Metropolitan Research, Inc., a consumer research organization, conducts surveys designed to evaluate a wide variety of products and services available to consumers. In one particular study, Metropolitan looked at consumer satisfaction with the performance of automobiles produced by a major Detroit manufacturer. A problemnaire sent to owners of one of the manufacturer's full-sized cars reviewed several complaints about early transmission problems. Nationwide, the population mean mileage, μ, is 80,000 miles, with a population standard deviation, σ, of 18,000 miles.
To learn more about the transmission failures, Metropolitan used a sample of actual transmission repairs provided by a transmission repair firm in the Detroit area. The data are shown in the CD file named "Auto" (Chapter 9 datasets). Use this data set to address the following problems.
1. Using the data provided above, what is the standard error of the mean? find out the probability that a simple random sample of 50 vehicles that will provide an estimate of the population mean mileage at transmission failure within +/- 4,000 miles of the population mean, μ. Clearly state your conclusions based on this information.
2. Use sample data contained in the dataset to develop point estimates of the population mean and the population standard deviation for mileage at transmission failure. Do these sample statistics contain any sampling error? If so, how large are the errors? Is there any way the research firm can limit the amount of sampling error when selecting a sample of vehicles?
3. At 95% confidence, what is the margin of error? What is the 95% confidence interval estimate of the population mean? Clearly state your conclusions based on this information. What would happen to the confidence interval if you increased the sample size to 200? Why? What are the new margin of error and confidence interval?
4. Use hypothesis testing to determine whether the sample data support the conclusion that the mean mileage of vehicles produced by this manufacturer is lower than the national average mileage at transmission failure. Be sure to clearly state your hypotheses. Use the p-value approach to conduct the hypothesis test using α = .005. Include the test statistic, p-values, rejection rule, and conclusions in your answer.
5. Assuming 95% confidence, what is the recommended sample size if the desired margin of error is 1,000 miles?