In a Random Sample of 500 People Aged $20 - 24,22 \%$

Multiple Choice

In a random sample of 500 people aged $20 - 24,22 \%$ were smokers. In a random sample of 450 people aged 25-29, 14\% were smokers. A $95 \%$ confidence interval for the difference between the proportion of 20-24 year olds and the proportion of 25-29 year olds who are smokers is $0.032 < \mathrm { p } _ { 1 } - \mathrm { p } _ { 2 } < 0.128$
Which of the following statements give a correct interpretation of this confidence interval?
I. We can be $95 \%$ confident that the interval $0.032$ to $0.128$ contains the true difference between the two population proportions.
II. There is a $95 \%$ chance that the true difference between the two population proportions lies between $0.032$ and $0.128$ .
III. If the process were repeated many times, each time selecting random samples of 500 people aged 20-24 and 450 people aged 25-29 and each time constructing a confidence interval for $\mathrm { p } _ { 1 }$ $p _ { 2 } , 95 \%$ of the time the true difference between the two population proportions will lie between $0.032$ and $0.128$ .
IV. If the process were repeated many times, each time selecting random samples of 500 people aged 20-24 and 450 people aged 25-29 and each time constructing a confidence interval for $p _ { 1 } -$ $\mathrm { p } _ { 2 } , 95 \%$ of the time the confidence interval limits will contain the true difference between the two population proportions.

A) I and IV
B) II and IV
C) I and III
D) II and III