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# Regression analysis & ANOVA

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Need step by step instructions of how to solve these problems:

2. The regression equation NetIncome = 2,277 + .0307 Revenue was fitted from a sample of 100 leading world companies (variables are in millions of dollars). (a) Interpret the slope. (b) Make a prediction of NetIncome when Revenue = 1,000.

3. The regression equation Credits = 15.4 â?' .07 Work was fitted from a sample of 21 statistics students. Credits is the number of college credits taken and Work is the number of hours worked per week at an outside job. (a) Interpret the slope. (b) Make a prediction of Credits when Work = 0 and when Work = 40. What do these predictions tell you?

4. Refer to the ANOVA table for this regression. (a) State the degrees of freedom for the F test for overall significance. (b) Use Appendix F to look up the critical value of F for ? = .05. (c) Calculate the F statistic. Is the regression significant overall? (d) Calculate R2 and R2adj, showing your formulas clearly.

Source d.f SS MS
Regression
3 1,196,410 398,803
Error 26 379,332 14,590
Total 29 1,575,742

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 ? = .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.

R² 0.202
Std. Error 6.816
n 35

ANOVA table
Source SS df MS F p-value
Regression 387.6959 1 387.6959 8.35 0.0068
Residual 1,533.0614 33 46.4564
Total 1,920.7573 34

Regression output
variables coefficients std. error t (df=33) p-value 95% lower 95% upper
Intercept 30.7963 6.4078 4.806 0.0000 17.7595 43.8331
Slope 0.0343 0.0119 2.889 0.0068 0.0101 0.0584

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 ? = .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.

R² 0.519
Std. Error 6.977
n 64

ANOVA table
Source SS df MS F p-value
Regression 3,260.0981 1 3,260.0981 66.97 1.90E-11
Residual 3,018.3339 62 48.6828
Total 6,278.4320 63

Regression output
variables coefficients std. error t (df=62) p-value 95% lower 95% upper
Intercept 6.5763 1.9254 3.416 0.0011 2.7275 10.4252
X1 0.0452 0.0055 8.183 1.90E-11 0.0342 0.0563

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.

Regression Analysisâ?"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 0.0000 6.318 0.722 0.724
2 0.0000 0.0000 5.334 0.802 0.805
3 0.0000 0.0242 0.0000 5.261 0.807 0.811
4 0.5764 0.0000 0.0311 0.0000 5.273 0.806 0.812
5 0.5937 0.0000 0.6289 0.0440 0.0000 5.287 0.805 0.812

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 0.000
Year 1,948 29.8 0.000
YearHire -826 -5.5 0.000
Race -2,093 -4.3 0.000
Rank -6,438 -22.3 0.000
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 1992â?"2003 only. Would a fitted trend be helpful in making a prediction for 2004? (e) Fit a trend model of your choice to the 1992â?"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?

US 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

#### Solution Summary

The solution provides step by step method for the calculation of regression model and ANOVA. Formula for the calculation and Interpretations of the results are also included.

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