Question

In an effort to test the hypothesis that the proportion of voters in the younger than 40 year old age bracket who will vote for a particular politician is different than the proportion voters in the above 40 age bracket, the following data was collected. The test statistics (z-score) of the test hypothesis at 0.10 level of significance would be___. (Specify your answer to the 2nd decimal.) Population Number Who Will Vote for Politician Sample Size Below 40 348 700 Above 40 290 650

Answer 1

`z=\frac{0.497-0.446}{\sqrt{0.473(1-0.473)((1)/(700)+(1)/(650))}}=1.88`

`p_v =2*P(Z>1.88)=0.0601`

Comparing the p value with the significance level given
`\alpha=0.1` we see that
`p_v<\alpha` so we can conclude that we have enough evidence to to reject the null hypothesis, and we can say that the proportions analyzed are significantly different.

Explanation:

Data given and notation

`X_(1)=348` represent the number of people Who Will Vote for Politician below 40

`X_(2)=290` represent the number of people Who Will Vote for Politician above 40

`n_(1)=700` sample below 40

`n_(2)=650` sample above 40

`p_(1)=(348)/(700)=0.497` represent the proportion estimated for Who Will Vote for Politician below 40

`p_(2)=(290)/(650)=0.446` represent the proportion estimated for Who Will Vote for Politician above 40

`\hat p` represent the pooled estimate of p

z would represent the statistic (variable of interest)

`p_v` represent the value for the test (variable of interest)

`\alpha=0.1` significance level given

Concepts and formulas to use

We need to conduct a hypothesis in order to check if is there is a difference between the two proportions, the system of hypothesis would be:

Null hypothesis:
`p_(1) = p_(2)`

Alternative hypothesis:
`p_(1) \\eq p_(2)`

We need to apply a z test to compare proportions, and the statistic is given by:

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

Where
`\hat p=(X_(1)+X_(2))/(n_(1)+n_(2))=(348+290)/(700+650)=0.473`

z-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.

Calculate the statistic

Replacing in formula (1) the values obtained we got this:

`z=\frac{0.497-0.446}{\sqrt{0.473(1-0.473)((1)/(700)+(1)/(650))}}=1.88`

Statistical decision

Since is a two sided test the p value would be:

`p_v =2*P(Z>1.875)=0.0601`

Comparing the p value with the significance level given
`\alpha=0.1` we see that
`p_v<\alpha` so we can conclude that we have enough evidence to to reject the null hypothesis, and we can say that the proportions analyzed are significantly different.