Question

19. In a random sample of 250 students, we found that 75 work out 4 or more times a week. Find the 95% confidence interval for the proportion of students who work out 4 or more times a week.

Answer 1

The 95% confidence interval for the proportion of students who work out 4 or more times a week is (0.2432, 0.3568).

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 z-score that has a p-value of
`1 - (\alpha)/(2)`.

In a random sample of 250 students, we found that 75 work out 4 or more times a week.

This means that
`n = 250, \pi = (75)/(250) = 0.3`

95% confidence level

So
`\alpha = 0.05`, z is the value of Z that has a p-value 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)} = 0.3 - 1.96\sqrt{(0.3*0.7)/(250)} = 0.2432`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.3 + 1.96\sqrt{(0.3*0.7)/(250)} = 0.3568`

The 95% confidence interval for the proportion of students who work out 4 or more times a week is (0.2432, 0.3568).