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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.

User Rodu
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Answer:

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).

User Aung
by
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