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# Regression Analysis

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Assistance to use algebraic and graphing concepts to solve problems involving real data.

Exercise 1.3

39. Internet Recruiting The percentage of fortune Global 500 firms that actively recruited
workers on the internet from 1998 through 2000 can be modeled by P(x) = 26.5x -
194.5 percent, where x is the number of years after 1990.

a. What is the slope of the graph of this function?
b. Interpret the slope as a rate of change.
( Source: Time, August 16, 1999)

49. Internet Users The percent of U.S. population with Internet access can be modeled by
the function

y = 5.7x + 14.61

Where x is the number of years after 1995.
a. Find the slope and the y-intercept of the graph of this equation.
b. What interpretation could be given to the y-intercept?
c. What interpretation could be given to the slope?
( Source: Jupiter Media Metrix)

from 1975 to 1996 can be modeled by the equation y =277.318x-1424.766,
where y is measured in millions of dollars and x is the number of years from 1970.
If this model remains accurate, in what years did the spending exceed \$6 billions?

Exercise 1.6

27. Smoking Cessation The data in the table below gives the percent of adults over age 20 who once smoked and quit smoking between 1965 and 1990.
a . Using an input equal to the number of years after 1960, write a linear model for the data.
b. In what year does the model indicate that the output is 39%?
c. Discuss the reliability of the answer to part (b).

Number of year after 1960 Adult who have quit smoking (%)
5 29.8
6 29.3
10 35.2
15 36
16 37
17 36.6
18 38.3
19 39
20 38.9
23 41.8
25 45
27 44.9
30 48.4
(Source: Indiana Tobacco Control Center, US. Centers for Disease control)

33. Marriage rate The marriage rate (per 1000 unmarried woman) is given by the table on the next page for the years 1960 to 1997.
a. Write the linear equation that models the marriage rate as a function of the years after 1950.
b. What does the model indicate the marriage rate to be in 1985?
c. For what year does the model indicate that the rate fell below 50 per 1000?
Year Marriage Rate ( marriage per 1000 unmarried woman)
1960 73.5
1970 76.5
1980 61.4
1990 54.5
1991 54.2
1992 53.3
1993 52.3
1994 51.5
1995 50.8
1996 49.7
1997 49.4

45. Prison Sentences The National Center for Policy Analysis calculates expected prison time by multiplying the probabilities of being arrested, of being prosecuted, and of being convicted by the length of sentence if convicted. The expected time served (in days) for each serious crime committed is shown in the table below.
a. Use the data to create a linear equation to model expected prison time in days for a serious
crime committed is shown in the table below.
b. Use the model to find the expected prison time in 1075.

Year Expected prison Time (Days)
1970 10.1
1980 10.6
1985 13.2
1990 18.0
1992 18.5
1993 17.2
1994 19.5
1995 20.2
1996 21.7

https://brainmass.com/math/interpolation-extrapolation-and-regression/regression-analysis-266527

#### Solution Summary

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

Source: Time, August 16, 1999, Jupiter Media Matrix, Federal Trade Commission, Indiana Tobacco Control Center, US. Centers for Disease control

\$2.19

## 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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