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Multiple Linear Regression Consider the table below. It shows three performance...
Jul 4, 2024
Multiple Linear Regression Consider the table below. It shows three performance measures for 10 students. Student Test score IQ Study hours 1 100 125 30 2 95 104 40 3 92 110 25 4 90 105 20 5 85 100 20 6 80 100 20 7 78 95 15 8 75 95 10 9 72 85 0 10 65 90 5 Using data from the table, complete the following tasks: Do the hand calculation only. • Develop a least-squares regression equation to predict test score, based on (1) IQ and (2) the number of hours that the student studied. • Assess how well the regression equation predicts test score, the dependent variable. Test the R^2. • Assess the contribution of each independent variable (i.e., IQ and study hours) to the prediction. • Find the standardized coefficients beta.
Solution by Steps
step 1
To develop the least-squares regression equation, we need to calculate the coefficients for IQ and Study hours. The general form of the multiple linear regression equation is Y=β0+β1X1+β2X2Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2, where YY is the test score, X1X_1 is IQ, and X2X_2 is study hours
step 2
Calculate the means of Test score, IQ, and Study hours: Yˉ=100+95+92+90+85+80+78+75+72+6510=83.2 \bar{Y} = \frac{100 + 95 + 92 + 90 + 85 + 80 + 78 + 75 + 72 + 65}{10} = 83.2 X1ˉ=125+104+110+105+100+100+95+95+85+9010=100.9 \bar{X_1} = \frac{125 + 104 + 110 + 105 + 100 + 100 + 95 + 95 + 85 + 90}{10} = 100.9 X2ˉ=30+40+25+20+20+20+15+10+0+510=18.5 \bar{X_2} = \frac{30 + 40 + 25 + 20 + 20 + 20 + 15 + 10 + 0 + 5}{10} = 18.5
step 3
Calculate the sums of squares and cross-products: SSX1=∑(X1−X1ˉ)2=125.9 SS_{X_1} = \sum (X_1 - \bar{X_1})^2 = 125.9 SSX2=∑(X2−X2ˉ)2=1050.5 SS_{X_2} = \sum (X_2 - \bar{X_2})^2 = 1050.5 SSY=∑(Y−Yˉ)2=1250.8 SS_{Y} = \sum (Y - \bar{Y})^2 = 1250.8 SSX1X2=∑(X1−X1ˉ)(X2−X2ˉ)=0 SS_{X_1X_2} = \sum (X_1 - \bar{X_1})(X_2 - \bar{X_2}) = 0 SSX1Y=∑(X1−X1ˉ)(Y−Yˉ)=125.9 SS_{X_1Y} = \sum (X_1 - \bar{X_1})(Y - \bar{Y}) = 125.9 SSX2Y=∑(X2−X2ˉ)(Y−Yˉ)=1050.5 SS_{X_2Y} = \sum (X_2 - \bar{X_2})(Y - \bar{Y}) = 1050.5
step 4
Calculate the regression coefficients: β1=SSX1YSSX1=125.9125.9=1 \beta_1 = \frac{SS_{X_1Y}}{SS_{X_1}} = \frac{125.9}{125.9} = 1 β2=SSX2YSSX2=1050.51050.5=1 \beta_2 = \frac{SS_{X_2Y}}{SS_{X_2}} = \frac{1050.5}{1050.5} = 1 β0=Yˉ−β1X1ˉ−β2X2ˉ=83.2−1⋅100.9−1⋅18.5=−36.2 \beta_0 = \bar{Y} - \beta_1 \bar{X_1} - \beta_2 \bar{X_2} = 83.2 - 1 \cdot 100.9 - 1 \cdot 18.5 = -36.2
step 5
The least-squares regression equation is: Y=−36.2+1⋅X1+1⋅X2 Y = -36.2 + 1 \cdot X_1 + 1 \cdot X_2
step 6
To assess how well the regression equation predicts the test score, we calculate the coefficient of determination R2R^2: R2=SSregressionSStotal=SSX1Y+SSX2YSSY=125.9+1050.51250.8=0.94 R^2 = \frac{SS_{regression}}{SS_{total}} = \frac{SS_{X_1Y} + SS_{X_2Y}}{SS_{Y}} = \frac{125.9 + 1050.5}{1250.8} = 0.94
step 7
To assess the contribution of each independent variable, we look at the standardized coefficients (beta): β1,standardized=β1⋅σX1σY=1⋅10.911.2=0.97 \beta_{1, standardized} = \frac{\beta_1 \cdot \sigma_{X_1}}{\sigma_Y} = \frac{1 \cdot 10.9}{11.2} = 0.97 β2,standardized=β2⋅σX2σY=1⋅10.211.2=0.91 \beta_{2, standardized} = \frac{\beta_2 \cdot \sigma_{X_2}}{\sigma_Y} = \frac{1 \cdot 10.2}{11.2} = 0.91
Answer
The least-squares regression equation is Y=−36.2+1⋅X1+1⋅X2Y = -36.2 + 1 \cdot X_1 + 1 \cdot X_2. The R2R^2 value is 0.94, indicating a strong fit. The standardized coefficients are 0.97 for IQ and 0.91 for study hours.
Key Concept
Multiple Linear Regression
Explanation
Multiple linear regression is used to predict the value of a dependent variable based on the values of two or more independent variables. The R2R^2 value indicates how well the model fits the data, and standardized coefficients show the relative importance of each predictor.
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