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20.1 Forrest Tucker, the head statistician for the National Parks Service, believes that park usage as measured by number of visitors (Y) is a function of the number of people who live within 200 miles of the park (X), the number of camping hookups available (X2), and the mean annual temperature at the park (X3). For a sample of 200 parks under Forrest's supervision, the following regression is calculated: 8 147 + .0212X1 + 15.4X2 + 186X3 $6,0157 = 12.4 sb = 10.4 R2 = .50 Adj. R2 = .43 For this regression, what can you tell Forrest? Write a one-page memo with your assessment. 20.2 If Janice Position-Classification, personnel officer for the Bureau of Forms, can forecast agency separations 6 months from now, she can plan recruitment efforts to replace these people. Janice believes that separations 6 months from now are determined by the number of agency people passed over for promotion (X1), the number of agency people 64 years old or older (X2), and the ratio of government salaries to private sector salaries (X3). Using regression, Janice finds the following: 8 + .35X1 + .54X2 - 271X3 sb,==.0031 sb,==0.136 263 Sylx = 54 R2 = .89 Adj. R2 = .85 N = 214 Write a one-page memo explaining the results, and then forecast the number of separations if 418 people are passed over for promotion, 327 people are 64 years old or older, and government salaries equal those in the private sector. 20.5 Lieutenant Colonel Syl Verleaf is placed in charge of base security at all 240 mili- tary bases in Europe. Verleaf believes that the crime rate is positively correlated to the size of the base, the percentage of troops without high school degrees, and the number of women on base. Verleaf's statistician finds the following: 47.3 + .031X1 + 2.4X2 - .065X3 = .0021 Sby =3.0 = .0027 Sylx = 17.1 R==.80 Adj. R2 = .78 where X is the number of troops on the base, X2 is the percentage of troops with- out high school degrees, X3 is the number of women on the base, and Y is the number of serious crimes in a month. Interpret all the regression coefficients, R2 and the intercept. What is the most important independent variable? Ramstein Air Base has 15,000 troops, 42% of its troops have no high school degree, and there are 3,000 women on base. What is your best estimate of the number of crimes per month for this base? 21.1 The Director of Economic Development in Potto Gulch, Wisconsin, is inter- ested in studying the relationship between business activity and spending on social welfare programs. Specifically, she hypothesizes that as business activ- ity increases the boost to the local economy (mainly through job creation) should result in lower spending on programs for the poor. To test this hy- pothesis, the director gathers annual data for the last 11 years on three vari- ables. The dependent variable is annual spending on social welfare programs (in millions of dollars). The director feels that the number of business per- mits issued each year is a good indicator of economic activity and selects this as the first independent variable (BPRMT). The second independent variable is the number of residents in Potto Gulch (POP). The regression output is displayed below. Model Summary Std. Error of Model R R2 Adj. R2 the Estimate 1 .791 .626 .532 800,770.013 Predictors: (Constant), BPRMT, POP. ANOVA Model Sum of Squares df Mean Square F Significance Regression 8,568,393,095,762.110 2 4,284,196,547,881.058 6,681 .020 1 Residual 5,129,860,913,328.790 8 641,232,614,166.099 Total 13,698,254,009,090.910 10 Predictors: (Constant), BPRMT, POP. Dependent variable: EXPENDIT. Coefficients Unstandardized Standardized Coefficients Coefficients Model B Std. Error Beta t Significance (Constant) 3,655,278.783 8,482,054.890 .431 .678 1 POP 109.855 100.721 .293 1.091 .307 BPRMT -5,049.119 2,339.638 -.581 -2.158 .063 Dependent variable: EXPENDIT. (a) Interpret the slopes and intercept for this model. What substantive conclu- sion can the director make about the relationship between business activity and spending on social welfare programs? (b) The director is concerned about the drop in the explanatory power of the model indicated by the adjusted R2 value. What is probably the reason why the adjusted R2 is lower than R2? (c) The director has located data on the actual number of Potto Gulch citizens receiving social welfare assistance each year. If the director includes this new variable in the model, which of the current independent variables should be removed from the model? Explain. 21.2 The director of the Northern Tennessee Association of Local Governments is conducting a study on differences in average property taxes (in thousands of dollars) that residents in 15 local communities pay on their homes. The direc- tor believes that variations in property taxes are a function of two variables: the number of government employees on staff in each city (EMPLOYEES) and city size, measured in square miles (SIZE). The director uses SPSS to generate the fol- lowing regression, where the average property tax per residence is the dependent variable. Model Summary Std. Error of Model R R2 Adj. R2 the Estimate 1 .983 .967 .961 112.85532 Predictors: (Constant), SIZE, EXPLOYEES ANOVA Model Sum of Squares df Mean Square F Significance Regression 4,421,115.713 2 2,210,557.857 173.563 .000a 1 Residual 152,835.887 12 12,736.324 Total 4,573,951.600 14 Predictors: (Constant), SIZE EMPLOYEES beDendent variable: PTAX. Coefficients Unstandardized Standardized Coefficients Coefficients Model B Std. Error Beta t Significance (Constant) 938.736 336.866 2.787 .016 1 Employees 8.225 2.631 .564 3.126 .009 Size 47.532 19.794 - -.430 2.380 .035 Dependent variable: PTAX. (a) Interpret the intercept, slopes, and R2 values. (b) Interpret the standardized coefficients. Explain how the standardized coefficients differ in magnitude from the unstandardized coefficients.

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