A random sample of 50 suspension helmets used by motorcycle riders and automobil

Question

A random sample of 50 suspension helmets used by motorcycle riders and automobile race-car drivers was subjected to an impact test, and some damage was observed on 18 of these helmets. (a) Find a 95% two sided confidence interval on the true proportion of helmets, denoted by p, that would show damage from this test and give an interpretation. (b) Find 95% lower and upper one-sided confidence bounds on the true proportion of helmets, denoted by p, that would show damage from this test and give interpretation. (c) Using the point estimate of p from the 50 helmets, how many helmets must be tested to be 95% confident that the marginal error E is less than 0.02? (d) How large must the sample be if we wish to be at least 95% confident that the marginal error E is less than 0.02 regardless of the true value of p?

Explanation / Answer

a) here p=18/50=0.36

and std error =(p(!-p)/n)1/2 =0.0679

for 95% CI, z=1.96

hence confidence interval =sample proportion -/+ z*std error =0.2270 ; 0.4930

b) for one sided interval' z=1.6449

hence one sided lower and upper bound =0.2483 ; 0.4717

c)margin of error E=0.02

and for 95% CI, z=1.96

hence sample size =p(1-p)(Z/E)2 =~2213

d) for estimated p =0.5 will provide maximum sample size

hence sample size =~2401

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