Answer to Question #296946 in Statistics and Probability for jsjaiia

Question

A presidential candidate asks a polling organization to conduct a nationwide survey to determine the percentage of potential voters who would vote for him over his rival presidential candidate. Out of 2 500 respondents, 915 said that they would vote for him. If 35% of the of the potential votersvote for his rival, is this significantly different from the percentage of potential voters of the candidate who requested the survey? Use 5% level of significance

Answer 1

Group 1(Potential voters supporting him)

$n_1=2500\\x_1=915\\\hat p_1={x_1\over n_1}={915\over2500}=0.366$

Group 2(Potential voters supporting his rival)

$n_2=2500\\\hat p_2=0.35\implies x_2=0.35\times2500=875$

Now,

$\hat p={x_1+x_2\over n_1+n_2}={915+875\over5000}=0.358$

Hypotheses,

$H_0:p_1=p_2\\vs\\H_1:p_1\not=p_2$

The test statistic is,

$Z={\hat p_1-\hat p_2\over\sqrt{\hat p(1-\hat p)({1\over n_1}+{1\over n_2})}}={0.016\over0.0136}=1.18$

The critical value is, $Z_{\alpha\over2}=Z_{0.05\over2}=Z_{0.025}=1.96$.

The null hypothesis is rejected if, $|Z|\gt Z_{0.025}$   

Since $|Z|=1.18\lt Z_{0.025}=1.96$, we fail to reject the null hypothesis and conclude that there is no sufficient evidence to show that the difference in percentages for the two candidates is significant at 5% significance level.