Question

Suppose a sample of 80 with a sample proportion of 0.58 is taken from a population. Which of the following is the approximate 95% confidence interval for the population parameter?

Answer 1

The 95% confidence interval for the population parameter is (0.4718, 0.6882).

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:

`n = 80, p = 0.58`

95% confidence interval

So
`\alpha = 0.05`, z is the value of Z that has a pvalue of
`1 - (0.05)/(2) = 0.975`, so
`z = 1.96`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.58 - 1.96\sqrt{(0.58*0.42)/(80)} = 0.4718`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.58 + 1.96\sqrt{(0.58*0.42)/(80)} = 0.6882`

The 95% confidence interval for the population parameter is (0.4718, 0.6882).