Report

Adapted by Peter Au, George Brown College McGraw-Hill Ryerson Copyright © 2011 McGraw-Hill Ryerson Limited. Part 1 Basic Multiple Regression Part 2 Using Squared and Interaction Terms Part 3 Dummy Variables and Advanced Statistical Inferences (Optional) Copyright © 2011 McGraw-Hill Ryerson Limited 12-2 12.1 The Multiple Regression Model 12.2 Model Assumptions and the Standard Error 12.3 The Least Squares Estimates and Point Estimation and Prediction 12.4 R2 and Adjusted R2 12.5 The Overall F Test 12.6 Testing the Significance of an Independent Variable 12.7 Confidence and Prediction Intervals Copyright © 2011 McGraw-Hill Ryerson Limited 12-3 12.8 The Quadratic Regression Model (Optional) 12.9 Interaction (Optional) Copyright © 2011 McGraw-Hill Ryerson Limited 12-4 12.10 Using Dummy Variables to Model Qualitative Independent Variables 12.11 The Partial F Test: Testing the Significance of a Portion of a Regression Model Copyright © 2011 McGraw-Hill Ryerson Limited 12-5 Part 1 Copyright © 2011 McGraw-Hill Ryerson Limited 12-6 • Simple linear regression uses one independent variable to explain the dependent variable • Some relationships are too complex to be described using a single independent variable • Multiple regression models use two or more independent variables to describe the dependent variable • This allows multiple regression models to handle more complex situations • There is no limit to the number of independent variables a model can use • Like simple regression, multiple regression has only one dependent variable Copyright © 2011 McGraw-Hill Ryerson Limited 12-7 • The linear regression model relating y to x1, x2,…, xk is y = my|x1,x2,…,xk + e = b0 + b1x1 + b2x2 + … + bkxk +e where • my|x1,x2,…,xk + e = b0 + b1x1 + b2x2 + … + bkxk is the mean value of the dependent variable y when the values of the independent variables are x1, x2,…, xk • β0, β1,β2, … βkare the regression parameters relating the mean value of y to x1, x2,…, xk • ɛ is an error term that describes the effects on y of all factors other than the independent variables x1, x2,…, xk Copyright © 2011 McGraw-Hill Ryerson Limited 12-8 • Consider the following data table that relates two independent variables x1 and x2 to the dependent variable y (table 12.1) Data x1 x2 y 1 28.0 18 12.4 2 28.0 14 11.7 3 32.5 24 12.4 4 39.0 22 10.8 5 45.9 8 9.4 6 57.8 16 9.5 7 58.1 1 8.0 8 62.5 0 7.5 Copyright © 2011 McGraw-Hill Ryerson Limited 12-9 Copyright © 2011 McGraw-Hill Ryerson Limited 12-10 • The plot shows that y tends to decrease in a straight-line fashion as x1 increases • This suggests that if we wish to predict y on the basis of x1 only, the simple linear regression model y = β0 + β1x1 + ɛ relates y to x1 Copyright © 2011 McGraw-Hill Ryerson Limited 12-11 Copyright © 2011 McGraw-Hill Ryerson Limited 12-12 • This plot shows that y tends to increase in a straight-line fashion as x2 increases • This suggests that if we wish to predict y on the basis of x2 only, the simple linear regression model y = β0 + β1x2 + ɛ Copyright © 2011 McGraw-Hill Ryerson Limited 12-13 L01 • The experimental region is defined to be the range of the combinations of the observed values of x1 and x2 Copyright © 2011 McGraw-Hill Ryerson Limited 12-14 L01 • The mean value of y when IV1 (independent variable one) is x1 and IV2 is x2 is μy|x1, x2 (mu of y given x1 and x2 • Consider the equation μy|x1, x2 = β0 + β1x1 + β2x2, which relates mean y values to x1 and x2 • This is a linear equation with two variables, geometrically this equation is the equation of a plane in three-dimensional space Copyright © 2011 McGraw-Hill Ryerson Limited 12-15 L01 Copyright © 2011 McGraw-Hill Ryerson Limited 12-16 L02 • We need to make certain assumptions about the error term ɛ • At any given combination of values of x1, x2, . . . , xk, there is a population of error term values that could occur Copyright © 2011 McGraw-Hill Ryerson Limited 12-17 L02 • The model is y = my|x1,x2,…,xk + e = b0 + b1x1 + b2x2 + … + bkxk +e • Assumptions for multiple regression are stated about the model error terms, e’s Copyright © 2011 McGraw-Hill Ryerson Limited 12-18 L02 1. 2. 3. 4. Mean of Zero Assumption The mean of the error terms is equal to 0 Constant Variance Assumption The variance of the error terms s2 is, the same for every combination values of x1, x2,…, xk Normality Assumption The error terms follow a normal distribution for every combination values of x1, x2,…, xk Independence Assumption The values of the error terms are statistically independent of each other Copyright © 2011 McGraw-Hill Ryerson Limited 12-19 SSE e Copyright © 2011 McGraw-Hill Ryerson Limited 2 i 2 ˆ (y i y i ) 12-20 • This is the point estimate of the residual variance s2 • This formula is slightly different from simple regression s MSE 2 Copyright © 2011 McGraw-Hill Ryerson Limited SSE n- k 1 12-21 • This is the point estimate of the residual standard deviation s • MSE is from last slide • This formula too is slightly different from simple regression s MSE SSE n- k 1 • n-(k+1) is the number of degrees of freedom associated with the SSE Copyright © 2011 McGraw-Hill Ryerson Limited 12-22 • Using Table 12.6 • Compute the SSE to be s MSE 2 SSE n- (k 1) Copyright © 2011 McGraw-Hill Ryerson Limited 0 . 674 83 0.1348 s s 2 0 . 1348 0.3671 12-23 L03 • Estimation/prediction equation yˆ b 0 b 1 x 01 b 2 x 02 ... b k x 0 k • is the point estimate of the mean value of the dependent variable when the values of the independent variables are x1, x2,…, xk • It is also the point prediction of an individual value of the dependent variable when the values of the independent variables are x1, x2,…, xk • b0, b1, b2,…, bk are the least squares point estimates of the parameters b0, b1, b2,…, bk • x01, x02,…, x0k are specified values of the independent predictor variables x1, x2,…, xk Copyright © 2011 McGraw-Hill Ryerson Limited 12-24 • A formula exists for computing the least squares model for multiple regression • This formula is written using matrix algebra and is presented in Appendix F available on Connect • In practice, the model can be easily computed using Excel, MegaStat or many other computer packages Copyright © 2011 McGraw-Hill Ryerson Limited 12-25 Copyright © 2011 McGraw-Hill Ryerson Limited 12-26 Copyright © 2011 McGraw-Hill Ryerson Limited 12-27 1. Total variation is given by the formula (y i y ) 2 2. Explained variation is given by the formula ( yˆ i y ) 2 3. Unexplained variation is given by the formula 2 ˆ (y y ) i i 4. Total variation is the sum of explained and unexplained variation 5. R2 is the ratio of explained variation to total variation R 2 Explained variation Total variation Copyright © 2011 McGraw-Hill Ryerson Limited 12-28 L04 • The multiple coefficient of determination, R2, is the proportion of the total variation in the n observed values of the dependent variable that is explained by the multiple regression model Copyright © 2011 McGraw-Hill Ryerson Limited 12-29 • The multiple correlation coefficient R is just the square root of R2 • With simple linear regression, r would take on the sign of b1 • There are multiple bi’s in a multiple regression model • For this reason, R is always positive • To interpret the direction of the relationship between the x’s and y, you must look to the sign of the appropriate bi coefficient Copyright © 2011 McGraw-Hill Ryerson Limited 12-30 • Adding an independent variable to multiple regression will always raise R2 • R2 will rise slightly even if the new variable has no relationship to y • The adjusted R2 corrects for this tendency in R2 • As a result, it gives a better estimate of the importance of the independent variables k n 1 2 R R n 1 n (k 1) 2 • The bar notation indicates adjusted R2 Copyright © 2011 McGraw-Hill Ryerson Limited 12-31 • Excel Multiple Regression Output from Table 12.1 n k Explained variation Total variation R 2 24.87502 25.54875 0 . 97363 Copyright © 2011 McGraw-Hill Ryerson Limited 2 8 1 2 R 0 . 97363 0 . 963081 8 1 8 (2 1) 12-32 • Hypothesis • H0: b1= b2 = …= bk = 0 versus • Ha: At least one of b1, b2,…, bk ≠ 0 • Test Statistic F(model) (Explained variation) /k (Unexplain ed variation) /[n - (k 1)] • Reject H0 in favor of Ha if: • F(model) > Fa* or • p-value < a *F is based on k numerator and n-(k+1) denominator degrees of freedom a Copyright © 2011 McGraw-Hill Ryerson Limited 12-33 • Test Statistic F(model) (Explained variation) /k (Unexplain ed variation) /[n - (k 1)] 24 . 8751 / 2 0 . 6737 /( 8 3) 92 . 33 • F-test at a = 0.05 level of significance • Fa is based on 2 numerator and 5 denominator degrees of freedom F(model) 92 . 33 5 . 79 F.05 and p - value 0 . 000 0 . 001 a • Reject H0 at a=0.05 level of significance Copyright © 2011 McGraw-Hill Ryerson Limited 12-34 • The F test tells us that at least one independent variable is significant • The natural question is which one(s)? • That question will be addressed in the next section Copyright © 2011 McGraw-Hill Ryerson Limited 12-35 • A variable in a multiple regression model is not likely to be useful unless there is a significant relationship between it and y • Significance Test Hypothesis • H0: bj = 0 versus • Ha: bj ≠ 0 Copyright © 2011 McGraw-Hill Ryerson Limited 12-36 • If the regression assumptions hold, we can reject H0: bj = 0 at the a level of significance (probability of Type I error equal to a) if and only if the appropriate rejection point condition holds • Or, equivalently, if the corresponding p-value is less than a Copyright © 2011 McGraw-Hill Ryerson Limited 12-37 Alternative Ha: βj ≠ 0 Reject H0 If |t| > tα/2* Ha: βj > 0 t > tα Ha: βj < 0 t < –tα * That p Value Twice area under t distribution right of |t| Area under t distribution right of t Area under t distribution left of t is t > tα/2 or t < –tα/2 tα/2, tα, and all p values are based on n - (k + 1) degrees of freedom Copyright © 2011 McGraw-Hill Ryerson Limited 12-38 • Test Statistic t= bj s bj • A 100(1-α)% confidence interval for βj is [b j ta 2 sb j ] • ta, ta/2 and p-values are based on n – (k+1) degrees of freedom Copyright © 2011 McGraw-Hill Ryerson Limited 12-39 • It is customary to test the significance of every independent variable in a regression model • If we can reject H0: bj = 0 at the 0.05 level of significance, then we have strong evidence that the independent variable xj is significantly related to y • If we can reject H0: bj = 0 at the 0.01 level of significance, we have very strong evidence that the independent variable xj is significantly related to y • The smaller the significance level a at which H0 can be rejected, the stronger is the evidence that xj is significantly related to y Copyright © 2011 McGraw-Hill Ryerson Limited 12-40 • Whether the independent variable xj is significantly related to y in a particular regression model is dependent on what other independent variables are included in the model • That is, changing independent variables can cause a significant variable to become insignificant or cause an insignificant variable to become significant • This issue is addressed in a later section on multicollinearity Copyright © 2011 McGraw-Hill Ryerson Limited 12-41 • A sales manager evaluates the performance of sales representatives by using a multiple regression model that predicts sales performance on the basis of five independent variables • x1 = number of months the representative has been employed by the company • x2 = sales of the company’s product and competing products in the sales territory (market potential) • x3 = dollar advertising expenditure in the territory • x4 = weighted average of the company’s market share in the territory for the previous four years • x5 = change in the company’s market share in the territory over the previous four years • y = β0 + β 1x1 + β 2x2 + β 3x3 + β 4x4 + β 5x5 + ɛ Copyright © 2011 McGraw-Hill Ryerson Limited 12-42 • Using MegaStat a regression model was computed using collected data Sbj • The p values associated with Time, MktPoten, Adver, and MktShare are all less than 0.01, we have very strong evidence that these variables are significantly related to y and, thus, are important in this model • The p value associated with Change is 0.0530, suggesting weaker evidence that this variable is important Copyright © 2011 McGraw-Hill Ryerson Limited 12-43 L06 • The point on the regression line corresponding to a particular value of x01, x02,…, x0k, of the independent variables is yˆ b 0 b 1 x 01 b 2 x 02 ... b k x 0 k • It is unlikely that this value will equal the mean value of y for these x values • Therefore, we need to place bounds on how far the predicted value might be from the actual value • We can do this by calculating a confidence interval for the mean value of y and a prediction interval for an individual value of y Copyright © 2011 McGraw-Hill Ryerson Limited 12-44 L06 • Both the confidence interval for the mean value of y and the prediction interval for an individual value of y employ a quantity called the distance value • With simple regression, we were able to calculate the distance value fairly easily • However, for multiple regression, calculating the distance value requires matrix algebra • See Appendix F on Connect for more details Copyright © 2011 McGraw-Hill Ryerson Limited 12-45 L06 • Assume that the regression assumptions hold • The formula for a 100(1-a) confidence interval for the mean value of y is as follows: [ yˆ t a /2 s ( y yˆ ) ] s ( y yˆ ) s Distance value • This is based on n-(k+1) degrees of freedom Copyright © 2011 McGraw-Hill Ryerson Limited 12-46 • Assume that the regression assumptions hold • The formula for a 100(1-a) prediction interval for an individual value of y is as follows: [ yˆ t a /2 s yˆ ], s yˆ s 1 + Distance value • This is based on n-(k+1) degrees of freedom Copyright © 2011 McGraw-Hill Ryerson Limited 12-47 Data Sales Time MktPoten Adver MktShare 3669.88 43.10 74065.11 4582.88 3473.95 108.13 58117.30 5539.78 5.51 0.15 2295.10 13.82 21118.49 2950.38 10.91 -0.72 4675.56 186.18 68521.27 2243.07 8.27 0.17 6125.96 161.79 57805.11 7747.08 9.15 0.50 2.51 Change 0.34 2134.94 8.94 37806.94 402.44 5.51 0.15 5031.66 365.04 50935.26 3140.62 8.54 0.55 3367.45 220.32 35602.08 2086.16 7.07 -0.49 6519.45 127.64 46176.77 8846.25 12.54 1.24 4876.37 105.69 42053.24 5673.11 8.85 0.31 2468.27 57.72 36829.71 2761.76 5.38 0.37 2533.31 23.58 33612.67 1991.85 5.43 -0.65 2408.11 13.82 21412.79 1971.52 8.48 0.64 2337.38 13.82 20416.87 1737.38 7.80 1.01 4586.95 86.99 36272.00 10694.20 10.34 0.11 2729.24 165.85 23093.26 8618.61 5.15 0.04 3289.40 116.26 26879.59 7747.89 6.64 0.68 2800.78 42.28 39571.96 4565.81 5.45 0.66 3264.20 52.84 51866.15 6022.70 6.31 -0.10 3453.62 165.04 58749.82 3721.10 6.35 -0.03 1741.45 10.57 23990.82 860.97 7.37 -1.63 2035.75 13.82 25694.86 3571.51 8.39 -0.43 1578.00 8.13 23736.35 2845.50 5.15 0.04 4167.44 58.54 34314.29 5060.11 12.88 0.22 2799.97 21.14 22809.53 3552.00 9.14 -0.74 Copyright © 2011 McGraw-Hill Ryerson Limited 12-48 • Using The Sales Territory Performance Case • The point prediction of the sales corresponding to; • • • • • TIME = 85.42 MktPoten = 35182.73 Adver = 7281.65 Mothered = 9.64 Change = 0.28 • Using the regression model from before; • ŷ = -1,113.7879 + 3.6121(85.42) + 0.0421(35,182.73) + 0.1289(7,281.65) + 256.9555(9.64) + 324.5334(0.28) = 4,181.74 (that is, 418,174 units) • This point prediction is given at the bottom of the MegaStat output in Figure 12.7, which we repeat here: Copyright © 2011 McGraw-Hill Ryerson Limited 12-49 Copyright © 2011 McGraw-Hill Ryerson Limited 12-50 L06 • 95% Confidence Interval [ yˆ t a /2 s Distance [4181.74 value ] ( 2.093)( 430 . 232 ) 0.109 ] [ 4181 . 74 296 . 829 ] [ 3884 . 91, 4478 . 58 ] • 95% Prediction Interval [ yˆ t a /2 s 1 Distance [4181.74 value ] ( 2.093)( 430 . 232 ) 1 0.109 ] [ 4181 . 74 948 . 137 ] [ 3233 . 60 , 5129 . 88 ] Copyright © 2011 McGraw-Hill Ryerson Limited 12-51 Part 2 Copyright © 2011 McGraw-Hill Ryerson Limited 12-52 • One useful form of linear regression is the quadratic regression model • Assume that we have n observations of x and y • The quadratic regression model relating y to x is y = b0 + b1x + b2x2 + e, where • b0 + b1x + b2x2 is the mean value of the dependent variable y when the value of the independent variable is x • b0, b1, and b2 are unknown regression parameters relating the mean value of y to x • e is an error term that describes the effects on y of all factors other than x and x2 Table of Contents Copyright © 2011 McGraw-Hill Ryerson Limited Next Section Next Part 12-53 Copyright © 2011 McGraw-Hill Ryerson Limited 12-54 • Even though the quadratic model employs the squared term x2 and, as a result, assumes a curved relationship between the mean value of y and x, this model is a linear regression model • This is because b0 + b1x + b2x2 expresses the mean value y as a linear function of the parameters b0, b1, and b2 • As long as the mean value of y is a linear function of the regression parameters, we have a linear regression model Copyright © 2011 McGraw-Hill Ryerson Limited 12-55 • The human resources department administers a stress questionnaire to 15 employees in which people rate their stress level on a 0 (no stress) to 4 (high stress) scale • Work performance was measured as the average number of projects completed by the employee per year, averaged over the last five years Copyright © 2011 McGraw-Hill Ryerson Limited 12-56 Copyright © 2011 McGraw-Hill Ryerson Limited 12-57 ^ y 25 . 7152 4 . 9762 x 1 . 01905 x Copyright © 2011 McGraw-Hill Ryerson Limited 2 12-58 • We have only looked at the simple case where we have y and x • That gave us the following quadratic regression model y = b0 + b1x + b2x2 + e • However, we are not limited to just two terms • The following would also be a valid quadratic regression model y = b0 + b1x1 + b2x12 + b3x2 + b4x3 + e Copyright © 2011 McGraw-Hill Ryerson Limited 12-59 • Multiple regression models often contain interaction variables • These are variables that are formed by multiplying two independent variables together • For example, x1·x2 • In this case, the x1·x2 variable would appear in the model along with both x1 and x2 • We use interaction variables when the relationship between the mean value of y and one of the independent variables is dependent on the value of another independent variable Table of Contents Copyright © 2011 McGraw-Hill Ryerson Limited Next Section Next Part 12-60 • Consider a company that runs both radio and television ads for its products • It is reasonable to assume that raising either ad amount would raise sales • However, it is also reasonable to assume that the effectiveness of television ads depends, in part, on how often consumers hear the radio ads • Thus, an interaction variable would be appropriate Copyright © 2011 McGraw-Hill Ryerson Limited 12-61 Copyright © 2011 McGraw-Hill Ryerson Limited 12-62 Copyright © 2011 McGraw-Hill Ryerson Limited 12-63 Copyright © 2011 McGraw-Hill Ryerson Limited 12-64 • These last two figures imply that the more is spent on one type of advertising, the smaller the slope for the other type of advertising • The is, the slope of one line depends on the value on the other variable • That says that there is interaction between x1 and x2 Copyright © 2011 McGraw-Hill Ryerson Limited 12-65 • Froid Frozen Foods Experiment Copyright © 2011 McGraw-Hill Ryerson Limited 12-66 • It is fairly easy to construct data plots to check for interaction when a careful experiment is carried out • It is often not possible to construct the necessary plots with less structured data • If an interaction is suspected, we can include the interactive term and see if it is significant Copyright © 2011 McGraw-Hill Ryerson Limited 12-67 • When an interaction term (say x1x2) is important to a model, it is the usual practice to leave the corresponding linear terms (x1 and x2) in the model no matter what their p-values Copyright © 2011 McGraw-Hill Ryerson Limited 12-68 Part 3 Copyright © 2011 McGraw-Hill Ryerson Limited 12-69 • So far, we have only looked at including quantitative data in a regression model • However, we may wish to include descriptive qualitative data as well • For example, might want to include the sex of respondents • We can model the effects of different levels of a qualitative variable by using what are called dummy variables • Also known as indicator variables Copyright © 2011 McGraw-Hill Ryerson Limited 12-70 • A dummy variable always has a value of either 0 or 1 • For example, to model sales at two locations, would code the first location as a zero and the second as a 1 • Operationally, it does not matter which is coded 0 and which is coded 1 Copyright © 2011 McGraw-Hill Ryerson Limited 12-71 • Suppose that Electronics World, a chain of stores that sells audio and video equipment, has gathered the data in Table 12.13 • These data concern store sales volume in July of last year (y, measured in thousands of dollars), the number of households in the store’s area (x, measured in thousands), and the location of the store Copyright © 2011 McGraw-Hill Ryerson Limited 12-72 • Location Dummy Variable DM Copyright © 2011 McGraw-Hill Ryerson Limited 1 if a store is in a mall location 0 otherwise 12-73 Copyright © 2011 McGraw-Hill Ryerson Limited 12-74 Copyright © 2011 McGraw-Hill Ryerson Limited 12-75 • Consider having three categories, say A, B, and C • Cannot code this using one dummy variable • A=0, B=1, and C=2 would be invalid • Assumes the difference between A and B is the same as B and C • We must use multiple dummy variables • Specifically, a categories requires a-1 dummy variables • For A, B, and C, would need two dummy variables • x1 is 1 for A, zero otherwise • x2 is 1 for B, zero otherwise • If x1 and x2 are zero, must be C • This is why the third dummy variable is not needed Copyright © 2011 McGraw-Hill Ryerson Limited 12-76 • Geometrical Interpretation of the Sales Volume Model y = β0 1 β1x + β2DM + β3xDM + ɛ Copyright © 2011 McGraw-Hill Ryerson Limited 12-77 Copyright © 2011 McGraw-Hill Ryerson Limited 12-78 • So far, have only considered dummy variables as standalone variables • Model so far is y = b0 + b1x + b2D + e, where D is dummy variable • However, can also look at interaction between dummy variable and other variables • That model would take the for y = b0 + b1x + b2D + b3xD+ e • With an interaction term, both the intercept and slope are shifted Copyright © 2011 McGraw-Hill Ryerson Limited 12-79 • So far, we have seen dummy variables used to code categorical variables • Dummy variables can also be used to flag unusual events that have an important impact on the dependent variable • These unusual events can be one-time events • Impact of a strike on sales • Impact of major sporting event coming to town • Or they can be reoccurring events • Hot temperatures on soft drink sales • Cold temperatures on coat sales Copyright © 2011 McGraw-Hill Ryerson Limited 12-80 • So far, we have looked at testing single slope coefficients using t test • We have also looked at testing all the coefficients at once using F test • The partial F test allows us to test the significance of any set of independent variables in a regression model Copyright © 2011 McGraw-Hill Ryerson Limited 12-81 • We can use this F test to test the significance of a portion of a regression mode Copyright © 2011 McGraw-Hill Ryerson Limited 12-82 • The model: y = b0 + b1x + b2DM + b3DD + e • DM and DD are dummy variables • This called the complete model • Will now look at just the reduced model: y = b0 + b1x + e • Hypothesis to test • H0: b2 = b3 = 0 verus Ha: At least one of b2 and b3 does not equal zero • The SSE for the complete model is SSEC = 443.4650 • The SSE for the reduced model is SSER = 2,467.8067 Copyright © 2011 McGraw-Hill Ryerson Limited 12-83 L05 F SSE R / k g SSE c / n k 1 SSE 2 ,467 . 8067 c 443 . 4650 / 2 443 . 4650 / 15 4 25 . 1066 • We compare F with F.01 = 7.21 • Based on k – g = 2 numerator degrees of freedom • And n – (k + 1) = 11 denominator degrees of freedom • Note that k – g denotes the number of regression parameters set to 0 • Since F = 25.1066 > 7.21 we reject the null hypothesis at a = 0.01 • We conclude that it appears as though at least two locations have different effects on mean sales volume Copyright © 2011 McGraw-Hill Ryerson Limited 12-84 • The multiple regression model employs at least 2 independent variables to relate to the dependent variable • Some ways to judge a models overall utility are; standard error, multiple coefficient of determination, adjusted multiple coefficient of determination, and the overall F test • Square terms can be used to model quadric relationships while cross product terms can be used to model interaction relationships • Dummy variables can use used to model qualitative independent variables • The partial F test can be used to evaluate a portion of the regression model Copyright © 2011 McGraw-Hill Ryerson Limited 12-85 Numerator df =2 7.21 Denominator df = 11 Copyright © 2011 McGraw-Hill Ryerson Limited Return 12-86