Question

You have recently been hired as the head of political polling in the country of Forsythia, a country with a fairly large population. You recently performed a poll for the upcoming presidential election. There are two candidates; the incumbent candidate and the opposition candidate. You randomly polled some Forsythian voters to determine the likelihood of the incumbent winning. You asked voters if they plan to vote for the incumbent or the opposition in the upcoming election. If they say the opposition, you record a 0. If they say the incumbent, you record a 1.

The following is the data you recorded:

0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1

Question

1.) For a 95% confidence level the margin of error is 9.37%.

Construct a confidence interval for both candidates:

2.)Do the confidence intervals overlap?

3.) If we were to change our confidence level to 99%, what would you expect to happen to our margin of error?

4.) If we were to change our confidence level to 99%, what would you expect to happen to our confidence intervals?

5.) If we were to change our confidence level to 80%, what would you expect to happen to our margin of error?

Answer 1

total number of voters = 114

voters supporting incumbent = 60 and supporting opposition = 54

proportion for supporting incumbent = 60/114 = 0.5263

converting to percent, we get 52.63%

proportion for supporting opposition = 54/114 = 0.4737

converting to percent, we get 47.37%

(1) Margin of error = 9.37%

We know that confidence interval = (mean-margin of error, mean+margin of error)

For incumbent, mean = 52.63%

So, confidence interval = (52.63-9.37, 52.63+9.37) = 43.26% to 62.00%

For opposition, mean = 47.37%

So, confidence interval = (47.37-9.37, 47.37+9.37) = 38.00% to 56.74%

(2) yes, confidence intervals overlap because lower limit of confidence interval for incumbent is less than upper limit for opposition confidence interval.

(3) Margin of error depends upon the critical value. If we change confidence level from 95% to 99%, the critical z value changes from 1.96 to 2.576. Thus, the margin of error will increase as we increase the confidence level.

(4) As we know that margin of error increases by increasing the confidence level and confidence interval is dependent upon margin of error. So, we can say that the width of confidence interval will increases due to increase in confidence level as margin of error increases.

(5) Margin of error depends upon the critical value. If we change confidence level from 95% to 80%, the critical z value changes from 1.96 to 1.28. Thus, the margin of error will decrease as we decrease the confidence level.