# Regression, Time series and hypothesis testing

Chapter Exercises

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 α = .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 .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 twotailed

test for zero slope, and use Appendix D to find the critical value at α = .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 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 yourchoice 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.

https://brainmass.com/statistics/regression-analysis/regression-time-series-and-hypothesis-testing-218796

#### Solution Summary

The solution provides step by step method for the calculation of regression, trend analysis, hypothesis testing, and confidence interval problems . Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.