Complete the following:
Q1. The average price of a gallon of unleaded regular gasoline was reported to be $2.34 in northern Kentucky. Use this price as the population mean, and assume the population standard deviation is $0.20.
a. What is the probability that the mean price for a sample of 30 service stations is within $0.03 of the population mean?
b. What is the probability that the mean price for a sample of 50 service stations is within $0.03 of the population mean?
c. What is the probability that the mean price for a sample of 100 service stations is within $0.03 of the population mean?
d. Which if any, of the sample sizes in part a, b, and c would you recommend to have at least a .95 probability that the sample mean is within $0.03 of the population mean?
Q2. People end up tossing 12% of what they buy at the grocery store (Reader's Digest, March, 2009). Assume this is the true population proportion and that you plan to take a sample survey of 540 grocery shoppers to further investigate their behavior.
a. Show the sampling distribution of ( p¯ ), the proportion of groceries thrown out by your sample respondents.
b. What is the probability that the sample proportion will be within ±.03 of the population proportion?
c. What is the probability that your survey will provide a sample proportion within ±.015 of the population proportion?
Q3. 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 districution of p¯ .
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 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)?