A poll finds that 54% of the 600 people polled favor the incumbent. Shortly after the poll is taken, it is disclosed that the incumbent had an extramarital affair. A new poll finds that 50% of the 1030 polled now favor the incumbent. We want to know if his support has decreased. The test statistic is

Shortly after the poll is taken, it is disclosed that the incumbent had an extramarital affair. A new poll finds that 50% of the 1030 polled now favor the incumbent.

Let [tex]p_1[/tex] = population proportion of people who favor the incumbent in the first poll

[tex]p_2[/tex] = population proportion of people who favor the incumbent in the second poll

So, Null Hypothesis, [tex]H_0[/tex] : [tex]p_1\geq p_2[/tex] {means that his support has increased or remained same after the second poll}

Alternate Hypothesis, [tex]H_0[/tex] : [tex]p_1 < p_2[/tex] {means that his support has decreased after the second poll}

The test statistics that would be used here is Two-sample z test for proportions;

T.S. = [tex]\frac{(\hat p_1-\hat p_2)-(p_1-p_2)}{\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2} } }[/tex] ~ N(0,1)

where, [tex]\hat p_1[/tex] = sample proportion of people who favor the incumbent in first poll = 54%

[tex]\hat p_1[/tex] = sample proportion of people who favor the incumbent in second poll = 50%

[tex]n_1[/tex] = sample of people in first poll = 600

[tex]n_2[/tex] = sample of people in second poll = 1030

So, the test statistics = [tex]\frac{(0.54-0.50)-(0)}{\sqrt{\frac{0.54(1-0.54)}{600}+\frac{0.50(1-0.50)}{1030} } }[/tex]

= 1.561

Hence, the value of z test statistics is 1.561.