NARRBEGIN: Real Estate Builder Real Estate Builder A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below.SUMMARY OUTPUT \[\begin{array}{l} \text { Regression Statistics }\\ \begin{array} { l l } \text { Multiple R } & 0.865 \\ \text { R. Square} & 0.748 \\ \text { Adjusted R Square } & 0.726 \\ \text {Standard Error } & 5.195 \\ \text { Observations } & 50 \end{array} \end{array}\] ANOVA $\begin{array}{l|ccccc} & d f & \text { SS } & M S & F & \text { Signif F } \\ \hline \text { Regression } & & 3605.7736 & 901.4434 & & 0.0001 \\ \text { Residual } & & 1214.2264 & 26.9828 & \\ \text { Total } & 49 & 4820.0000 & & \end{array}$ \[\begin{array} { | l | c c c c | } \hline & \text { Coeff } & \text { st.error } & \boldsymbol { t }\boldsymbol { state } & \boldsymbol { P } \text {-value } \\ \hline \text { Intercept } & - 1.6335 & 5.807 \mathrm { 8 } & - 0.281 & 0 .798 \\ \text { Family Income } & 0.4485 & 0.1137 & 3.9545 & 0 .0003 \\ \text { Family Size } & 4.2615 & 0.8062 & 5.286 & 0 .0001 \\ \text { Education } & - 0.6517 & 0.4319 & - 1.509 & 0 .1383 \\ \hline \end{array}\] NARREND -{Real Estate Builder Narrative} Which of the following values for the level of significance is the smallest for which the regression model as a whole is significant: $\alpha$ = .00005, .001, .01, and .05?

The answer of NARRBEGIN: Real Estate Builder Real Estate Builder A real...

Consider the following statistics of a multiple regression model: n = 25, k = 5, b1 = -6.31, and s$\varepsilon$ = 2.98. Can we conclude at the 1% significance level that x1 and y are linearly related?

@#LAT-DLM& vs. @#LAT-DLM& Rejection region: | t |

NARRBEGIN: Life Expectancy Life Expectancy An actuary wanted to develop a model to predict how long individuals will live. After consulting a number of physicians, she collected the age at death (y), the average number of hours of exercise per week (xS1U1B11S1U1B0), the cholesterol level (xS1U1B12S1U1B0), and the number of points that the individual's blood pressure exceeded the recommended value (xS1U1B13S1U1B0). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below.THE REGRESSION EQUATION IS y = 55.8 + 1.79xS1U1B11S1U1B0 - 0.021xS1U1B12S1U1B0 -0.061xS1U1B13S1U1B0 \[\begin{array} { | c | c c c | } \hline \text { predictor } & Coef & SUDev & T \\ \hline \text { Constant } & 55.8 & 11.8 & 4.729 \\ \boldsymbol { \lambda } _ { 1 } & 1.79 & 0.44 & 4.068 \\ \lambda _ { 2 } & - 0.021 & 0.011 & - 1.909 \\ \lambda _ { 3 } & - 0.016 & 0.014 & - 1.143 \\ \hline \end{array}\] \[S = 9.47 \quad R - S q = 22.5 \%\] ANALYSIS OF VARIANCE \[\begin{array} { | l | c c c c | } \hline \text { Source of Variation } & \boldsymbol { d } & \mathbf { SS } & \boldsymbol { M S } & \boldsymbol { F } \\ \hline \text { Repression } & 3 & 936 & 312 & 3.477 \\ \text { Error } & 36 & 3230 & 89.722 & \\ \hline \text { Tatol } & 39 & 4166 & & \\ \hline \end{array}\] NARREND -{Life Expectancy Narrative} Interpret the coefficient b3.

b 3 = 0.016. This tells us for each additi

NARRBEGIN: Student's Final Grade Student's Final Grade A statistics professor investigated some of the factors that affect an individual student's final grade in her course. She proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ , where y is the final grade (out of 100 points), xS1U1B11S1U1B0 is the number of lectures skipped, xS1U1B12S1U1B0 is the number of late assignments, and xS1U1B13S1U1B0 is the midterm exam score (out of 100). The professor recorded the data for 50 randomly selected students. The computer output is shown below.THE REGRESSION EQUATION IS $\hat { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ \[\begin{array} { | c | c c c | } \hline \text { PredictOr } & \text { Coef } & \text { SuDsv } & T \\ \hline \text { Constant } & 41.6 & 17.8 & 2.337 \\ x _ { 1 } & - 3.18 & 1.66 & - 1.916 \\ x _ { 2 } & - 1.17 & 1.13 & - 1.035 \\ x _ { 3 } & 0.63 & 0.13 & 4.846 \\ \hline \end{array}\] \[S = 13.74 \quad R - S q = 30.0 \%\] ANALYSIS OF VARIANCE \[\begin{array} { | l | c c c c | } \hline \text { Source of Variation } & \boldsymbol { d } & \mathbf { 5 8 } & \boldsymbol { M S } & \boldsymbol { F } \\ \hline \text { Regression } & 3 & 3716 & 1238.667 & 6.558 \\ \text { Error } & 46 & 8688 & 188.870 & \\ \hline \text { Total } & 49 & 12404 & & \\ \hline \end{array}\] NARREND -{Student's Final Grade Narrative} Does this data provide enough evidence at the 1% significance level to conclude that the final grade and the midterm exam score are positively linearly related?

@#LAT-DLM& vs. @#LAT-DLM& Rejection region: t > t 0.01,46 @#LAT-DLM& 2.412 Tes

NARRBEGIN: Student's Final Grade Student's Final Grade A statistics professor investigated some of the factors that affect an individual student's final grade in her course. She proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ , where y is the final grade (out of 100 points), xS1U1B11S1U1B0 is the number of lectures skipped, xS1U1B12S1U1B0 is the number of late assignments, and xS1U1B13S1U1B0 is the midterm exam score (out of 100). The professor recorded the data for 50 randomly selected students. The computer output is shown below.THE REGRESSION EQUATION IS $\hat { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ \[\begin{array} { | c | c c c | } \hline \text { PredictOr } & \text { Coef } & \text { SuDsv } & T \\ \hline \text { Constant } & 41.6 & 17.8 & 2.337 \\ x _ { 1 } & - 3.18 & 1.66 & - 1.916 \\ x _ { 2 } & - 1.17 & 1.13 & - 1.035 \\ x _ { 3 } & 0.63 & 0.13 & 4.846 \\ \hline \end{array}\] \[S = 13.74 \quad R - S q = 30.0 \%\] ANALYSIS OF VARIANCE \[\begin{array} { | l | c c c c | } \hline \text { Source of Variation } & \boldsymbol { d } & \mathbf { 5 8 } & \boldsymbol { M S } & \boldsymbol { F } \\ \hline \text { Regression } & 3 & 3716 & 1238.667 & 6.558 \\ \text { Error } & 46 & 8688 & 188.870 & \\ \hline \text { Total } & 49 & 12404 & & \\ \hline \end{array}\] NARREND -{Student's Final Grade Narrative} Does this data provide enough evidence at the 5% significance level to conclude that the final grade and the number of late assignments are negatively linearly related?

@#LAT-DLM& vs. @#LAT-DLM& Rejection region: t 0.05,46 @#LAT-DLM& -1.679 T

Exam 17: Multiple Regression

NARRBEGIN: Real Estate Builder Real Estate Builder A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below.SUMMARY OUTPUT Regression Statistics Multiple R 0.865 R. Square 0.748 Adjusted R Square 0.726 Standard Error 5.195 Observations 50 ANOVA df SS MS F Signif F Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff st.error -value Intercept -1.6335 5.807 -0.281 0.798 Family Income 0.4485 0.1137 3.9545 0.0003 Family Size 4.2615 0.8062 5.286 0.0001 Education -0.6517 0.4319 -1.509 0.1383 NARREND -{Real Estate Builder Narrative} Which of the following values for the level of significance is the smallest for which the regression model as a whole is significant: $\alpha$ = .00005, .001, .01, and .05?

(Short Answer)

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Question 41

The range of the values of the Durbin-Watson statistic d is ____________________.

(Essay)

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Question 42

Consider the following statistics of a multiple regression model: n = 25, k = 5, b₁ = -6.31, and s_{$\varepsilon$} = 2.98. Can we conclude at the 1% significance level that x₁ and y are linearly related?

(Essay)

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Question 43

In testing the significance of a multiple regression model with three independent variables, the null hypothesis is $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 }$ .

(True/False)

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Question 44

NARRBEGIN: Life Expectancy Life Expectancy An actuary wanted to develop a model to predict how long individuals will live. After consulting a number of physicians, she collected the age at death (y), the average number of hours of exercise per week (x₁), the cholesterol level (x₂), and the number of points that the individual's blood pressure exceeded the recommended value (x₃). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below.THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁ - 0.021x₂ -0.061x₃ predictor Coef SUDev T Constant 55.8 11.8 4.729 1.79 0.44 4.068 -0.021 0.011 -1.909 -0.016 0.014 -1.143 $S = 9.47 \quad R - S q = 22.5 \%$ ANALYSIS OF VARIANCE Source of Variation Repression 3 936 312 3.477 Error 36 3230 89.722 Tatol 39 4166 NARREND -{Life Expectancy Narrative} Interpret the coefficient b₃.

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Question 45

Small values of the Durbin-Watson statistic d (d < 2) indicate a negative first-order autocorrelation.

(True/False)

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Question 46

Multicollinearity is present when there is a high degree of correlation between the independent variables included in the regression model.

(True/False)

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Question 47

For a multiple regression model, the following statistics are given: Total variation in y = 500, SSE = 80, and n = 25. Then, the coefficient of determination is:

(Multiple Choice)

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Question 48

In multiple regression analysis, the ratio MSR/MSE yields the:

(Multiple Choice)

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Question 49

A high value of the coefficient of determination significantly above 0 in multiple regression, accompanied by insignificant t-statistics on all parameter estimates, very often indicates a high correlation between independent variables in the model.

(True/False)

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Question 50

NARRBEGIN: Student's Final Grade Student's Final Grade A statistics professor investigated some of the factors that affect an individual student's final grade in her course. She proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ , where y is the final grade (out of 100 points), x₁ is the number of lectures skipped, x₂ is the number of late assignments, and x₃ is the midterm exam score (out of 100). The professor recorded the data for 50 randomly selected students. The computer output is shown below.THE REGRESSION EQUATION IS $\hat { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ PredictOr Coef SuDsv T Constant 41.6 17.8 2.337 -3.18 1.66 -1.916 -1.17 1.13 -1.035 0.63 0.13 4.846 $S = 13.74 \quad R - S q = 30.0 \%$ ANALYSIS OF VARIANCE Source of Variation Regression 3 3716 1238.667 6.558 Error 46 8688 188.870 Total 49 12404 NARREND -{Student's Final Grade Narrative} Does this data provide enough evidence at the 1% significance level to conclude that the final grade and the midterm exam score are positively linearly related?

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Question 51

In reference to the equation $\hat { y } = 1.86 - 0.51 x _ { 1 } + 0.60 x _ { 2 }$ , the value 0.60 is the average change in y per unit change in x₂, regardless of the value of x₁.

(True/False)

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Question 52

NARRBEGIN: Student's Final Grade Student's Final Grade A statistics professor investigated some of the factors that affect an individual student's final grade in her course. She proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ , where y is the final grade (out of 100 points), x₁ is the number of lectures skipped, x₂ is the number of late assignments, and x₃ is the midterm exam score (out of 100). The professor recorded the data for 50 randomly selected students. The computer output is shown below.THE REGRESSION EQUATION IS $\hat { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ PredictOr Coef SuDsv T Constant 41.6 17.8 2.337 -3.18 1.66 -1.916 -1.17 1.13 -1.035 0.63 0.13 4.846 $S = 13.74 \quad R - S q = 30.0 \%$ ANALYSIS OF VARIANCE Source of Variation Regression 3 3716 1238.667 6.558 Error 46 8688 188.870 Total 49 12404 NARREND -{Student's Final Grade Narrative} Does this data provide enough evidence at the 5% significance level to conclude that the final grade and the number of late assignments are negatively linearly related?

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Question 53

If the Durbin-Watson statistic has a value close to 4, which assumption is violated?

(Multiple Choice)

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Question 54

One of the consequences of multicollinearity in multiple regression is biased estimates on the slope coefficients.

(True/False)

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Question 55

Some of the requirements for the error variable in a multiple regression model are that the standard deviation is a(n) and the errors are .

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Question 56

In regression analysis, the total variation in the dependent variable y, measured by $\sum \left( y _ { i } - \bar { y } \right) ^ { 2 }$ , can be decomposed into two parts: the explained variation, measured by SSR, and the unexplained variation, measured by SSE.

(True/False)

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Question 57

A multiple regression equation has a coefficient of determination of 0.81. Then, the percentage of the variation in y that is explained by the regression equation is 90%.

(True/False)

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Question 58

In reference to the equation $\hat { y } = - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 }$ , the value 0.12 is the average change in y per unit change in x₁, when x₂ is held constant.

(True/False)

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Question 59

In a multiple regression model, the error variable $\varepsilon$ is assumed to have a mean of:

(Multiple Choice)

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Question 60

The range of the values of the Durbin-Watson statistic d is ____________________.

Consider the following statistics of a multiple regression model: n = 25, k = 5, b₁ = -6.31, and s_{$\varepsilon$} = 2.98. Can we conclude at the 1% significance level that x₁ and y are linearly related?

In testing the significance of a multiple regression model with three independent variables, the null hypothesis is $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 }$ .

Small values of the Durbin-Watson statistic d (d < 2) indicate a negative first-order autocorrelation.

Multicollinearity is present when there is a high degree of correlation between the independent variables included in the regression model.

For a multiple regression model, the following statistics are given: Total variation in y = 500, SSE = 80, and n = 25. Then, the coefficient of determination is:

In multiple regression analysis, the ratio MSR/MSE yields the:

A high value of the coefficient of determination significantly above 0 in multiple regression, accompanied by insignificant t-statistics on all parameter estimates, very often indicates a high correlation between independent variables in the model.

In reference to the equation $\hat { y } = 1.86 - 0.51 x _ { 1 } + 0.60 x _ { 2 }$ , the value 0.60 is the average change in y per unit change in x₂, regardless of the value of x₁.

If the Durbin-Watson statistic has a value close to 4, which assumption is violated?

One of the consequences of multicollinearity in multiple regression is biased estimates on the slope coefficients.

Some of the requirements for the error variable in a multiple regression model are that the standard deviation is a(n) and the errors are .

In regression analysis, the total variation in the dependent variable y, measured by $\sum \left( y _ { i } - \bar { y } \right) ^ { 2 }$ , can be decomposed into two parts: the explained variation, measured by SSR, and the unexplained variation, measured by SSE.

A multiple regression equation has a coefficient of determination of 0.81. Then, the percentage of the variation in y that is explained by the regression equation is 90%.

In reference to the equation $\hat { y } = - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 }$ , the value 0.12 is the average change in y per unit change in x₁, when x₂ is held constant.

In a multiple regression model, the error variable $\varepsilon$ is assumed to have a mean of:

What Is Statistics

Graphical Descriptive Techniques I

Graphical Descriptive Techniques II

Numerical Descriptive Techniques

Data Collection and Sampling

Probability

Random Variables and Discrete Probability Distributions

Continuous Probability Distributions

Sampling Distributions

Introduction to Estimation

Introduction to Hypothesis Testing

Inference About a Population

Inference About Comparing Two Populations

Analysis of Variance

Chi-Squared Tests Optional

Simple Linear Regression and Correlation

Filters

Exam 17: Multiple Regression

The range of the values of the Durbin-Watson statistic d is ____________________.

Consider the following statistics of a multiple regression model: n = 25, k = 5, b1 = -6.31, and s ε\varepsilonε = 2.98. Can we conclude at the 1% significance level that x1 and y are linearly related?

In testing the significance of a multiple regression model with three independent variables, the null hypothesis is H0:β1=β2=β3H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 }H0​:β1​=β2​=β3​ .

Small values of the Durbin-Watson statistic d (d < 2) indicate a negative first-order autocorrelation.

Multicollinearity is present when there is a high degree of correlation between the independent variables included in the regression model.

For a multiple regression model, the following statistics are given: Total variation in y = 500, SSE = 80, and n = 25. Then, the coefficient of determination is:

In multiple regression analysis, the ratio MSR/MSE yields the:

A high value of the coefficient of determination significantly above 0 in multiple regression, accompanied by insignificant t-statistics on all parameter estimates, very often indicates a high correlation between independent variables in the model.

In reference to the equation y^=1.86−0.51x1+0.60x2\hat { y } = 1.86 - 0.51 x _ { 1 } + 0.60 x _ { 2 }y^​=1.86−0.51x1​+0.60x2​ , the value 0.60 is the average change in y per unit change in x2, regardless of the value of x1.

If the Durbin-Watson statistic has a value close to 4, which assumption is violated?

One of the consequences of multicollinearity in multiple regression is biased estimates on the slope coefficients.

Some of the requirements for the error variable in a multiple regression model are that the standard deviation is a(n) ____________________ and the errors are ____________________.

In regression analysis, the total variation in the dependent variable y, measured by ∑(yi−yˉ)2\sum \left( y _ { i } - \bar { y } \right) ^ { 2 }∑(yi​−yˉ​)2 , can be decomposed into two parts: the explained variation, measured by SSR, and the unexplained variation, measured by SSE.

A multiple regression equation has a coefficient of determination of 0.81. Then, the percentage of the variation in y that is explained by the regression equation is 90%.

In reference to the equation y^=−0.80+0.12x1+0.08x2\hat { y } = - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 }y^​=−0.80+0.12x1​+0.08x2​ , the value 0.12 is the average change in y per unit change in x1, when x2 is held constant.

In a multiple regression model, the error variable ε\varepsilonε is assumed to have a mean of:

What Is Statistics

Graphical Descriptive Techniques I

Graphical Descriptive Techniques II

Numerical Descriptive Techniques

Data Collection and Sampling

Probability

Random Variables and Discrete Probability Distributions

Continuous Probability Distributions

Sampling Distributions

Introduction to Estimation

Introduction to Hypothesis Testing

Inference About a Population

Inference About Comparing Two Populations

Analysis of Variance

Chi-Squared Tests Optional

Simple Linear Regression and Correlation

Filters

Consider the following statistics of a multiple regression model: n = 25, k = 5, b₁ = -6.31, and s_{$\varepsilon$} = 2.98. Can we conclude at the 1% significance level that x₁ and y are linearly related?

In testing the significance of a multiple regression model with three independent variables, the null hypothesis is $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 }$ .

In reference to the equation $\hat { y } = 1.86 - 0.51 x _ { 1 } + 0.60 x _ { 2 }$ , the value 0.60 is the average change in y per unit change in x₂, regardless of the value of x₁.

Some of the requirements for the error variable in a multiple regression model are that the standard deviation is a(n) and the errors are .

In regression analysis, the total variation in the dependent variable y, measured by $\sum \left( y _ { i } - \bar { y } \right) ^ { 2 }$ , can be decomposed into two parts: the explained variation, measured by SSR, and the unexplained variation, measured by SSE.

In reference to the equation $\hat { y } = - 0.80 + 0.12 x _ { 1 } + 0.08 x _ { 2 }$ , the value 0.12 is the average change in y per unit change in x₁, when x₂ is held constant.

In a multiple regression model, the error variable $\varepsilon$ is assumed to have a mean of: