Question

The 2010 General Social Survey asked the question: "After an average work day, about how many hours do you have to relax or pursue activities that you enjoy?" to a random sample of 1155 Americans. A 90% confidence interval for the mean number of hours spent relaxing or pursuing activities they enjoy was [1.17, 1.83].

(a) Interpret this interval in context of the data: "There is a 90% chance that the average number of hours spent by Americans relaxing after an average day of work is between 1.17 hours and 1.83 hours."

(b) Suppose another set of researchers reported a confidence interval with a larger margin of error based on the same sample of 1155 Americans. How does their confidence level compare to the confidence level of the interval stated above? .

(c) Suppose next year a new survey asking the same question is conducted, and this time the sample size is 2700. Assuming that the sampled standard deviation does not change, what will be the new margin of error of the 90% confidence interval constructed based on data from the new survey?

Answer 1

the formula for calculating the confidence interval of the mean is

The Z distribution may change to T if the sample size is low enough.

But you can see that the Margin of error is directly proportional to the standard deviation and inversely proportional to the sample size.

a) It states that we are 90% confident that the average number of hours spent by Americans relaxing after an average day of work is between 1.17 hours and 1.83 hours.

b) A large margin of error states that the confidence interval will be broader.

As the width of the confidence interval is twice the margin of error.

c) first we will find the sample standard deviation from part 1.

n is 1155 above

and for 90% confidence

for new sample size the new margin of error =