Question

In a recent Pew Research poll, 2142 Republicans (or Republican leaning) and 2965 Democrats (or Democratic leaning) were asked if they were in favor of making private gun sales and sales at gun shows subject to background checks. 79% of the Republicans and 91% of the Democrats said yes.

a) Set up an appropriate hypotheses to test if this difference is statistically significant.

b)What is the value of the test statistic?

c) Is it significant at ? = .01?

Answer 1

Two-Proportion Z test

The following information is provided: (a) Sample 1 - The sample size is N1 = 2142, and the sample proportion is p^1​​=0.79 (b) Sample 2 - The sample size is N2 = 2965, and the sample proportion is p^2​​=0.91 and the significance level is α=0.01 Pooled Proportion The value of the pooled proportion is computed as (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p1 = p2 Ha: p1 ≠ p2 This corresponds to a Two-tailed test, for which a z-test for two population proportions needs to be conducted. (2a) Critical Value Based on the information provided, the significance level is α=0.01, therefore the critical value for this Two-tailed test is Zc​=2.5758. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Two-tailed test is |Z|>2.5758 i.e. Z>2.5758 or Z<-2.5758 (3) Test Statistics The z-statistic is computed as follows: (4) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is p =P(|Z|>12.1837)=0 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that |Z|=12.1837 > Zc​=2.5758, it is then concluded that the null hypothesis is rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0, and since p=0≤0.01, it is concluded that the null hypothesis is rejected. (6) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p1 is different than p2, at the 0.01 significance level.

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