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Exam 14: Introduction to Multiple Regression

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Question 71

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TABLE 14-16 The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: $\begin{array}{lr}\hline\text {Regression Statistics } \\\hline \text {Multiple R }& 0.7930 \\\text {R Square} & 0.6288 \\\text {Adjusted R Square} & 0.6029 \\\text {Standard Error }& 10.4570 \\\text {Observations }& 47 \\\hline\end{array}$ ANOVA $\begin{array}{lccccc}\hline &\text { d f } &\text { SS }& \text { MS }& \text { F } & \text { Significance F} \\\hline \text { Regression }& 3 & 7965.08 & 2655.03 & 24.2802 & 2.3853 \mathrm{E}-09 \\\text { Residual} & 43 & 4702.02 & 109.35 & & \\\text { Total }& 46 & 12667.11 & & & \\\hline\end{array}$ $\begin{array}{lrrrrrr}\hline &\text { Coeffs} & \text { Stnd Err} &\text { t Stat} &\text { p -value }&\text { Lower 95\%} \text { Upper 95\%} \\\hline\text { Intercept }& -753.4225 & 101.1149 & -7.4511 & 2.88 \mathrm{E}-09 & -957.3401 & -549.5050 \\\%\text { Attend }& 8.5014 & 1.0771 & 7.8929 & 6.73 \mathrm{E}-10 & 6.3292 & 10.6735 \\\text { Salary} & 6.85 \mathrm{E}-07 & 0.0006 & 0.0011 & 0.9991 & -0.0013 & 0.0013 \\\text { Spending }& 0.0060 & 0.0046 & 1.2879 & 0.2047 & -0.0034 & 0.0153 \\\hline\end{array}$ -Referring to Table 14-16, what are the lower and upper limits of the 95% confidence interval estimate for the effect of a one dollar increase in instructional spending per pupil on average percentage of students passing the proficiency test?

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Question 72

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TABLE 14-17 The marketing manager for a nationally franchised lawn service company would like to study the characteristics that differentiate home owners who do and do not have a lawn service. A random sample of 30 home owners located in a suburban area near a large city was selected; 15 did not have a lawn service (code 0) and 15 had a lawn service (code 1). Additional information available concerning these 30 home owners includes family income (Income, in thousands of dollars), lawn size (Lawn Size, in thousands of square feet), attitude toward outdoor recreational activities (Attitude 0 = unfavorable, 1 = favorable), number of teenagers in the household (Teenager), and age of the head of the household (Age). The Minitab output is given below: $\begin{array}{lccccccrr} & & & & \text { Odds } & \text { 95 \% CI } \\\text {Predictor }& \text {Coef }&\text { SE Coef }&\text { Z }& \text {P} &\text { Ratio} &\text { Lower} & \text {Upper }\\\text {Constant }& -70.49 & 47.22 & -1.49 & 0.135 & & & \\\text {Income }& 0.2868 & 0.1523 & 1.88 & 0.060 & 1.33 & 0.99 & 1.80 \\\text {Lawn Size} & 1.0647 & 0.7472 & 1.42 & 0.154 & 2.90 & 0.67 & 12.54 \\\text {Attitude} & -12.744 & 9.455 & -1.35 & 0.178 & 0.00 & 0.00 & 326.06 \\\text {Teenager} & -0.200 & 1.061 & -0.19 & 0.850 & 0.82 & 0.10 & 6.56 \\\text {Age} & 1.0792 & 0.8783 & 1.23 & 0.219 & 2.94 & 0.53 & 16.45\end{array}$ Log-Likelihood = -4.890 Test that all slopes are zero: G = 31.808, DF = 5, P-Value = 0.000 Goodness-of-Fit Tests $\begin{array}{lrrr}\text { Method } & \text { Chi-Square } & \text { DF } & \text { P } \\ \text { Pearson } & 9.313 & 24 & 0.997 \\ \text { Deviance } & 9.780 & 24 & 0.995 \\ \text { Hosmer-Lemeshow } & 0.571 & 8 & 1.000\end{array}$ -Referring to Table 14-17, what is the p-value of the test statistic when testing whether Income makes a significant contribution to the model in the presence of the other independent variables?

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TABLE 14-10 You worked as an intern at We Always Win Car Insurance Company last summer. You notice that individual car insurance premiums depend very much on the age of the individual, the number of traffic tickets received by the individual, and the population density of the city in which the individual lives. You performed a regression analysis in EXCEL and obtained the following information: Regression Analysis $\begin{array}{lr}\hline\text {Regression Statistics } \\\hline\text { Multiple R} & 0.63 \\\text {R Square }& 0.40 \\\text {Adjusted R Square} & 0.23 \\\text {Standard Error }& 50.00 \\\text {Observations} & 15.00 \\\hline\end{array}$ ANOVA $\begin{array}{lccccc}\hline & \text { d f} & \text {SS} & \text {MS} & \text {F } & \text {Significance F } \\\hline \text {Regression} & 3 & & 5994.24 & 2.40 & 0.12 \\\text {Residual }& 11 & 27496.82 & & & \\\text {Total }& 45479.54 & & & \\\hline\end{array}$ $\begin{array}{lccrrrr}\hline & \text {oefficients}&\text { Standard Error }& \text { t Stat} &\text { p-value }& \text {Lower 99.0\% }& \text {Upper 99.0 \% } \\\hline\text { Intercept} & 123.80 & 48.71 & 2.54 & 0.03 & -27.47 & 275.07 \\\text {AGE }& -0.82 & 0.87 & -0.95 & 0.36 & -3.51 & 1.87 \\\text {TICKETS}& 21.25 & 10.66 & 1.99 & 0.07 & -11.86 & 54.37 \\\text {DENSITY} & -3.14 & 6.46 & -0.49 & 0.64 & -23.19 & 16.91 \\\hline\end{array}$ -Referring to Table 14-10, the residual mean squares (MSE) that are missing in the ANOVA table should be ______.

TABLE 14-5 A microeconomist wants to determine how corporate sales are influenced by capital and wage spending by companies. She proceeds to randomly select 26 large corporations and record information in millions of dollars. The Microsoft Excel output below shows results of this multiple regression. $\begin{array}{l}\text { Regression Statistics }\\\begin{array} { l r } \hline \text { Multiple R } & 0.830 \\\text { R Square } & 0.689 \\\text { Adjusted R Square } & 0.662 \\\text { Standard Error } & 17501.643 \\\text { Observations } & 26\end{array}\end{array}$ ANOVA $\begin{array}{lcrrrr}\hline & \text { d f }&\text { S S } & \text { M S } & \text { F }&\text { Significance F } \\\hline \text {Regression} & 2 & 15579777040 & 7789888520 & 25.432 & 0.0001 \\\text {Residual }& 23 & 7045072780 & 306307512 & & \\\text {Total }& 25 & 22624849820 & & & \\\hline\end{array}$ $\begin{array} { l c c c c } & \text { Coefficients } & \text { Standard Error} & \text { t Stat } & \text { p-value } \\\text { Intercept } & 15800.0000 & 6038.2999 & 2.617 & 0.0154 \\\text { C apital } & 0.1245 & 0.2045 & 0.609 & 0.5485 \\\text { W ages } & 7.0762 & 1.4729 & 4.804 & 0.0001 \\\hline\end{array}$ -Referring to Table 14-5, what is the p-value for testing whether Capital has a negative influence on corporate sales?

TABLE 14-16 The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: $\begin{array}{lr}\hline\text {Regression Statistics } \\\hline \text {Multiple R }& 0.7930 \\\text {R Square} & 0.6288 \\\text {Adjusted R Square} & 0.6029 \\\text {Standard Error }& 10.4570 \\\text {Observations }& 47 \\\hline\end{array}$ ANOVA $\begin{array}{lccccc}\hline &\text { d f } &\text { SS }& \text { MS }& \text { F } & \text { Significance F} \\\hline \text { Regression }& 3 & 7965.08 & 2655.03 & 24.2802 & 2.3853 \mathrm{E}-09 \\\text { Residual} & 43 & 4702.02 & 109.35 & & \\\text { Total }& 46 & 12667.11 & & & \\\hline\end{array}$ $\begin{array}{lrrrrrr}\hline &\text { Coeffs} & \text { Stnd Err} &\text { t Stat} &\text { p -value }&\text { Lower 95\%} \text { Upper 95\%} \\\hline\text { Intercept }& -753.4225 & 101.1149 & -7.4511 & 2.88 \mathrm{E}-09 & -957.3401 & -549.5050 \\\%\text { Attend }& 8.5014 & 1.0771 & 7.8929 & 6.73 \mathrm{E}-10 & 6.3292 & 10.6735 \\\text { Salary} & 6.85 \mathrm{E}-07 & 0.0006 & 0.0011 & 0.9991 & -0.0013 & 0.0013 \\\text { Spending }& 0.0060 & 0.0046 & 1.2879 & 0.2047 & -0.0034 & 0.0153 \\\hline\end{array}$ -Referring to Table 14-16, the alternative hypothesis H1 : At least one of þj × 0 for j = 1, 2, 3 implies that percentage of students passing the proficiency test is related to all of the explanatory variables.

TABLE 14-16 The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing) , daily average of the percentage of students attending class (% Attendance) , average teacher salary in dollars (Salaries) , and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: $\begin{array}{lr}\hline\text {Regression Statistics } \\\hline \text {Multiple R }& 0.7930 \\\text {R Square} & 0.6288 \\\text {Adjusted R Square} & 0.6029 \\\text {Standard Error }& 10.4570 \\\text {Observations }& 47 \\\hline\end{array}$ ANOVA $\begin{array}{lccccc}\hline &\text { d f } &\text { SS }& \text { MS }& \text { F } & \text { Significance F} \\\hline \text { Regression }& 3 & 7965.08 & 2655.03 & 24.2802 & 2.3853 \mathrm{E}-09 \\\text { Residual} & 43 & 4702.02 & 109.35 & & \\\text { Total }& 46 & 12667.11 & & & \\\hline\end{array}$ $\begin{array}{lrrrrrr}\hline &\text { Coeffs} & \text { Stnd Err} &\text { t Stat} &\text { p -value }&\text { Lower 95\%} \text { Upper 95\%} \\\hline\text { Intercept }& -753.4225 & 101.1149 & -7.4511 & 2.88 \mathrm{E}-09 & -957.3401 & -549.5050 \\\%\text { Attend }& 8.5014 & 1.0771 & 7.8929 & 6.73 \mathrm{E}-10 & 6.3292 & 10.6735 \\\text { Salary} & 6.85 \mathrm{E}-07 & 0.0006 & 0.0011 & 0.9991 & -0.0013 & 0.0013 \\\text { Spending }& 0.0060 & 0.0046 & 1.2879 & 0.2047 & -0.0034 & 0.0153 \\\hline\end{array}$ -Referring to Table 14-16, which of the following is the correct null hypothesis to test whether daily average of the percentage of students attending class has any effect on percentage of students passing the proficiency test?

TABLE 14-8 A financial analyst wanted to examine the relationship between salary (in $1,000) and 4 variables: age (X₁ = Age), experience in the field (X₂ = Exper), number of degrees (X₃ = Degrees), and number of previous jobs in the field (X₄ = Prevjobs). He took a sample of 20 employees and obtained the following Microsoft Excel output: $\text {SUMMARY OUTPUT}$ $\begin{array}{ll}\hline \text { Regression Statistics } \\\hline \text { Multiple R }& 0.992 \\ \text { R Square} & 0.984 \\ \text { Adjusted R Square} & 0.979 \\ \text { Standard Error }& 2.26743 \\ \text { Observations} & 20 \\\hline\end{array}$ ANOVA $\begin{array}{lccclc}\hline & \text {d f }& \text { SS }& \text { M S } & \text { F } & \text {Significance F } \\\hline \text { Regression} & 4 & 4609.83164& 1152.45791 & 224.160& 0.0001 \\ \text {Residual} &15& 77.11836 & 5.14122& & \\ \text {Total} & 19 & 4686.95000 & & & \\\hline\end{array}$ $\begin{array}{lrrrr}\hline & \text {Coefficients} &\text { Standard Error} & \text { t Stat } & \text { p -value} \\\hline \text { Intercept }& -9.611198 & 2.77988638 & -3.457 & 0.0035 \\\text { Age }& 1.327695 & 0.11491930 & 11.553 & 0.0001 \\\text { Exper }& -0.106705 & 0.14265559 & -0.748 & 0.4660 \\\text { Degrees} & 7.311332 & 0.80324187 & 9.102 & 0.0001 \\\text { Prevjobs }& -0.504168 & 0.44771573 & -1.126 & 0.2778 \\\hline\end{array}$ -Referring to Table 14-8, the value of the coefficient of multiple determination, r² Y.1234, is____

TABLE 14-3 An economist is interested to see how consumption for an economy (in $ billions) is influenced by gross domestic product ($ billions) and aggregate price (consumer price index) . The Microsoft Excel output of this regression is partially reproduced below. SUMMARY OUTPUT $\begin{array}{lc}{\text { Regression Statistics }} \\\hline \text { Multiple R } & 0.991 \\\text { R Square } & 0.982 \\\text { Adjusted R Square }& 0.976 \\\text { Standard Error } & 0.299 \\\text { Observations } & 10 \\\hline\end{array}$ ANOVA $\begin{array}{lrrrrr}\hline & \text { d f}& \text { SS } & \text {MS } & \text { F } & \text {Significance F } \\\hline \text { Regression} & 2 & 33.4163 & 16.7082 & 186.325 & 0.0001 \\ \text {Residual }& 7 & 0.6277 & 0.0897 & & \\ \text {Total} & 9 & 34.0440 & & & \\\hline\end{array}$ $\begin{array}{lcccr}\hline & \text {Coefficients} & \text { Standard Error} & \text { t Stat }& \text { p -value} \\\hline \text {Intercept }& -0.0861 & 0.5674 & -0.152 & 0.8837 \\ \text {GDP} & 0.7654 & 0.0574 & 13.340 & 0.0001 \\ \text {Price }& -0.0006 & 0.0028 & -0.219 & 0.8330 \\\hline\end{array}$ -Referring to Table 14-3, to test whether aggregate price index has a positive impact on consumption, the p-value is

TABLE 14-5 A microeconomist wants to determine how corporate sales are influenced by capital and wage spending by companies. She proceeds to randomly select 26 large corporations and record information in millions of dollars. The Microsoft Excel output below shows results of this multiple regression. $\begin{array}{l}\text { Regression Statistics }\\\begin{array} { l r } \hline \text { Multiple R } & 0.830 \\\text { R Square } & 0.689 \\\text { Adjusted R Square } & 0.662 \\\text { Standard Error } & 17501.643 \\\text { Observations } & 26\end{array}\end{array}$ ANOVA $\begin{array}{lcrrrr}\hline & \text { d f }&\text { S S } & \text { M S } & \text { F }&\text { Significance F } \\\hline \text {Regression} & 2 & 15579777040 & 7789888520 & 25.432 & 0.0001 \\\text {Residual }& 23 & 7045072780 & 306307512 & & \\\text {Total }& 25 & 22624849820 & & & \\\hline\end{array}$ $\begin{array} { l c c c c } & \text { Coefficients } & \text { Standard Error} & \text { t Stat } & \text { p-value } \\\text { Intercept } & 15800.0000 & 6038.2999 & 2.617 & 0.0154 \\\text { C apital } & 0.1245 & 0.2045 & 0.609 & 0.5485 \\\text { W ages } & 7.0762 & 1.4729 & 4.804 & 0.0001 \\\hline\end{array}$ -Referring to Table 14-5, which of the independent variables in the model are significant at the 5% level?

TABLE 14-4 A real estate builder wishes to determine how house size (House) is influenced by family income (Income) , family size (Size) , and education of the head of household (School) . House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is in years. The builder randomly selected 50 families and ran the multiple regression. Microsoft Excel output is provided below: $\begin{array}{ll} & \text { Regression Stuistics } \\\hline \text { Multiple R } & 0.865 \\\text { R Square } & 0.748 \\\text { Adjusted R Square } & 0.726 \\\text { Standard Error } & 5.195 \\\text { Observations } &50 \\\hline\end{array}$ ANOVA $\begin{array}{lrrrrr}\hline & \text { d f }& \text {S S } & \text {M S } & \text {F } & \text {Significance F } \\\hline \text {Regression} & & 3605.7736 & 1201.9245 & &0.0000 \\\text {Residual} & & 1214.2264 & 26.3962 & \\Total & 49 & 4820.0000 & & & \\\hline\end{array}$ $\begin{array}{lcccc}\hline & \text { Coefficients} & \text {Standard Error} & \text {t Stat }& \text {p -value} \\\hline \text {Intercept }& -1.6335 & 5.8078 & -0.281 & 0.7798 \\ \text {Income} & 0.4485 & 0.1137 & 3.9545 & 0.0003 \\ \text {Size} & 4.2615 & 0.8062 & 5.286 & 0.0001 \\ \text {School }& -0.6517 & 0.4319 & -1.509 & 0.1383 \\\hline\end{array}$ -Referring to Table 14-4, when the builder used a simple linear regression model with house size (House) as the dependent variable and education (School) as the independent variable, he obtained an r² value of 23.0%. What additional percentage of the total variation in house size has been explained by including family size and income in the multiple regression?

TABLE 14-17 The marketing manager for a nationally franchised lawn service company would like to study the characteristics that differentiate home owners who do and do not have a lawn service. A random sample of 30 home owners located in a suburban area near a large city was selected; 15 did not have a lawn service (code 0) and 15 had a lawn service (code 1). Additional information available concerning these 30 home owners includes family income (Income, in thousands of dollars), lawn size (Lawn Size, in thousands of square feet), attitude toward outdoor recreational activities (Attitude 0 = unfavorable, 1 = favorable), number of teenagers in the household (Teenager), and age of the head of the household (Age). The Minitab output is given below: $\begin{array}{lccccccrr} & & & & \text { Odds } & \text { 95 \% CI } \\\text {Predictor }& \text {Coef }&\text { SE Coef }&\text { Z }& \text {P} &\text { Ratio} &\text { Lower} & \text {Upper }\\\text {Constant }& -70.49 & 47.22 & -1.49 & 0.135 & & & \\\text {Income }& 0.2868 & 0.1523 & 1.88 & 0.060 & 1.33 & 0.99 & 1.80 \\\text {Lawn Size} & 1.0647 & 0.7472 & 1.42 & 0.154 & 2.90 & 0.67 & 12.54 \\\text {Attitude} & -12.744 & 9.455 & -1.35 & 0.178 & 0.00 & 0.00 & 326.06 \\\text {Teenager} & -0.200 & 1.061 & -0.19 & 0.850 & 0.82 & 0.10 & 6.56 \\\text {Age} & 1.0792 & 0.8783 & 1.23 & 0.219 & 2.94 & 0.53 & 16.45\end{array}$ Log-Likelihood = -4.890 Test that all slopes are zero: G = 31.808, DF = 5, P-Value = 0.000 Goodness-of-Fit Tests $\begin{array}{lrrr}\text { Method } & \text { Chi-Square } & \text { DF } & \text { P } \\ \text { Pearson } & 9.313 & 24 & 0.997 \\ \text { Deviance } & 9.780 & 24 & 0.995 \\ \text { Hosmer-Lemeshow } & 0.571 & 8 & 1.000\end{array}$ -Referring to Table 14-17, what is the p-value of the test statistic when testing whether the model is a good-fitting model?

TABLE 14-11 A logistic regression model was estimated in order to predict the probability that a randomly chosen university or college would be a private university using information on average total Scholastic Aptitude Test score (SAT) at the university or college, the room and board expense measured in thousands of dollars (Room/Brd), and whether the TOEFL criterion is at least 550 (Toefl550 = 1 if yes, 0 otherwise.) The dependent variable, Y, is school type (Type = 1 if private and 0 otherwise). The Minitab output is given below: Logistic Regression Table $\begin{array}{lrrrrrrr} & & & && \text { Odds } & \text { 95: CI } \\\text { Predictor } & {\text { Coef }} & \text { SE Coef } & Z &{\text { P }} & \text { Ratio } & \text { Lower } & \text { Upper } \\\text { Constant } &-27.118&6 .696& -4.05 & 0.000 & & & \\\text { SAT } & 0.015 & 0.004666 & 3.17 & 0.002 & 1.01 & 1.01 & 1.02 \\\text { Toefl550 } & -0.390 & 0.9538 & -0.41 & 0.682 & 0.68 & 0.10 & 4.39 \\\text { Room/Brd } & 2.078 & 0.5076 & 4.09 & 0.000 & 7.99 & 2.95 & 21.60\end{array}$ Log-Likelihood = -21.883 Test that all slopes are zero: G = 62.083, DF = 3, P-Value = 0.000 Goodness-of-Fit Tests $\begin{array}{lrcr}\text { Method } & \text { Chi-Square } & \text { DF } & \text { P } \\ \text { Pearson } & 143.551 & 76 & 0.000 \\ \text { Deviance } & 43.767 & 76 & 0.999 \\ \text { Hosmer-Lemeshow } & 15.731 & 8 & 0.046\end{array}$ TABLE 14-10 Regression Analysis $\begin{array}{lr}\hline\text {Regression Statistics } \\\hline\text { Multiple R} & 0.63 \\\text {R Square }& 0.40 \\\text {Adjusted R Square} & 0.23 \\\text {Standard Error }& 50.00 \\\text {Observations} & 15.00 \\\hline\end{array}$ ANOVA $\begin{array}{lccccc}\hline & \text { d f} & \text {SS} & \text {MS} & \text {F } & \text {Significance F } \\\hline \text {Regression} & 3 & & 5994.24 & 2.40 & 0.12 \\\text {Residual }& 11 & 27496.82 & & & \\\text {Total }& 45479.54 & & & \\\hline\end{array}$ $\begin{array}{lccrrrr}\hline & \text {oefficients}&\text { Standard Error }& \text { t Stat} &\text { p-value }& \text {Lower 99.0\% }& \text {Upper 99.0 \% } \\\hline\text { Intercept} & 123.80 & 48.71 & 2.54 & 0.03 & -27.47 & 275.07 \\\text {AGE }& -0.82 & 0.87 & -0.95 & 0.36 & -3.51 & 1.87 \\\text {TICKETS}& 21.25 & 10.66 & 1.99 & 0.07 & -11.86 & 54.37 \\\text {DENSITY} & -3.14 & 6.46 & -0.49 & 0.64 & -23.19 & 16.91 \\\hline\end{array}$ -Referring to Table 14-11, what is the estimated probability that a school with an average SAT score of 1250, a TOEFL criterion that is at least 550, and the room and board expense of 5 thousand dollars will be a private school?

TABLE 14-7 The department head of the accounting department wanted to see if she could predict the GPA of students using the number of course units (credits) and total SAT scores of each. She takes a sample of students and generates the following Microsoft Excel output: $\text {SUMMARY OUTPUT}$ $\begin{array}{ll}\hline \text { Regression Statistics } \\\hline \text { Multiple R }& 0.916 \\ \text { R Square} & 0.839 \\ \text { Adjusted R Square} & 0.732 \\ \text { Standard Error }& 0.24685 \\ \text { Observations} & 6 \\\hline\end{array}$ ANOVA $\begin{array}{lccclc}\hline & \text {d f }& \text { SS }& \text { M S } & \text { F } & \text {Significance F } \\\hline \text { Regression} & 2 & 0.95219 & 0.47610 & 7.813 & 0.0646 \\ \text {Residual} & 3 & 0.18281 & 0.06094 & & \\ \text {Total} & 5 & 1.13500 & & & \\\hline\end{array}$ $\begin{array}{lrcrr}\hline & \text {Coefficients }& \text {Standard Error} & \text {t Stat } & \text { p -value} \\\hline \text { Intercept }& 4.593897 & 1.13374542 & 4.052 & 0.0271 \\ \text {Units }& -0.247270 & 0.06268485 & -3.945 & 0.0290 \\ \text {SAT Total }& 0.001443 & 0.00101241 & 1.425 & 0.2494 \\\hline\end{array}$ -Referring to Table 14-7, the department head wants to use a t test to test for the significance of the coefficient of X1. The value of the test statistic is_____ .

TABLE 14-10 You worked as an intern at We Always Win Car Insurance Company last summer. You notice that individual car insurance premiums depend very much on the age of the individual, the number of traffic tickets received by the individual, and the population density of the city in which the individual lives. You performed a regression analysis in EXCEL and obtained the following information: Regression Analysis $\begin{array}{lr}\hline\text {Regression Statistics } \\\hline\text { Multiple R} & 0.63 \\\text {R Square }& 0.40 \\\text {Adjusted R Square} & 0.23 \\\text {Standard Error }& 50.00 \\\text {Observations} & 15.00 \\\hline\end{array}$ ANOVA $\begin{array}{lccccc}\hline & \text { d f} & \text {SS} & \text {MS} & \text {F } & \text {Significance F } \\\hline \text {Regression} & 3 & & 5994.24 & 2.40 & 0.12 \\\text {Residual }& 11 & 27496.82 & & & \\\text {Total }& 45479.54 & & & \\\hline\end{array}$ $\begin{array}{lccrrrr}\hline & \text {oefficients}&\text { Standard Error }& \text { t Stat} &\text { p-value }& \text {Lower 99.0\% }& \text {Upper 99.0 \% } \\\hline\text { Intercept} & 123.80 & 48.71 & 2.54 & 0.03 & -27.47 & 275.07 \\\text {AGE }& -0.82 & 0.87 & -0.95 & 0.36 & -3.51 & 1.87 \\\text {TICKETS}& 21.25 & 10.66 & 1.99 & 0.07 & -11.86 & 54.37 \\\text {DENSITY} & -3.14 & 6.46 & -0.49 & 0.64 & -23.19 & 16.91 \\\hline\end{array}$ -Referring to Table 14-10, the standard error of the estimate is_____ .

TABLE 14-16 The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: $\begin{array}{lr}\hline\text {Regression Statistics } \\\hline \text {Multiple R }& 0.7930 \\\text {R Square} & 0.6288 \\\text {Adjusted R Square} & 0.6029 \\\text {Standard Error }& 10.4570 \\\text {Observations }& 47 \\\hline\end{array}$ ANOVA $\begin{array}{lccccc}\hline &\text { d f } &\text { SS }& \text { MS }& \text { F } & \text { Significance F} \\\hline \text { Regression }& 3 & 7965.08 & 2655.03 & 24.2802 & 2.3853 \mathrm{E}-09 \\\text { Residual} & 43 & 4702.02 & 109.35 & & \\\text { Total }& 46 & 12667.11 & & & \\\hline\end{array}$ $\begin{array}{lrrrrrr}\hline &\text { Coeffs} & \text { Stnd Err} &\text { t Stat} &\text { p -value }&\text { Lower 95\%} \text { Upper 95\%} \\\hline\text { Intercept }& -753.4225 & 101.1149 & -7.4511 & 2.88 \mathrm{E}-09 & -957.3401 & -549.5050 \\\%\text { Attend }& 8.5014 & 1.0771 & 7.8929 & 6.73 \mathrm{E}-10 & 6.3292 & 10.6735 \\\text { Salary} & 6.85 \mathrm{E}-07 & 0.0006 & 0.0011 & 0.9991 & -0.0013 & 0.0013 \\\text { Spending }& 0.0060 & 0.0046 & 1.2879 & 0.2047 & -0.0034 & 0.0153 \\\hline\end{array}$ -Referring to Table 14-16, what is the p-value of the test statistic when testing whether instructional spending per pupil has any effect on percentage of students passing the proficiency test?

TABLE 14-16 The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing) , daily average of the percentage of students attending class (% Attendance) , average teacher salary in dollars (Salaries) , and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: $\begin{array}{lr}\hline\text {Regression Statistics } \\\hline \text {Multiple R }& 0.7930 \\\text {R Square} & 0.6288 \\\text {Adjusted R Square} & 0.6029 \\\text {Standard Error }& 10.4570 \\\text {Observations }& 47 \\\hline\end{array}$ ANOVA $\begin{array}{lccccc}\hline &\text { d f } &\text { SS }& \text { MS }& \text { F } & \text { Significance F} \\\hline \text { Regression }& 3 & 7965.08 & 2655.03 & 24.2802 & 2.3853 \mathrm{E}-09 \\\text { Residual} & 43 & 4702.02 & 109.35 & & \\\text { Total }& 46 & 12667.11 & & & \\\hline\end{array}$ $\begin{array}{lrrrrrr}\hline &\text { Coeffs} & \text { Stnd Err} &\text { t Stat} &\text { p -value }&\text { Lower 95\%} \text { Upper 95\%} \\\hline\text { Intercept }& -753.4225 & 101.1149 & -7.4511 & 2.88 \mathrm{E}-09 & -957.3401 & -549.5050 \\\%\text { Attend }& 8.5014 & 1.0771 & 7.8929 & 6.73 \mathrm{E}-10 & 6.3292 & 10.6735 \\\text { Salary} & 6.85 \mathrm{E}-07 & 0.0006 & 0.0011 & 0.9991 & -0.0013 & 0.0013 \\\text { Spending }& 0.0060 & 0.0046 & 1.2879 & 0.2047 & -0.0034 & 0.0153 \\\hline\end{array}$ -Referring to Table 14-16, which of the following is a correct statement?

A regression had the following results: SST = 82.55, SSE = 29.85. It can be said that 63.84% of the variation in the dependent variable is explained by the independent variables in the regression.

A regression had the following results: SST = 82.55, SSE = 29.85. It can be said that 73.4% of the variation in the dependent variable is explained by the independent variables in the regression.

The coefficient of multiple determination measures the fraction of the total variation in the dependent variable that is explained by the regression plane.

TABLE 14-5 A microeconomist wants to determine how corporate sales are influenced by capital and wage spending by companies. She proceeds to randomly select 26 large corporations and record information in millions of dollars. The Microsoft Excel output below shows results of this multiple regression. $\begin{array}{l}\text { Regression Statistics }\\\begin{array} { l r } \hline \text { Multiple R } & 0.830 \\\text { R Square } & 0.689 \\\text { Adjusted R Square } & 0.662 \\\text { Standard Error } & 17501.643 \\\text { Observations } & 26\end{array}\end{array}$ ANOVA $\begin{array}{lcrrrr}\hline & \text { d f }&\text { S S } & \text { M S } & \text { F }&\text { Significance F } \\\hline \text {Regression} & 2 & 15579777040 & 7789888520 & 25.432 & 0.0001 \\\text {Residual }& 23 & 7045072780 & 306307512 & & \\\text {Total }& 25 & 22624849820 & & & \\\hline\end{array}$ $\begin{array} { l c c c c } & \text { Coefficients } & \text { Standard Error} & \text { t Stat } & \text { p-value } \\\text { Intercept } & 15800.0000 & 6038.2999 & 2.617 & 0.0154 \\\text { C apital } & 0.1245 & 0.2045 & 0.609 & 0.5485 \\\text { W ages } & 7.0762 & 1.4729 & 4.804 & 0.0001 \\\hline\end{array}$ -Referring to Table 14-5, what is the p-value for Capital?