A random sample of 50 suspension helmets used by motorcycle riders and race-car

Question

A random sample of 50 suspension helmets used by motorcycle riders and race-car drivers was subjected to an impact test, and on 18 of these helmets some damage was observed. a) Compute a 95% confidence interval for the true proportion of helmets of this type that would show damage from this impact test. b) In past samples, it has been found that at most 40% of helmets suffered damage as a result of this test. How many helmets must be tested to be 95% confident that the true value of p is within 2% of our point estimate?

Explanation / Answer

a)Since n*^p= 50*0.36= 18 and n*(1-^p) = 50*(1-0.36) = 32 are both at least greater than 5, then n is considered to be large and hence the sampling distribution of all possible sample proportions of damaged helments-^p will follow the z distribution.

A 95% confidence interval on the true proportion of helmets that would show damage from this test
= [^p (+-)[^p(1-^p)/(n-1)]
= [0.36 (+-)[0.36(1-0.36)/(50-1)]
= [0.36 (+-)0.2304/49]
= [0.2914,0.4286]

So, we are 95% confident that the true proportion of helmets that would show damage from this test should be any value lie between 0.2914 and 0.4286.

b) Let n be the number of helments that must be tested to be 95% confident that the error bound is less than 0.02.

Our given confidence level = 95%
100(1-alpha)% = 95%
1-alpha = 0.95
alpha = 0.05

Using ^p = 0.36 to estimate p, then
n= p(1-p)*[z(alpha/2)/B]^2
n = 0.36*(1-0.36)*[z(0.05/2)/0.02]^2
n = 0.36*(1-0.36)*[z0.025/0.02]^2
n = 0.2304*(1.96/0.02)^2
n 2213

So, around 2213 helments must be tested to be 95% confident that the error in estimating the true value of p is less than 0.02.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.