Data Analysis for Managers

[1] An earlier study claims that U.S. adults spend an average of 114 minutes with their families per day. A recently taken sample of 25 adults showed that they spend an average of 109 minutes per day with their families. The sample standard deviation is 11 minutes. Assume that the time spent by adults with their families has an approximate normal distribution. We wish to test whether the mean time spent currently by all adults with their families is less than 114 minutes a day.

1. Set up the null and alternative hypotheses.
2. Select the distribution to use. Explain briefly why you selected it.
3. Using the 2% significance level, determine the rejection and nonrejection regions based on your hypotheses in a). State the critical value.
4. Calculate the value of the test statistic.
5. Does the sample information support that the mean time spent currently by all adults with their families is less than 114 minutes a day? Explain your conclusion in words.

[2] In March 16, 1998, issue of Fortune magazine, the results of a survey of 2,221 MBA students from across the United States conducted by the Stockholm-based academic consulting firm Universum showed that only 20 percent of MBA students expect to stay at their first job five years or more. (Source: Shalley Branch, "MBAs: What Do They Really Want," Fortune (March 16, 1998), p.167.)

1. Assuming that a random sample was employed, test hypotheses at the 2% level of significance for the proportion 25% of all U.S. MBA students who expect to stay at their first job five years or more.
2. Based on the interval from a), is there strong evidence that less than one-fourth of all U.S. MBA students expect to stay?

[3] Suppose that the federal government proposes to give a substantial tax break to automakers producing midsize cars that get a mean mileage that exceeds 31 mpg. Letting  be a midsize car's mean mileage, the government will award the tax credit if an automaker is able to reject H ₀ :  \[\le \] 31 in favor of H ₁ :  > 31 at the 0.05 level of significance. Suppose that the sample of 49 mileages has mean \[\bar{x}\] = 31.5531 and standard deviation s = 0.7992. Use these sample results and hypothesis testing to test H ₀ versus H ₁ at the 0.05 level of significance.

1. Will the automaker be awarded the tax break? Justify your answer.
2. Would the tax break have been awarded if the federal government had set  equal to 0.01? Justify your answer.

1. If you wish to conduct a hypothesis test to determine whether the check verification system reduces the probability that a bad check will be cashed, what should you choose for the null hypothesis? The alternative hypothesis?
2. Does your alternative hypothesis in part a) imply a one-or two-tailed test? Explain.
3. Select the distribution to test the hypotheses. Explain briefly why you selected it.
4. Noting the data, what does your intuition tell you? Do you think that the check verification system is effective in reducing the proportion of bad checks that were cashed?
5. Conduct a hypothesis test based on your hypotheses in a), and state your conclusion. Test using =.05. Do the test conclusions agree with your intuition in part d)?

[5] The average retail price for bananas in 1994 was 46.0 cents per pound, as reported by the U.S. Department of Agriculture in Food Cost Review . Recently, a random sample of 15 markets gave the following prices for bananas in cents per pound.

We wish to conduct a hypothesis test to determine whether the current mean retail price for bananas is different from that of 1994 with a significance level of 0.05.

1. Set up the hypotheses.
2. Select the distribution to test the hypotheses. Explain briefly why you selected it.
3. Draw a graph indicating the critical value(s), and rejection region based on your hypotheses in a).
4. Calculate the value of the test statistic and the p-value.
5. Applying the hypothesis test, can we conclude that the current mean retail price for bananas is different from the 1994 mean of 46.0 cents per pound?

1. State the null and alternative hypotheses, letting  represent the mean waiting time under the new system.
2. Select the distribution to use. Explain briefly why you selected it.
3. Assuming that she wishes to test the claim at =0.05, determine the rejection and non rejection regions based on your hypotheses in a). State the critical value.
4. Calculate the value of the test statistic.
5. What do you conclude about whether the new system has reduced the mean waiting time to below six minutes? Explain your conclusion in words.

[7] A consulting agency was asked by a large insurance company to investigate if business majors were better salespersons. A sample of 40 salespersons with a business degree showed that they sold an average of 10 insurance policies per week with a standard deviation of 1.80 policies. Another sample of 45 salespersons with a degree other than business showed that they sold an average of 9.5 insurance policies per week with a standard deviation of 1.35 policies. Assuming that the independence assumption holds, and letting  ₁ = the mean insurance policies sold by salespersons with business majors, and  ₂ = the mean insurance policies sold by salespersons with non-business majors.

1. The values of the two population standard deviations are not known. Do you think that  ₁ and  ₂ are equal? Explain briefly if you need to make such an assumption in order to use the pooled-variance t test.
2. Set up the null and alternative hypotheses that should be used to attempt to justify that the mean sales by business majors is higher than the mean sales by non-business majors.
3. Test the hypotheses that you set up in b) and whatever assumption you made in a) with  = 0.05. Interpret the results of this test.

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