# Statistics Problem

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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 questionnaire sent to owners of one of the manufacturer's full-sized cars reviewed several complaints about early transmission problems. Nationwide, the population mean mileage is 80,000 miles, with a population standard deviation 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 Attached file named 'Auto'. Use this data set to address the following questions.

1. Using the data provided above, what is the standard error of the mean? Calculate 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. 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 ± = .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? Would it be reasonable to obtain this sample size? Why/why not?

This is a statistical report, not simply a collection of different types of Excel output. It is not necessary to include the formulas you used or a copy of the dataset. When answering the questions above, make sure to include any relevant statistics and/or the results of your calculations. Like any other written report, you will want to start with an introductory paragraph or problem statement, and finish with a conclusion that summarizes the information presented.

When answering the questions, make sure to double-check your calculations. It is also important to fully address the question being asked. Be sure to include your results as well as a detailed explanation/interpretation of the information. Statistical analysis includes not just the ability to obtain the correct results, but also the ability to describe and interpret what the results mean. Instead of simply saying, 'Variable X has more variation' use your statistics to support your conclusions, Variable X has more variation because ...etc.

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This solution is comprised of detailed step-by-step calculations and analysis of the given problems related to Statistics and provides students with a clear perspective of the underlying concepts.

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- BSc, Banaras Hindu University

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