Question

A random sample of 50 suspension helmets used by motorcycle riders and automobile race-car drivers was subjected to an impact test, and on 18 of these helmets some damage was observed.

(a)Test the hypotheses H0: p = 0.3 versus H1: p ≠ 0.3 with α = 0.05, using the normal approximation.

(b)Find the P-value for this test.

(c)Find a 95% two-sided confidence interval on the true proportion of helmets of this type that would show damage from this test. Explain how this confidence interval can be used to test the hypothesis in part (a).

(d)Using the point estimate of p obtained from the preliminary sample of 50 helmets, how many helmets must be tested to be 95% confident in order to keep the associated β-error at 0.1 (or the power at 90%)?

Answer 1

(a)Test the hypotheses H0: p = 0.3 versus H1: p ≠ 0.3 with α = 0.05, using the normal approximation.

p̂ = 18/50 = 0.36

n = 50

p = 0.3

The test statistic, z = (p̂ - p)/√p(1-p)/n = (0.36 - 0.3)/√0.3(1-0.3)/50 = 0.93

(b)Find the P-value for this test.

The p-value is 0.3545.

Since the p-value (0.3545) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Therefore, we can conclude that p = 0.3.

(c)Find a 95% two-sided confidence interval on the true proportion of helmets of this type that would show damage from this test. Explain how this confidence interval can be used to test the hypothesis in part (a).

The 95% confidence interval will be:

= p ± z*(√p(1-p)/n

= 0.36 ± 1.96*(√0.36(1-0.36)/50

= (0.227, 0.493)

(d)Using the point estimate of p obtained from the preliminary sample of 50 helmets, how many helmets must be tested to be 95% confident in order to keep the associated β-error at 0.1 (or the power at 90%)?

n = 612

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