Question

Senator Lockhart is setting up his reelection campaign, and his campaign chair is commissioning a series of polls to measure sentiment among the electorate and start setting up strategies. In one poll, 175 women and 208 men were asked whether they were likely to vote to reelect Lockhart. 98 women said yes, as did 110 of the men. Is there a statistically significant difference between these two results? Use an alpha of 5%.

Answer 1

Solution :

1 represents women and 2 represents men.

For sample 1, we have that the sample size is N1​=175, the number of favorable cases is X1​=98, so then the sample proportion is

For sample 2, we have that the sample size is N2​=208, the number of favorable cases is X2​=110, so then the sample proportion is \

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(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.

(2) Rejection Region

The significance level is α=0.05, and the critical value for a two-tailed test is zc​=1.96.

The rejection region for this two-tailed test is R = { z : ∣z∣ > 1.96}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣z∣ = 0.61 ≤ zc ​= 1.96, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.5421, and since p = 0.5421 ≥ 0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p1​ is different than p2​, at the 0.05 significance level.

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