SOLUTION: In a random sample of 55 members of the Statsburgh Republican Party, 36 said they would support raising the speed limit in downtown Statsburgh to 45 miles per hour. In a random sam

Question 1188145: In a random sample of 55 members of the Statsburgh Republican Party, 36 said they would support raising the speed limit in downtown Statsburgh to 45 miles per hour. In a random sample of 90 members of the Statsburgh Democratic Party, 23 said they would support raising the speed limit.
Construct a 95% confidence interval for the difference between p1 , the proportion of Statsburgh Republicans who want to raise the speed limit and p2 , the proportion of Statsburgh Democrats who want to raise the speed limit.
Answer: We are 95% confident that __ < p1 − p2 < __.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
z-score for 95% confidence interval is equal to plus or minus 1.96

36 out of 55 republican party members said they would support raising the speed limit to 45 miles per hour.

23 out of 90 democratic party members said they would do the same.

p1 = x1/n1 = 36/55 = .654545454545.
p2 = x2/n2 = 23/90 = .255555555556.

x1 is the number of republicans that voted for the increase.
n1 i the number of republicans.

x2 is the number of democrats that voted for the increase.
n2 is the number of democrats.

p0 = (36 + 23) / (55 + 90) = .4068965517.

the pooled standard error = sqrt(p0*(1-p0)*(1/n1+1/n2).

this is equal to sqrt(.4068965517*(1-.4068965517)*(1/55+1/90)) = .0840792116.

the unpooled standard error = sqrt(p1 * (1-p1) / n1 + p2 * (1-p1)/n2).

this is equal to sqrt(.654545454545*(1-.654545454545/55 + .255555555556*(1-.255555555556/90) = .0788989809.

the pooled z-score = (p1-p2)/the pooled standard error.

this is equal to (.654545454545-.255555555556)/.0840792116 = 4.745404854.

the unpooled z-score = (.654545454545-.255555555556)/.0788989809 = 5.056971516.

since the critical z-score is plus or minus 1.96, these are extremely significant results, indicating that the two parties are nowhere near equal in their support for the 45 mile per hour speed limit.

the reference i used for this analysis can be found at https://sixsigmastudyguide.com/two-sample-test-of-proportions/

the pooled and unpooled z-scores were both significant, so there's no reason to have to decide whether the pooled or unpooled analysis was the one to use.

i'm not even sure which one is the right one in this case.
because the result is the same (significant), i haven't bothered to try to determine which option was the right option.
i did both analyses to determine whether it was important to know which option was the right option to use.
in this case, it wasn't.

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