Question

In 2008 the Better Business Bureau settled 75% of complaints they received (USA Today, March 2, 2009). Suppose you have been hired by the Better Business Bureau to investigate the complaints they received this year involving new car dealers. You plan to select a sample of new car dealer complaints to estimate the proportion of complaints the Better Business Bureau is able to settle. Assume the population proportion of complaints settled for new car dealers is.75, the same as the overall proportion of complaints settled in 2008. a) Suppose you select a sample of 450 complaints involving new car dealers. Show the sampling distribution of b) Based upon a 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 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)?

Answer 1

a)

The sampling distribution of sample proportion is

And standard deviation is

Answer: normal

b)

The z-score for $\hat{p}=p-0.04$ is

The z-score for $\hat{p}=p+0.04$ is

The required probability is

$P(p-0.04<hat{p}<p+0.04)=P(-1.96<z<1.96)=0.95$

(c)

The sampling distribution of sample proporiton will be approximately normal distribution with mean

$p=0.75$

and standard deviation

(d)

The z-score for $\hat{p}=p-0.04$ is

$z=rac{hat{p}-p}{sigma}=rac{p-0.04-0.75}{0.0306}=-1.31$

The z-score for $\hat{p}=p&plus;0.04$ is

The required probability is

$P(p-0.04<hat{p}<p+0.04)=P(-1.31<z<1.31)=0.8098$

(E)

Results are more precise for sample size 450. Probability is approximately 14% higher for sample size 450.