A random sample of 50 musicians was taken from a population of 400 to determine

Question

A random sample of 50 musicians was taken from a population of 400 to determine how many use ear protection. Twenty of the sampled musicians used ear protection. (a) Estimate the total number of musicians who use ear protection. (b) Find an approximate standard error for this estimate. (c) Of the 20 musicians who used ear protection, 12 are guitarists. Estimate the proportion of musicians who use ear protection who are guitarists. Give a standard error for this estimate. (d) Estmate the total number of musicians who use ear protection who are guitarists. Give a standard error for this estimate.

Explanation / Answer

(3)

in a sample of 50 musicians, 20 used ear protection.

Hence p = 20/50 = 2/5

(A)

Estimated number of musicians who used ear protection are 2/5 * 400 = 160

(B)

standard error = sqrt(p*(1-p)/n) = sqrt(0.4*0.6/400) = 0.0245

(C)

proportion of musicians who used ear protection are guitarist = 12/50 = 0.24

There are estimated 400 musicians, out of these guitarist who use ear protection are 0.24*400 = 96.

standard error = sqrt(0.24*0.76/4000) = 0.0214

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