5.12 Mental health. The General Social Survey asked the question: “For how many days during the past 30 days was your mental health, which includes stress, depression, and problems with emotions, not good?" Based on responses from 1,151 US residents, the survey reported a 95% confidence interval of 3.40 to 4.24 days in 2010. (a) Interpret this interval in context of the data. (b) What does “95% confident" mean? Explain in the context of the application. (c) Suppose the researchers think a 99% confidence level would be more appropriate for this interval. Will this new interval be smaller or wider than the 95% confidence interval? (d) If a new survey were to be done with 500 Americans, do you think the standard error of the estimate be larger, smaller, or about the same.

MY Question is attached as an image below. 

I have ansswered it below, but I think I've got (b) wrong, not sure how to get to the right answer. 

Exercise 5.12

(a)

We are 95% sure that the true proportion of survey respondents’ answers of “not good” are between 3.4% and 4.24% with a mean of 3.82%. The mean is the midpoint of the confidence interval (3.4% + .42%).

(b)

95% confidence means that about 95% of the intervals contain the parameter, p. The Standard Error is .0011 or 1.1%.

> (sqrt(.0382*(1-.0382)/1151))*1.96

[1] 0.01107369 = Standard Error

> 1.96*.011

[1] 0.02156

> .0382+0.02156

[1] 0.05976

> .0382-0.02156

[1] 0.01664

(c)

The interval at 99% will be wider. At 99% the standard deviation from the mean will be -2.58 to +2.58 , versus -1.96 to +1.96 at the 95% confidence interval.  

(d)

The standard error of estimate will be larger.  A bigger sample tends to provide a more precise point estimate than a smaller sample. In this case there is a “larger net” with this smaller sample due to less than ½ as many Americans (500) participating in the survey.