Question

A random sample of 500 adult residents of Maricopa County found that 356 were in favor of increasing the highway speed limit to 75 mph, while another sample of 400 adult residents of Pima County found that 297 were in favor of the increased speed limit. Do these data indicate that there is a difference in the support for in increasing the speed limit between the residents of the two counties? Use α = 0.05.a) do these data indicate that there is a difference in thesupport for increasing the speed limit between residents of the twocounties? Use α = 0.05. What is the P-value forthis test?

b) construct a 95% confidence interval on the difference inthe two proportions. Provide a practical interpretation ofthis interval.
Please show formulas

Answer 1

`z=1.04`

`p_v =2*P(Z>1.04)\approx 0.298`

Comparing the p value with the significance level given
`\alpha=0.05` we see that
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can't say that the proportion of people in favor of the increased speed limit not differs significantly between the two groups.

Explanation:

1) Data given and notation

`X_(PC)=297` represent the number of residents of Pima County in favor of the increased speed limit

`X_(MC)=356` represent the number of residents of Maricopa County in favor of the increased speed limit

`n_(PC)=400` sample of Pima County selected

`n_(MC)=500` sample of Maricopa County selected

`p_(PC)=(297)/(400)=0.743` represent the proportion of residents of Pima County in favor of the increased speed limit

`p_(MC)=(356)/(500)=0.712` represent the proportion of residents of Maricopa County in favor of the increased speed limit

z would represent the statistic (variable of interest)

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

`\alpha=0.05` significance level given

2) Concepts and formulas to use

We need to conduct a hypothesis in order to check if is there is a difference in the support for in increasing the speed limit between the residents of the two counties, the system of hypothesis would be:

Null hypothesis:
`p_(PC) - p_(MC)=0`

Alternative hypothesis:
`p_(PC) - \mu_(MC) \\eq 0`

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

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

Where
`\hat p=(X_(PC)+X_(MC))/(n_(PC)+n_(MC))=(297+356)/(400+500)=0.726`

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.

3) Calculate the statistic

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

`z=\frac{0.743-0.712}{\sqrt{0.726(1-0.726)((1)/(400)+(1)/(500))}}=1.04`

4) Statistical decision

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

`p_v =2*P(Z>1.04)\approx 0.298`

Comparing the p value with the significance level given
`\alpha=0.05` we see that
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can't say that the proportion of people in favor of the increased speed limit not differs significantly between the two groups.