Question

A subsample of the 2006 MTF survey suggests that adolescents are generally concerned with social issues. In fact, 70% of the 1,482 respondents who answered the question reported that they either sometimes or often think about social issues. Estimate 95% and 99% confidence intervals the percentage of all adolescents who sometimes or often think about social issues. Please round to a whole number.

Answer 1

a) (68%,72%)

b) (67%,73%)

Explanation:

We are given the following in the question:

Sample size, n = 1482

Proportion of reported that they either sometimes or often think about social issues =

`\hat{p} = 70\% = 0.7`

a) 95% Confidence interval:

`\hat{p}\pm z_(stat)\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}`

`z_(critical)\text{ at}~\alpha_(0.05) = 1.96`

Putting the values, we get:

`0.7\pm 1.96(\sqrt{(0.7(1-0.7))/(1482)})\\\\ = 0.7\pm 0.023\\\\=(0.6767,0.7233)\approx (68\%,72\%)`

b) 99% Confidence interval:

`\hat{p}\pm z_(stat)\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}`

`z_(critical)\text{ at}~\alpha_(0.01) = 2.58`

Putting the values, we get:

`0.7\pm 2.58(\sqrt{(0.7(1-0.7))/(1482)})\\\\ = 0.7\pm 0.0301\\\\=(0.6693,0.7307)\approx (67\%,73\%)`