Chapter 7, Problem 55SE

Thriving In June 2012, a Gallup Poll showed that when a sample of Americans were asked whether they were thriving, struggling, or suffering, about $54\%$ said they were thriving.

a. Assuming the sample size was 1000, how many in the sample said they were thriving?

b. Is the sample size large enough to apply the Central Limit Theorem? Explain. Assume the other conditions are met.

c. Find a $95\%$ confidence interval for the proportion that said they were thriving.

d. Find the width of the interval you found in part b by subtracting the lower limit from the upper limit. Round your answer to the nearest whole percent.

e. Now assume the sample size was 4000. How many would have said they were thriving?

f. Find a $95\%$ confidence interval for the proportion that said they were thriving using the numbers from part e.

g. What is the width of the interval you found in part f?

h. When the sample size is multiplied by 4 (as it was here, going from 1000 to 4000), is the interval width decreased by a factor of 4? In other words, is the interval for the larger sample size one-fourth $\left(1/4\right)$ of the interval for the smaller sample size? If not, what is it divided by?