# Various quantities related to performance

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DIRECTIONS: Make sure your responses are neat and readable. Homework that is difficult to grade due to messiness will be returned ungraded. Provide all your answers in the blanks. Round off final answers to two decimal places, if appropriate. Excel must be used for the statistical analysis.

1. Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the car was set to 60 miles per hour by cruise control, and the mpg were recorded at random times.

The data for mpg values from the experiment is given in Excel under MILEAGE sheet.

a. Find the mean and the standard deviation of 20 samples? (2p)

b. Based on a 95% confidence level, what is the margin of error for the mean estimate? (2p)

c. Given the margin of error computed in part (b), give a 95% confidence interval for μ, the mean highway mpg for this vehicle. The vehicle sticker information for the vehicle states a highway average of 27 mpg. Are the results of this experiment consistent with the vehicle sticker? (6p)

2. The U.S. Department of Agriculture (USDA) uses sample surveys to produce important economic estimates. One pilot study estimated durum wheat prices in July and in January using independent samples of wheat producers in the two months.

The data is given in Excel under WHEAT PRICES sheet.

Although there is variation among prices within each month, the top four prices are all from July and the four lowest prices are from January. You will run a significance test to check that if the difference between months is just by chance. You’ll use α = 0.05 for significance test. Assume that the two Normal population distributions have the same standard deviation.

a. Write out the null and alternative hypothesis associated with the research question. (2p)

b. What type of statistical test will you use to answer the proposed research question? (Note: Is this a z-test or a t-test? (2p)

c. What is the critical value at the 0.05 level of significance? Be sure and include whether this critical value is a z or t value and, if appropriate, include the degrees of freedom associated with this statistical test. (4p)

d. What is the calculated value of the test statistic? (4p)

e. What decision should be made about the null hypothesis? In other words, should you reject or retain the null hypothesis? (3p)

g. Provide a brief conclusion regarding your findings. Use your powerpoint lecture slides for writing out the interpretation of your results. (5p)

Excel output (10p)

3. In what ways are companies that fail different from those that continue to do business? To answer this question, one study compared various characteristics of active and failed retail firms. One of the variables was the cash flow margin. Roughly speaking, this is a measure of how efficiently a company converts its sales dollars to cash and is a key profitability measure. The higher the percent, the more profitable the company.

The data for 101 companies is given in Excel under ACTIVE VERSUS FAILED RETAIL COMPANIES sheet.

You will compare the mean cash flow margin for the two groups of firms using a significance test. Take Group 1 to be the firms that were active and Group 2 to be those that failed. The question of interest is whether the mean cash flow margin is different for the two groups.

We assume that the population variances are equal and α = 0.01 for significance test.

a. Write out the null and alternative hypothesis associated with the research question. (Note: is this a one or two tailed test?) (2p)

b. What type of statistical test will you use to answer the proposed research question? (Note: Is this a z-test or a t-test? (2p)

c. What critical value is associated with the 0.01 level of significance? Be sure and include whether this critical value is a z or t value and, if appropriate, include the degrees of freedom associated with this statistical test. (4p)

d. What is the calculated value of the test statistic? (4p)

e. What decision should be made regarding the null hypothesis? (3 p)

f. Provide a brief conclusion regarding your findings. Use your powerpoint lecture slides for writing out the interpretation of your results. (5 p)

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