Question

15. In a random sample of 60 adults, 55% of them reported having a fear of public speaking. (1)(1.5 points) Are the assumptions for constructing a confidence interval for the population proportion satisfied? Explain. (2)(5 points) Construct a 99% confidence interval for the population proportion of adults who have a fear of public speaking. (3)(2.5 points) Interpret the 99% confident interval. (4)(3.5 points) Estimate the sample size needed so that a 99% confidence interval will have a margin of error of 0.04. (5)(1.5 point) If the sample size were n = 100, would the margin of error be larger or smaller?___________

Answer 1

1) We have here:

np = 60*0.55 = 33 >= 10
n(1-p) = 60*(1 - 0.55) = 27 >= 10

Therefore the assumption is satisfied here. And we can use the normal distribution assumption here.

2) From standard normal tables, we have:
P(-2.576 < Z < 2.576) = 0.99

Therefore the confidence interval here is obtained as:

This is the required 99% confidence interval here.

3) The interpretation of the confidence interval here is that there is a 99% confidence that the true population proportion of adults who reported having a fear of public speaking lies in the above interval.

4) The margin of error here is given to be 0.04. Therefore the sample size here is computed as:

Therefore 1027 is the required sample size here.

5) For a lower sample size, the margin of error would have been larger, because the margin of error is inversely proportional to the square root of sample size. Therefore the margin of error would be larger here.