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# Statistics Problems

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1. A university wants to estimate the average amount of money that students spend on books in a semester. It takes a random sample of 45 students and finds that the average amount of money spent was \$282 with a standard deviation of \$21. Find a 95% confidence interval estimate for the true mean amount of money spent on books in a semester.

2. A local public television station, which broadcasts educational programs, is soliciting contributions from its audience during pledge week. The master of ceremonies states that the average contribution of its members is \$47. do you think that this average is the mean mode or median? Why?

3. A survey found that 30% of pet owners had their pets bathed professionally rather than doing it themselves. If 18 pet owners are randomly selected find the probability that exactly 5 people have their pets professionally bathed.

4. The Statistical Bulletin published by the Metropolitan life insurance company reported 2% of all Americans birth results in twins. If a random sample of 8000 births is taken, find the mean, variance and standard deviation of the number of births that would result in twins.

5. It was found that 60% of Americans victims of health care fraud are senior citizens. If 10 victims are randomly selected, find the probability that exactly 3 are senior citizens.

6. Find the mean, mode, median, variance and standard deviation of the following data:
35
45
30
35
40
25

7. A survey of 30 adults found that the mean age of a person's primary vehicle is 5.6 years. Assuming the standard deviation of the population is .8 years, find the 95% confidence interval of the population mean.

8. Find everything to the left of Z= 2.42

9. Find everything to the right of Z= -1.68

10. Find everything in between -1.68 and 2.42

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## Statistics Problems - Regression Analysis, Autocorrelation, Multicollinearity

1. Suppose an appliance manufacturer is doing a regression analysis, using quarterly time-series data, of the factors affecting its sales of appliances. A regression equation was estimated between appliance sales (in dollars) as the dependent variable and disposable personal income and new housing starts as the independent variables. The statistical tests of the model showed large t-values for both independent variables, along with a high r2 value. However, analysis of the residuals indicated that substantial autocorrelation was present.

a. What are some of the possible causes of this autocorrelation?

b. How does this autocorrelation affect the conclusions concerning the significance of the individual explanatory variables and the overall explanatory power of the regression model?

c. Given that a person uses the model for forecasting future appliance sales, how does this autocorrelation affect the accuracy of these forecasts?

d. What techniques might be used to remove this autocorrelation from the model?

2. Suppose the appliance manufacturer discussed in Exercise 1 also developed another model, again using time-series data, where appliance sales was the dependent variable and disposable personal income and retail sales of durable goods were the independent variables. Although the r2 statistic is high, the manufacturer also suspects that serious multicollinearity exists between the two independent variables.

a. In what ways does the presence of this multicollinearity affect the results of the regression analysis?

b. Under what conditions might the presence of multicollinearity cause problems in the use of this regression equation in designing a marketing plan for appliance sales?

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