been met.