Question

A survey of businesses in a particular state found that out of 150 surveyed, 51 were owned by women. We want to estimate the true proportion of businesses owned by women.

Construct a 95% confidence interval. Round your answers to 4 decimal places.

Answer 1

The 95% confidence interval for the true proportion of businesses owned by women is (0.2763, 0.4037).

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)`.

A survey of businesses in a particular state found that out of 150 surveyed, 51 were owned by women.

This means that
`n = 150, \pi = (51)/(150) = 0.34`

95% confidence level

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)} = 0.34 - 1.96\sqrt{(0.34*0.66)/(150)} = 0.2763`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.34 + 1.96\sqrt{(0.34*0.66)/(150)} = 0.4037`

The 95% confidence interval for the true proportion of businesses owned by women is (0.2763, 0.4037).