In 2008, the BBB settled 75% of complaints it recieved. Supposed they hired you

Question

In 2008, the BBB settled 75% of complaints it recieved. Supposed they hired you to investigate the complaints it received involving new car dealers. You plan to select a sample of new car dealer complaints to estimate the proportion of complaints the BBB is able to settle. Assume the population proportion of complaints settled for new car dealers is .75, the same as the overall propoprtion of complaints settled in 2008.

a) suppose you select a sample of 450 complaints involving new car dealers. Show the sampling of p.

b) based upon the sample of 450 complaints, what is the probability that the sample proportion will be within .04 of the population proportion?

c) suppose you select a sample of 200 complaints involving new car dealers. Show the sampling distribution of p.

d) Based upon the smaller sample of only 200 complaints, what is the probability that the sample proportion will be within .04 of the population proportion?

e) As measured by the increase in probability, how much do you gain in precision by taking the larger sample in part (b)?

Explanation / Answer

a) p=0.75

sample mean = 450*0.75 = 337.5

b) 0.04 = z(a/2)*sqrt(0.75*0.25/450)

z(a/2) = 0.04/sqrt(0.75*0.25/450) = 1.96

probability that the sample proportion will be within .04 of the population proportion=0.95

c) p=0.75

sample mean= 200*0.75 = 150

d) z=0.04/sqrt(0.75*0.25/200) = 1.31

probability that the sample proportion will be within .04 of the population proportion=0.81

e) as sample size increasesprobability that the sample proportion will be within .04 of the population proportion increases

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