Question

Given the sample data below, test the claim that the proportion of male voters who plan to vote Republican at the next presidential election is 15 percentage points more than the percentage of female voters who plan to vote Republican. Assume all requirements are met. Use the P-value method of testing hypotheses.

Men: n1 = 250, x1 = 146
Women: n2 = 202, x2 = 103

Answer 1

`z=\frac{0.584-0.510 -0.15}{\sqrt{0.551(1-0.551)((1)/(250)+(1)/(202))}}=-1.615`

The p value would be given by:

`p_v =P(Z<-1.615)= 0.0531`

The p value if is compared with a significance of 5% we see that is higher than 0.05 so then at this significance level we have enough evidence to conclude that the true difference in the proportion of males in favor is 15% more than the % of females in favor

Explanation:

Information provided

`X_(1)=146` represent the number of male in favor

`X_(2)=103` represent the number of female in favor

`n_(1)=250` sample of males selected

`n_(2)=203` sample of females selected

`p_(1)=(146)/(250)=0.584` represent the proportion of males in favor

`p_(2)=(103)/(202)=0.510` represent the proportion of females in favor

`\hat p` represent the pooled estimate of p

z would represent the statistic

`p_v` represent the value

System of hypothesis

We want to test if the proportion of males in favor is 0.15 point above the proportion of females, the system of hypothesis would be:

Null hypothesis:
`p_(1) - p_(2) \geq 0.15`

Alternative hypothesis:
`p_(1) - p_(2) <0.15`

The statistic is given by:

`z=\frac{p_(1)-p_(2) -0.15}{\sqrt{\hat p (1-\hat p)((1)/(n_(1))+(1)/(n_(2)))}}` (1)

Where
`\hat p=(X_(1)+X_(2))/(n_(1)+n_(2))=(146+103)/(250+202)=0.551`

Replacing the info given we got:

`z=\frac{0.584-0.510 -0.15}{\sqrt{0.551(1-0.551)((1)/(250)+(1)/(202))}}=-1.615`

The p value would be given by:

`p_v =P(Z<-1.615)= 0.0531`

The p value if is compared with a significance of 5% we see that is higher than 0.05 so then at this significance level we have enough evidence to conclude that the true difference in the proportion of males in favor is 15% more than the % of females in favor