Question

4. According to statistics reported on IN-CORP a surprising number of motor vehicles are not covered by insurance. Sample results, consistent with the IN-CORP report, showed 46 of 200 vehicles were not covered by insurance. a. What is the point estimate of the proportion of vehicles not covered by insurance? b. Develop a 95% confidence interval for the population proportion.

Answer 1

a) 0.23

b) The 95% confidence interval for the population proportion is (0.1717, 0.2883).

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)`.

Point estimate

The point estimate is:

`\pi = (46)/(200) = 0.23`

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)} = 0.23 - 1.96\sqrt{(0.23*0.77)/(200)} = 0.1717`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.23 + 1.96\sqrt{(0.23*0.77)/(200)} = 0.2883`

The 95% confidence interval for the population proportion is (0.1717, 0.2883).