Question

The National Assessment of Educational Progress (NAEP) includes a "long-term trend" study that tracks reading and mathematics skills over time, and obtains demographic information. In the 2012 study, a random sample of 9000 17-year-old students was selected.24 The NAEP sample used a multistage design, but the overall effect is quite similar to an SRS of 17-year-olds who are still in school. In the sample, 51% of students had at least one parent who was a college graduate. Estimate, with 99% confidence, the proportion of all 17-year-old students in 2012 who had at least one parent graduate from college.

Answer 1

The 99% confidence interval estimate for the proportion of all 17-year-old students in 2012 who had at least one parent graduate from college is (0.4964, 0.5236).

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence level of
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm z\sqrt{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

Sample of 9000, 51% of students had at least one parent who was a college graduate.

This means that
`n = 9000, \pi = 0.51`

99% confidence level

So
`\alpha = 0.01`, z is the value of Z that has a pvalue of
`1 - (0.01)/(2) = 0.995`, so
`Z = 2.575`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.51 - 2.575\sqrt{(0.51*0.49)/(9000)} = 0.4964`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.51 + 2.575\sqrt{(0.51*0.49)/(9000)} = 0.5236`

The 99% confidence interval estimate for the proportion of all 17-year-old students in 2012 who had at least one parent graduate from college is (0.4964, 0.5236).