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. less-than-or-equal-to p less-than-or-equal-to

(b) Using the point estimate of p obtained from the preliminary sample of 60 helmets, how many helmets must be tested to be 95% confident that the error in estimating the true value of p is less than 0.02? n equals

(c) How large must the sample be if we wish to be at least 95% confident that the error in estimating p is less than 0.02, regardless of the true value of p? n equals

Answer 1

a)

Sample proportion = 17 / 60 = 0.283

95% confidence interval for p is

- Z * sqrt ( ( 1 - ) / n) < p < + Z * sqrt ( ( 1 - ) / n)

0.283 - 1.96 * sqrt( 0.283 * 0.717 / 60) < p < 0.283 + 1.96 * sqrt( 0.283 * 0.717 / 60)

0.169 < p < 0.397

95% CI is ( 0.169 , 0.397 )

b)

Sample size = Z² * p( 1 - p) / E²

= 1.96² * 0.283 * 0.717 / 0.02²

= 1948.75

n = 1949 (Rounded up to nearest integer)

c)

When prior estimate of proportion is not specified is not specified then p = 0.50

Sample size = Z² * p( 1 - p) / E²

= 1.96² * 0.50 * 0.50 / 0.02²

= 2401