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 ( $x _ { 1 }$ ), the cholesterol level ( $x _ { 2 }$ ), and the number of points by which the individual's blood pressure exceeded the recommended value ( $x _ { 3 }$ ). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below: THE REGRESSION EQUATION IS $y =$ $55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$ \[\begin{array} { | c | c c c | } \hline \text { Predictor } & \text { Coef } & \text { StDev } & \mathrm { T } \\ \hline \text { Constant } & 55.8 & 11.8 & 4.729 \\ x _ { 1 } & 1.79 & 0.44 & 4.068 \\ x _ { 2 } & - 0.021 & 0.011 & - 1.909 \\ x _ { 3 } & - 0.016 & 0.014 & - 1.143 \\ \hline \end{array}\] S = 9.47 R-Sq = 22.5%. \[\begin{array}{l} \text { ANALYSIS OF VARIANCE }\\ \begin{array} { | l | c c c c | } \hline \text { Source of Variation } & \text { df } & \text { SS } & \text { MS } & \text { F } \\ \hline \text { Regression } & 3 & 936 & 312 & 3.477 \\ \text { Error } & 36 & 3230 & 89.722 & \\ \hline \text { Total } & 39 & 4166 & & \\ \hline \end{array} \end{array}\] What is the coefficient of determination? What does this statistic tell you?

@#IMG-DLM& 0.225. This means that 22.5% of the var

Pop-up coffee vendors have been popular in the city of Adelaide in 2013. A vendor is interested in knowing how temperature (in degrees Celsius) and number of different pastries and biscuits offered to customers impacts daily hot coffee sales revenue (in $00's). A random sample of 6 days was taken, with the daily hot coffee sales revenue and the corresponding temperature and number of different pastries and biscuits offered on that day, noted. Excel output for a multiple linear regression is given below: \[\begin{array} { | c | c | r | } \hline \text { Coffee sales revenue } & \text { Temperature } & \text { Pastries/biscuits } \\ \hline 6.5 & 25 & 7 \\ \hline 10 & 17 & 13 \\ \hline 5.5 & 30 & 5 \\ \hline 4.5 & 35 & 6 \\ \hline 3.5 & 40 & 3 \\ \hline 28 & 9 & 15 \\ \hline \end{array}\] \(\begin{array}{|l|r|r|r|r|r|c|}\hline \text { SUMMARV } \\ \text { OUTPUT } \\ \hline \text { Regression } \\ \text { Stotistics } \\ \hline \text { Multiple R } & 0.87 \\ \hline \text { R Square } & 0.75 \\ \hline \begin{array}{l} \text { Adjusted R } \\ \text { Square } \end{array} & 0.59 \\ \hline \text { Standard Error } & 5.95 \\ \hline \text { Otservations } & 6.00 \\ \hline\\ \hline \text { ANOVA }\\ \hline & & & & & \text { Significonce } \\ & \text { df } & S S & M S & F & F \\ \hline \text { Regression } & 2.00 & 322.14 & 161.07 & 4.55 & 0.12 \\\hline \text { Residual } & 3.00 & 106.20 & 35.40 \\ \hline \text { Total } & 5.00 & 428.33 & \\ \hline\\ \hline & \text { Coefficients } & {\begin{array}{c} \text { Standard } \\ \text { Error } \end{array}} & {\text { tStot }} & \text { Pvolue } & \text { Lower 95\% } & \begin{array}{c} \text { Upper } \\ 95 \% \end{array} \\ \hline \text { Intercept } & 18.68 & 37.8 & 0.49 & 0.66 & -101.88 & 139.24 \\ \hline \text { Temperature } & -0.50 & 0.83 & -0.60 & 0.59 & -3.15 & 2.15 \\ \hline \text { Patries/bisouits } & 0.49 & 2.02 & 0.24 & 0.82 & -5.94 & 6.92 \\ \hline \end{array}\) Comment on the difference between the coefficient of determination and the Adjusted coefficient of determination.

r 2 = 0.75 Adjusted r 2 = 0.59 The

A multiple regression model has the form ŷ = b0 + b1x1 + b2x2. Which of the following best describes b2?

A) We estimate for each one unit increase in x2, that y will increase by b2 units, on average. B) Whilst holding x1 constant, we estimate for each one unit increase in x2, that y will increase by b2 units, on average. C) Whilst holding x2 constant, we estimate for each one unit increase in x1, that y will increase by b2 units, on average. D) Whilst holding x1 constant, we estimate for each one unit increase in x2, that y will increase by b1 units, on average. A) We estimate for each one unit increase in x2, that y will increase by b2 units, on average. B) Whilst holding x1 constant, we estimate for each one unit increase in x2, that y will increase by b2 units, on average. C) Whilst holding x2 constant, we estimate for each one unit increase in x1, that y will increase by b2 units, on average. D) Whilst holding x1 constant, we estimate for each one unit increase in x2, that y will increase by b1 units, on average.

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 ( $x _ { 1 }$ ), the cholesterol level ( $x _ { 2 }$ ), and the number of points by which the individual's blood pressure exceeded the recommended value ( $x _ { 3 }$ ). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below: THE REGRESSION EQUATION IS $y =$ $55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$ \[\begin{array} { | c | c c c | } \hline \text { Predictor } & \text { Coef } & \text { StDev } & \mathrm { T } \\ \hline \text { Constant } & 55.8 & 11.8 & 4.729 \\ x _ { 1 } & 1.79 & 0.44 & 4.068 \\ x _ { 2 } & - 0.021 & 0.011 & - 1.909 \\ x _ { 3 } & - 0.016 & 0.014 & - 1.143 \\ \hline \end{array}\] S = 9.47 R-Sq = 22.5%. \[\begin{array}{l} \text { ANALYSIS OF VARIANCE }\\ \begin{array} { | l | c c c c | } \hline \text { Source of Variation } & \text { df } & \text { SS } & \text { MS } & \text { F } \\ \hline \text { Regression } & 3 & 936 & 312 & 3.477 \\ \text { Error } & 36 & 3230 & 89.722 & \\ \hline \text { Total } & 39 & 4166 & & \\ \hline \end{array} \end{array}\] Is there enough evidence at the 10% significance level to infer that the model is useful in predicting length of life?

@#IMG-DLM& . @#IMG-DLM& At least one @#IMG-DLM& is not equal

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 ( $x _ { 1 }$ ), the cholesterol level ( $x _ { 2 }$ ), and the number of points by which the individual's blood pressure exceeded the recommended value ( $x _ { 3 }$ ). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below: THE REGRESSION EQUATION IS \[y =\] \[55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }\] \[\begin{array} { | c | c c c | } \hline \text { Predictor } & \text { Coef } & \text { StDev } & \mathrm { T } \\ \hline \text { Constant } & 55.8 & 11.8 & 4.729 \\ x _ { 1 } & 1.79 & 0.44 & 4.068 \\ x _ { 2 } & - 0.021 & 0.011 & - 1.909 \\ x _ { 3 } & - 0.016 & 0.014 & - 1.143 \\ \hline \end{array}\] S = 9.47 R-Sq = 22.5%. \[\begin{array}{l} \text { ANALYSIS OF VARIANCE }\\ \begin{array} { | l | c c c c | } \hline \text { Source of Variation } & \text { df } & \text { SS } & \text { MS } & \text { F } \\ \hline \text { Regression } & 3 & 936 & 312 & 3.477 \\ \text { Error } & 36 & 3230 & 89.722 & \\ \hline \text { Total } & 39 & 4166 & & \\ \hline \end{array} \end{array}\] Is there enough evidence at the 5% significance level to infer that the number of points by which the individual's blood pressure exceeded the recommended value and the age at death are negatively linearly related?

@#IMG-DLM& . @#IMG-DLM& @#IMG-DLM& 0. Rejection region: t <

Exam 19: Multiple Regression

For a multiple regression model, the following statistics are given: Total variation in y = SSY = 250, SSE = 50, k = 4, n = 20. The coefficient of determination adjusted for degrees of freedom is:

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Question 21

For the multiple regression model $For the multiple regression model = 40 + 15 x _ { 1 } - 10 x _ { 2 } + 5 x _ { 3 } , if x _ { 2 } were to increase by 5 units, holding x _ { 1 } and x _ { 3 } constant, the value of y would decrease by 50 units, on average.$ $= 40 + 15 x _ { 1 } - 10 x _ { 2 } + 5 x _ { 3 }$ , if $x _ { 2 }$ were to increase by 5 units, holding $x _ { 1 }$ and $x _ { 3 }$ constant, the value of $y$ would decrease by 50 units, on average.

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Question 22

Multicollinearity is a situation in which the independent variables are highly correlated with the dependent variable.

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Question 23

In a multiple regression model, the probability distribution of the error variable $\varepsilon$ is assumed to be:

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Question 24

In order to test the significance of a multiple regression model involving 4 independent variables and 25 observations, the number of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are 4 and 20, respectively.

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Question 25

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 ( $x _ { 1 }$ ), the cholesterol level ( $x _ { 2 }$ ), and the number of points by which the individual's blood pressure exceeded the recommended value ( $x _ { 3 }$ ). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below: THE REGRESSION EQUATION IS $y =$ $55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$ Predictor Coef StDev 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 R-Sq = 22.5%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 936 312 3.477 Error 36 3230 89.722 Total 39 4166 What is the coefficient of determination? What does this statistic tell you?

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Question 26

In multiple regression, the descriptor 'multiple' refers to more than one independent variable.

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Question 27

Pop-up coffee vendors have been popular in the city of Adelaide in 2013. A vendor is interested in knowing how temperature (in degrees Celsius) and number of different pastries and biscuits offered to customers impacts daily hot coffee sales revenue (in $00's). A random sample of 6 days was taken, with the daily hot coffee sales revenue and the corresponding temperature and number of different pastries and biscuits offered on that day, noted. Excel output for a multiple linear regression is given below: Coffee sales revenue Temperature Pastries/biscuits 6.5 25 7 10 17 13 5.5 30 5 4.5 35 6 3.5 40 3 28 9 15 SUMMARV OUTPUT Regression Stotistics Multiple R 0.87 R Square 0.75 Adjusted R Square 0.59 Standard Error 5.95 Otservations 6.00 ANOVA Significonce df SS MS F F Regression 2.00 322.14 161.07 4.55 0.12 Residual 3.00 106.20 35.40 Total 5.00 428.33 Coefficients Standard Error tStot Pvolue Lower 95\% Upper 95\% Intercept 18.68 37.8 0.49 0.66 -101.88 139.24 Temperature -0.50 0.83 -0.60 0.59 -3.15 2.15 Patries/bisouits 0.49 2.02 0.24 0.82 -5.94 6.92 Comment on the difference between the coefficient of determination and the Adjusted coefficient of determination.

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Question 28

In a multiple regression model, the following statistics are given: SSE = 100, $R ^ { 2 } = 0.995$ , k = 5, n = 15. The multiple coefficient of determination adjusted for degrees of freedom is:

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Question 29

The adjusted multiple coefficient of determination is adjusted for the number of independent variables and the sample size.

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Question 30

In multiple regression, when the response surface (the graphical depiction of the regression equation) hits every single point, the sum of squares for error SSE = 0, the standard error of estimate $S _ { \varepsilon }$ = 0, and the coefficient of determination $R ^ { 2 }$ = 1.

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Question 31

In a multiple regression analysis, when there is no linear relationship between each of the independent variables and the dependent variable, then:

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Question 32

A multiple regression model has the form ŷ = b₀+ b₁x₁ + b₂x₂. Which of the following best describes b₂?

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Question 33

In regression analysis, we judge the magnitude of the standard error of estimate relative to the values of the dependent variable, and particularly to the mean of y.

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Question 34

Which of the following statements is not true?

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Question 35

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 ( $x _ { 1 }$ ), the cholesterol level ( $x _ { 2 }$ ), and the number of points by which the individual's blood pressure exceeded the recommended value ( $x _ { 3 }$ ). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below: THE REGRESSION EQUATION IS $y =$ $55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$ Predictor Coef StDev 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 R-Sq = 22.5%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 936 312 3.477 Error 36 3230 89.722 Total 39 4166 Is there enough evidence at the 10% significance level to infer that the model is useful in predicting length of life?

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Question 36

In order to test the validity of a multiple regression model involving 4 independent variables and 35 observations, the numbers of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are:

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Question 37

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 ( $x _ { 1 }$ ), the cholesterol level ( $x _ { 2 }$ ), and the number of points by which the individual's blood pressure exceeded the recommended value ( $x _ { 3 }$ ). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below: THE REGRESSION EQUATION IS $y =$ $55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$ Predictor Coef StDev 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 R-Sq = 22.5%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 936 312 3.477 Error 36 3230 89.722 Total 39 4166 Is there enough evidence at the 5% significance level to infer that the number of points by which the individual's blood pressure exceeded the recommended value and the age at death are negatively linearly related?

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Question 38

A statistician wanted to determine whether the demographic variables of age, education and income influence the number of hours of television watched per week. A random sample of 25 adults was selected to estimate the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ . Where: y = number of hours of television watched last week. $x _ { 1 }$ = age. $x _ { 2 }$ = number of years of education. $x _ { 3 }$ = income (in $1000s). The computer output is shown below. THE REGRESSION EQUATION IS $y =$ $22.3 + 0.41 x _ { 1 } - 0.29 x _ { 2 } - 0.12 x _ { 3 }$ Predictor Coef StDev Constant 22.3 10.7 2.084 0.41 0.19 2.158 -0.29 0.13 -2.231 -0.12 0.03 -4.00 S = 4.51 R-Sq = 34.8%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 227 75.667 3.730 Error 21 426 20.286 Total 24 653 Interpret the coefficient $b _ { 2 }$ .

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Question 39

A multiple regression equation includes 5 independent variables, and the coefficient of determination is 0.64. The percentage of the variation in y that is explained by the regression equation is:

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Question 40

For a multiple regression model, the following statistics are given: Total variation in y = SSY = 250, SSE = 50, k = 4, n = 20. The coefficient of determination adjusted for degrees of freedom is:

Multicollinearity is a situation in which the independent variables are highly correlated with the dependent variable.

In a multiple regression model, the probability distribution of the error variable $\varepsilon$ is assumed to be:

In order to test the significance of a multiple regression model involving 4 independent variables and 25 observations, the number of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are 4 and 20, respectively.

In multiple regression, the descriptor 'multiple' refers to more than one independent variable.

In a multiple regression model, the following statistics are given: SSE = 100, $R ^ { 2 } = 0.995$ , k = 5, n = 15. The multiple coefficient of determination adjusted for degrees of freedom is:

The adjusted multiple coefficient of determination is adjusted for the number of independent variables and the sample size.

In multiple regression, when the response surface (the graphical depiction of the regression equation) hits every single point, the sum of squares for error SSE = 0, the standard error of estimate $S _ { \varepsilon }$ = 0, and the coefficient of determination $R ^ { 2 }$ = 1.

In a multiple regression analysis, when there is no linear relationship between each of the independent variables and the dependent variable, then:

A multiple regression model has the form ŷ = b₀+ b₁x₁ + b₂x₂. Which of the following best describes b₂?

In regression analysis, we judge the magnitude of the standard error of estimate relative to the values of the dependent variable, and particularly to the mean of y.

Which of the following statements is not true?

In order to test the validity of a multiple regression model involving 4 independent variables and 35 observations, the numbers of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are:

A multiple regression equation includes 5 independent variables, and the coefficient of determination is 0.64. The percentage of the variation in y that is explained by the regression equation is:

What Is Statistics

Types of Data, Data Collection and Sampling

Graphical Descriptive Methods Nominal Data

Graphical Descriptive Techniques Numerical Data

Numerical Descriptive Measures

Probability

Random Variables and Discrete Probability Distributions

Continuous Probability Distributions

Statistical Inference: Introduction

Sampling Distributions

Estimation: Describing a Single Population

Estimation: Comparing Two Populations

Hypothesis Testing: Describing a Single Population

Hypothesis Testing: Comparing Two Populations

Inference About Population Variances

Analysis of Variance

Additional Tests for Nominal Data: Chi-Squared Tests

Simple Linear Regression and Correlation

Model Building

Nonparametric Techniques

Statistical Inference: Conclusion

Time-Series Analysis and Forecasting

Index Numbers

Decision Analysis

Filters

Exam 19: Multiple Regression

For a multiple regression model, the following statistics are given: Total variation in y = SSY = 250, SSE = 50, k = 4, n = 20. The coefficient of determination adjusted for degrees of freedom is:

For the multiple regression model =40+15x1−10x2+5x3 = 40 + 15 x _ { 1 } - 10 x _ { 2 } + 5 x _ { 3 }=40+15x1​−10x2​+5x3​ , if x2x _ { 2 }x2​ were to increase by 5 units, holding x1x _ { 1 }x1​ and x3x _ { 3 }x3​ constant, the value of yyy would decrease by 50 units, on average.

Multicollinearity is a situation in which the independent variables are highly correlated with the dependent variable.

In a multiple regression model, the probability distribution of the error variable ε\varepsilonε is assumed to be:

In order to test the significance of a multiple regression model involving 4 independent variables and 25 observations, the number of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are 4 and 20, respectively.

In multiple regression, the descriptor 'multiple' refers to more than one independent variable.

In a multiple regression model, the following statistics are given: SSE = 100, R2=0.995R ^ { 2 } = 0.995R2=0.995 , k = 5, n = 15. The multiple coefficient of determination adjusted for degrees of freedom is:

The adjusted multiple coefficient of determination is adjusted for the number of independent variables and the sample size.

In multiple regression, when the response surface (the graphical depiction of the regression equation) hits every single point, the sum of squares for error SSE = 0, the standard error of estimate SεS _ { \varepsilon }Sε​ = 0, and the coefficient of determination R2R ^ { 2 }R2 = 1.

In a multiple regression analysis, when there is no linear relationship between each of the independent variables and the dependent variable, then:

A multiple regression model has the form ŷ = b0 + b1x1 + b2x2. Which of the following best describes b2?

In regression analysis, we judge the magnitude of the standard error of estimate relative to the values of the dependent variable, and particularly to the mean of y.

Which of the following statements is not true?

In order to test the validity of a multiple regression model involving 4 independent variables and 35 observations, the numbers of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are:

A multiple regression equation includes 5 independent variables, and the coefficient of determination is 0.64. The percentage of the variation in y that is explained by the regression equation is:

What Is Statistics

Types of Data, Data Collection and Sampling

Graphical Descriptive Methods Nominal Data

Graphical Descriptive Techniques Numerical Data

Numerical Descriptive Measures

Probability

Random Variables and Discrete Probability Distributions

Continuous Probability Distributions

Statistical Inference: Introduction

Sampling Distributions

Estimation: Describing a Single Population

Estimation: Comparing Two Populations

Hypothesis Testing: Describing a Single Population

Hypothesis Testing: Comparing Two Populations

Inference About Population Variances

Analysis of Variance

Additional Tests for Nominal Data: Chi-Squared Tests

Simple Linear Regression and Correlation

Model Building

Nonparametric Techniques

Statistical Inference: Conclusion

Time-Series Analysis and Forecasting

Index Numbers

Decision Analysis

Filters

In a multiple regression model, the probability distribution of the error variable $\varepsilon$ is assumed to be:

In a multiple regression model, the following statistics are given: SSE = 100, $R ^ { 2 } = 0.995$ , k = 5, n = 15. The multiple coefficient of determination adjusted for degrees of freedom is:

In multiple regression, when the response surface (the graphical depiction of the regression equation) hits every single point, the sum of squares for error SSE = 0, the standard error of estimate $S _ { \varepsilon }$ = 0, and the coefficient of determination $R ^ { 2 }$ = 1.

A multiple regression model has the form ŷ = b₀+ b₁x₁ + b₂x₂. Which of the following best describes b₂?