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Exam 16: Multiple Regression

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Question 61

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In multiple regression, the Durbin-Watson test is used to determine if there is autocorrelation in the regression model

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Question 62

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In a multiple regression analysis involving 40 observations and 5 independent variables, total variation in y = SST = 350 and SSE = 50. The coefficient of determination is:

A) 0.8408.
B) 0.8571.
C) 0.8469.
D) 0.8529.

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Question 63

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In a multiple regression problem involving 24 observations and three independent variables, the estimated regression equation is $\hat { y }$ = 72 + 3.2x1 + 1.5 x2 - x3. For this model, SST = 800 and SSE = 245. The value of the F-statistic for testing the significance of this model is 15.102.

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 ŷ = $55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$ $\begin{array} { | c | c c c | } \hline \text { Predictor } & \text { Coef } & \text { StDev } & \mathrm { T } \\\hline \text { Constant } & 55.8 & 11.8 & 4.729 \\x _ { 1 } & 1.79 & 0.44 & 4.068 \\x _ { 2 } & - 0.021 & 0.011 & - 1.909 \\x _ { 3 } & - 0.016 & 0.014 & - 1.143 \\\hline\end{array}$ se = 9.47 R2 = 22.5%. $\begin{array}{l}\text { ANALYSIS OF VARIANCE }\\\begin{array} { | l | c c c c | } \hline \text { Source of Variation } & \mathrm { df } & \text { SS } & \text { MS } & \text { F } \\\hline \text { Regression } & 3 & 936 & 312 & 3.477 \\\text { Error } & 36 & 3230 & 89.722 & \\\hline \text { Total } & 39 & 4166 & & \\\hline\end{array}\end{array}$ What is the coefficient of determination? What does this statistic tell you?

In a multiple regression model, the probability distribution of the error variable  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 standard error of estimate is defined by $S _ { \varepsilon } = \sqrt { SSE / ( n - k ) }$ , where n is the sample size and k is the number of independent variables.

For each x term in the multiple regression equation, the corresponding $\beta$ is referred to as a partial regression coefficient or slope of the independent variable.

To test the validity of a multiple regression model involving 2 independent variables, the null hypothesis is that:

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: $\begin{array} { | c | c | r | } \hline \text { Coffee sales revenue } & \text { Temperature } & \text { Pastries/biscuits } \\\hline 6.5 & 25 & 7 \\\hline 10 & 17 & 13 \\\hline 5.5 & 30 & 5 \\\hline 4.5 & 35 & 6 \\\hline 3.5 & 40 & 3 \\\hline 28 & 9 & 15 \\\hline\end{array}$ $\begin{array}{|l|r|l|l|l|l|l|}\hline \text { SUMMARY OUTPUT } & & & & & & \\\hline \text { Regression Statistios } & & & & & & \\\hline \text { Multiple R } & 0.87 & & & & & \\\hline \text { R Square } & 0.75 & & & & & \\\hline \text { Adjusted R Square } & 0.59 & & & & & \\\hline \text { Standard Error } & 5.95 & & & & & \\\hline \text { Observations } & 6.00 & & & & & \\\hline \\\hline \text { ANOVA } & & & & & \\\hline & {d f} & SS & M S & F & \text { Significance } F \\\hline \text { Regression } & 2.00 & 322.14 & 161.07 & 4.55 & 0.12 \\\hline \text { Residual } & 3.00 & 106.20 & 35.40 & & \\\hline \text { Total } & 5.00 & 428.33 & & & \\\hline \\\hline & \text { Coeffients } & \text { Standard Error } & \text { tStat } & \text { P-value } & {\text { Lower } 95 \%} & {\text { Upper } 95 \%} \\\hline \text { Intercept } & 18.68 & 37.88 & 0.49 & 0.66 & -101.88 & 139.24 \\\hline \text { Temperature } & -0.50 & 0.83 & -0.60 & 0.59 & -3.15 & 2.15 \\\hline \text { Pastries/biscuits } & 0.49 & 2.02 & 0.24 & 0.82 & -5.94 & 6.92 \\\hline\end{array}$ Interpret the intercept. Does this make sense?

In a multiple regression, a large value of the test statistic F indicates that most of the variation in y is explained by the regression equation, and that the model is useful; while a small value of F indicates that most of the variation in y is unexplained by the regression equation, and that the model is useless.

In multiple regression analysis involving 9 independent variables and 110 observations, the critical value of t for testing individual coefficients in the model will have:

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:

Which of the following is not true when we add an independent variable to a multiple regression model?

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.

An economist wanted to develop a multiple regression model to enable him to predict the annual family expenditure on clothes. After some consideration, he developed the multiple regression model: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ . Where: y = annual family clothes expenditure (in $1000s) $x _ { 1 }$ = annual household income (in $1000s) $x _ { 2 }$ = number of family members $x _ { 3 }$ = number of children under 10 years of age The computer output is shown below. THE REGRESSION EQUATION IS ŷ = $1.74 + 0.091 x _ { 1 } + 0.93 x _ { 2 } + 0.26 x _ { 3 }$ $\begin{array} { | c | c c c | } \hline \text { Predictor } & \text { Coef } & \text { StDev } & \mathrm { T } \\\hline \text { Constant } & 1.74 & 0.630 & 2.762 \\x _ { 1 } & 0.091 & 0.025 & 3.640 \\x _ { 2 } & 0.93 & 0.290 & 3.207 \\x _ { 3 } & 0.26 & 0.180 & 1.444 \\\hline\end{array}$ se = 2.06, R2 = 59.6%. $\begin{array}{l}\text { ANALYSIS OF VARIANCE }\\\begin{array} { | l | c c c c | } \hline \text { Source of Variation } & \mathrm { df } & \text { SS } & \text { MS } & \text { F } \\\hline \text { Regression } & 3 & 288 & 96 & 22.647 \\\text { Error } & 46 & 195 & 4.239 & \\\hline \text { Total } & 49 & 483 & & \\\hline\end{array}\end{array}$ What is the coefficient of determination? What does this statistic tell you?

A multiple regression model has the form ? = 24 - 0.001x1 + 3x2. As x1 increases by 1 unit, holding $x _ { 2 }$ constant, the value of y is estimated to decrease by 0.001units, on average.

In multiple regression, the problem of multicollinearity affects the t-tests of the individual coefficients as well as the F-test in the analysis of variance for regression, since the F-test combines these t-tests into a single test.

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 regression model involving 30 observations, the following estimated regression model was obtained: $\hat { y }$ = 60 + 2.8x1 + 1.2 x2 - x3. For this model, total variation in y = SST = 800 and SSE = 200. The value of the F-statistic for testing the validity of this model is: