Question

A researcher constructs a confidence interval for a population proportion using a sample of size 50. The value of pˆ is .3, and the resulting confidence interval is determined to be (.1323, .4677). What's the level of confidence for this interval?a. 80%b. 90%c. 95%d. 99%e. Not enough information to answer

Answer 1

d. 99%

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence interval
`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)`.

For this problem, we have that:

`p = 0.30, n = 50`

Let's start from the higher confidence levels, since the higher the confidence level, the higher the width of the interval.

d. 99%

So
`\alpha = 0.01`, z is the value of Z that has a pvalue of
`1 - (0.05)/(2) = 0.905`, so
`Z = 2.575`.

The lower limit is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.30 - 2.575\sqrt{(0.3*0.7)/(50)} = 0.133`

The upper limit is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.30 + 2.575\sqrt{(0.3*0.7)/(50)} = 0.467`

This is very close to the interval found, so d. is the correct answer