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# Applied Statistics in Business and Economics - Regression

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Prepare answers to the following assignments from the e-text, Applied Statistics in Business and Economics, by Doane and Seward:

12.48 In the following regression, X = weekly pay, Y = income tax withheld, and n = 35 McDonald's employees. (a) Write the fitted regression equation. (b) State the degrees of freedom for a two-tailed test for zero slope, and use Appendix D to find the critical value at alpha = .05. (c) What is your conclusion about the slope? (d) Interpret the 95 percent confidence limits for the slope. (e) Verify that F = t2 for the slope. (f) In your own words, describe the fit of this regression.

12.50 In the following regression, X = total assets (\$ billions), Y = total revenue (\$ billions), and n = 64 large banks. (a) Write the fitted regression equation. (b) State the degrees of freedom for a two tailed test for zero slope, and use Appendix D to find the critical value at alpha = .05. (c) What is your conclusion about the slope? (d) Interpret the 95 percent confidence limits for the slope. (e) Verify that F = t2 for the slope. (f) In your own words, describe the fit of this regression.

13.30 A researcher used stepwise regression to create regression models to predict BirthRate (births per 1,000) using five predictors: LifeExp (life expectancy in years), InfMort (infant mortality rate), Density (population density per square kilometer), GDPCap (Gross Domestic Product per capita), and Literate (literacy percent). Interpret these results. BirthRates2

Regression Analysis-Stepwise Selection (best model of each size)

153 observations
BirthRate is the dependent variable

p-values for the coefficients
Nvar LifeExp InfMort Density GDPCap Literate s Adj R2 R2
1 .0000 6.318 .722 .724
2 .0000 .0000 5.334 .802 .805
3 .0000 .0242 .0000 5.261 .807 .811
4 .5764 .0000 .0311 .0000 5.273 .806 .812
5 .5937 .0000 .6289 .0440 .0000 5.287 .805 .812

13.32 An expert witness in a case of alleged racial discrimination in a state university school of nursing introduced a regression of the determinants of Salary of each professor for each year during an 8-year period (n = 423) with the following results, with dependent variable Year (year in which the salary was observed) and predictors YearHire (year when the individual was hired), Race (1 if individual is black, 0 otherwise), and Rank (1 if individual is an assistant professor, 0 otherwise).

Interpret these results.

Variable Coefficient t p
Intercept -3,816,521 -29.4 .000
Year 1,948 29.8 .000
YearHire -826 -5.5 .000
Race -2,093 -4.3 .000
Rank -6,438 -22.3 .000
R2 = 0.811 R2
adj = 0.809 s = 3,318

14.16 (a) Plot the data on U.S. general aviation shipments. (b) Describe the pattern and discuss possible causes. (c) Would a fitted trend be helpful? Explain. (d) Make a similar graph for 1992-2003 only. Would a fitted trend be helpful in making a prediction for 2004? (e) Fit a trend model of your choice to the 1992-2003 data. (f) Make a forecast for 2004, using either the fitted trend model or a judgment forecast. Why is it best to ignore earlier years in this data set? Airplanes

U.S. Manufactured General Aviation Shipments, 1966-2003
Year Planes Year Planes Year Planes Year Planes
1966 15,587 1976 15,451 1986 1,495 1996 1,053
1967 13,484 1977 16,904 1987 1,085 1997 1,482
1968 13,556 1978 17,811 1988 1,143 1998 2,115
1969 12,407 1979 17,048 1989 1,535 1999 2,421
1970 7,277 1980 11,877 1990 1,134 2000 2,714
1971 7,346 1981 9,457 1991 1,021 2001 2,538
1972 9,774 1982 4,266 1992 856 2002 2,169
1973 13,646 1983 2,691 1993 870 2003 2,090
1974 14,166 1984 2,431 1994 881
1975 14,056 1985 2,029 1995 1,028

Source: U.S. Manufactured General Aviation Shipments, Statistical Databook 2003, General Aviation Manufacturers Association, used with permission.

Note: Methods of computation could include the usage of Excel®, SPSS®, Lotus®, SAS®, MINITAB®, or by hand computation

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#### Solution Summary

A complete, neat and step-by-step solution is provided in the attached files. It goes through hypothesis testing of each problem.

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## Applied Statistics in Business and Economics

Correlation & Regression Analysis

Do heavier cars use more gasoline? If so, can we predict the mileage rating of a car given its weight? Suppose 8 cars were randomly chosen and their weights (in hundreds of pounds) and mileage rating (mpg) are recorded.

Weight (x) MPG (y)
27 30
44 19
32 24
47 13
23 29
40 17
34 21
52 14

A) A) If we want to estimate gas consumption based on the weight of car, which variable is the dependent variable and which is the independent variable?
B) B) Draw a scatter diagram [first enter the data in Excel column A & B. Use MegaStat, correlation/regression, scatterplot; or use Excel, Tools, data analysis, regression function to answer questions c-e].
C) C) What is the value of coefficient of correlation (Use Megastat or Excel, Tools, Data Analysis, Regression function)? Is there a linear relationship between the two variables? If so, how strong is the relationship?
D) D) Determine the coefficient of determination (Use Megastat or Excel, Tools, Data Analysis, Regression function). Interpret the result.
E) Write the simple linear regression equation for the least square regression line that shows the relationship between weight of cars and gas consumption (use Megastat or Excel regression coefficients table).
F) Interpret the meanings of regression coefficients (slope and intercept) in the above equation.
G) Predict the mileage rating for a car that weighs 6,000 pounds?

Part 2. A sociologist claims that the success of students in college (measured by their GPA) is related to their family income. For a sample of 20 students, the coefficient of correlation is 0.40.
Using the alpha 0.01 level of significance, can we conclude that there is a positive association between the two variables - family income and students' GPA? What is the p-value? Interpret. Test using correlation test of hypothesis formula: [see the model example posted in Suepplementary Reading on Correction & Regression posted in Discusson/course materials forum].

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