Question

A random sample of 60 suspension helmets used by motorcycle riders and automobile race-car drivers was subjected to an impact test, and on 17 of these helmets some damage was observed. (a) Find a 95% two-sided confidence interval on the true proportion of helmets of this type that would show damage from this test. Round your answers to 3 decimal places.

Answer 1

The 95% confidence interval on the true proportion of helmets of this type that would show damage from this test is (0.169, 0.397).

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)`.

For this problem, we have that:

`n = 60, \pi = (17)/(60) = 0.283`

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)} = 0.283 - 1.96\sqrt{(0.283*0.717)/(60)} = 0.169`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.283 + 1.96\sqrt{(0.283*0.717)/(60)} = 0.397`

The 95% confidence interval on the true proportion of helmets of this type that would show damage from this test is (0.169, 0.397).