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(a) How does correlation analysis...

(a) How does correlation analysis differ from regression analysis? (b) What does a correlation coefficient reveal? (c) State the quick rule for a significant correlation and explain its limitations. (d) What sums are needed to calculate a correlation coefficient? (e) What are the two ways of testing a correlation coefficient for significance?

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 a = .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(NNN) NNN-NNNN1(NNN) NNN-NNNN8.35 .0068

Residual 1,533.0614 33 46.4564

Total 1,920.7573 34

Regression output confidence interval

variables coefficients std. error t (df =33) p-value 95% lower 95% upper

Intercept 30.7963 6.4078 4.806 .0000 17.7595 43.8331

Slope 0.0343 0.0119 2.889 .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 find the critical value at a = .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 confidence interval

variables coefficients std. error t (df =62) p-value 95% lower 95% upper

Intercept 6.5763 1.9254 3.416 .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. BirthRates2

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

153 observations

BirthRate is the dependent variable

p-values for the coefficients

Nvar &nbs p; 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 & nbsp; .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

.S. Manufactured General Aviation Shipments, 1966–2003

Year Planes

1966 15,587

1967 13,484

1968 13,556

1969 12,407

1970 7,277

1971 7,346

1972 9,774

1973 13,646

1974 14,166

1975 14,056

1976 15,451

1977 16,904

1978 17,811

1979 17,048

1980 11,877

1981 9,457

1982 4,266

1983 2,691

1984 2,431

1985 2,029

1986 1,495

1987 1,085

1988 1,143

1989 1,535

1990 1,134

1991 1,021

1992 856

1993 870

1994 881

1995 1,028

1996 1,053

1997 1,482

1998 2,115

1999 2,421

2000 2,714

2001 2,538

2002 2,169

2003 2,090

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 a = .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(NNN) NNN-NNNN1(NNN) NNN-NNNN8.35 .0068

Residual 1,533.0614 33 46.4564

Total 1,920.7573 34

Regression output confidence interval

variables coefficients std. error t (df =33) p-value 95% lower 95% upper

Intercept 30.7963 6.4078 4.806 .0000 17.7595 43.8331

Slope 0.0343 0.0119 2.889 .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 find the critical value at a = .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 confidence interval

variables coefficients std. error t (df =62) p-value 95% lower 95% upper

Intercept 6.5763 1.9254 3.416 .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. BirthRates2

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

153 observations

BirthRate is the dependent variable

p-values for the coefficients

Nvar &nbs p; 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 & nbsp; .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

.S. Manufactured General Aviation Shipments, 1966–2003

Year Planes

1966 15,587

1967 13,484

1968 13,556

1969 12,407

1970 7,277

1971 7,346

1972 9,774

1973 13,646

1974 14,166

1975 14,056

1976 15,451

1977 16,904

1978 17,811

1979 17,048

1980 11,877

1981 9,457

1982 4,266

1983 2,691

1984 2,431

1985 2,029

1986 1,495

1987 1,085

1988 1,143

1989 1,535

1990 1,134

1991 1,021

1992 856

1993 870

1994 881

1995 1,028

1996 1,053

1997 1,482

1998 2,115

1999 2,421

2000 2,714

2001 2,538

2002 2,169

2003 2,090

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