Regression analysis Please go to documents for "Appendix:MINITABoutput. "Use the Minitab output for the gasoline mileage data to answer questions that refer to output. You do not need the data, but it appears in Table B.3 in the 4th edition textbook. QUESTION 1 Adding more regressors to a regression model is always desirable because it may increase the . True False QUESTION 2 A prediction interval of a future response of y at an observation x is always wider than a confidence interval for the same observation of x. True False QUESTION 3 Adjusted will not necessarily increase when adding more regressors to a regression model. True False QUESTION 4 In a multiple regression, if the t-tests for individual regression coefficients show none of the coefficients are significant, then no regressors are useful. True False QUESTION 5 A variance inflation factor greater than 10 for a regressor x1 implies that x1 is linearly related to the other regressors. True False QUESTION 6 Normal probability plot of the observed y's is used to check the normality assumption of the errors. True False QUESTION 7 A lack-of-fit test requires that we have replicate observations on the response y for at least one level of x. True False QUESTION 8 Data transformation can be used when some of the model assumptions are violated. True False QUESTION 9 A predicted residual or prediction error is calculated for a row when the corresponding row is not used to estimate the coefficients of the model. True False QUESTION 10 A large implies that a point has high leverage. True False QUESTION 11 What are the units of the slope estimate 1 hat? miles per gallon cubic inches miles per gallon per cubic inch 1/(cubic inches) QUESTION 12 A one-unit change in x1 changes the estimated mean of y by how much? an increase of 0.0761 a decrease of 0.0761 an increase of 19.4 cannot be determined from the output QUESTION 13 What is the estimated ? 3.146 9.895 9.752 95.11 QUESTION 14 Calculate a 95% confidence interval for 1. (-0.13236, -0.0199) (-0.1938, 0.0416) (-0.24482, 0.09256) Not available QUESTION 15 In the output, no t-tests are significant at =0.05, but the F-test for regression has a p-value of approximately 0. The best explanation of these results is: These regressors are not useful predictors Multicollinearity is present All these regressors are useful predictors Only first variable x1 is a useful predictor QUESTION 16 Four assumptions for the multiple linear regression equation are: linearity, errors with constant variance, means zero, and normally distributed. To obtain estimates of the parameters, which assumptions are needed? Linearity, errors with means zero, and normally distributed Errors with constant variance, means zero, and normally distributed Linearity, errors with constant variance and normally distributed Linearity, errors with constant variance, means zero QUESTION 17 A failure of the linearity assumption is best detected by what plot? Normal probability plots of the residuals Residuals versus predicted y versus each x separately Plot of residuals in time sequence QUESTION 18 A failure of the nonconstant variance assumption is best detected by what plots? Normal probability plot Residuals versus predicted Residuals versus independent variables Both b and c QUESTION 19 If a row of data affects the prediction of its own y, but does not change other predictors very much, what influence measure would be most sensitive? COOK'S D DFFITS DFBETAS Both b and c QUESTION 20 Given that the following is the covariance matrix for the parameters in a multiple regression model with 3 parameters (one intercept and two slopes), what is the estimated standard error of 1 hat? 1.41 3 1.75 0.9 QUESTION 21 To calculate the Variance Inflation Factor for x3 in the regression model y on x1, x2, x3, one can use the Rsquared obtained from the regression model of x3 on x1 and x2 and the VIF is VIF = 1/(1-R2j) True False QUESTION 22 In a regression problem with n = 40 observations, and 4 parameters (including the intercept), what is the distribution of a deleted residual? Normal distribution t-distribution with 35 degrees of freedom t-distribution with 36 degrees of freedom none of the above QUESTION 23 Given a multiple regression problem with n = 30 rows and 3 predictors and an intercept, calculate the mean square for pure error. There are only 3 points with replicated y's and the y's are shown below. 18.25 21 2.25 3.6 QUESTION 24 If the x's are considered to be fixed numbers, in the 95% statement in the confidence interval what is assumed about the x's in hypothetical future sample? Observations of y are obtained at the same x values Observations of y are obtained at random x values Observations of y are obtained at a subset of the same x values Does not matter QUESTION 25 By adding any new regressors to a regression model, the R2 of the new model will not change will not decrease will not increase depends on which regressors are added Applied Regression Analysis Quiz: Minitab Output Results for: GASOLINE_MILEAGE_DATA_WITH_INDICATOR_VARIABLES.MTW y: miles per gallon x1: Displacement (cubic inches) x2: Horsepower (ft-lb) x3: Torque (ft-lb) Etc. Regression Analysis: y versus x1, x2, x3, x4, x5, x6, x7, x8, x9, x10 The regression equation is y = 19.4 - 0.0761 x1 - 0.0743 x2 + 0.121 x3 + 1.32 x4 + 5.98 x5 + 0.29 x6 - 3.40 x7 + 0.185 x8 - 0.409 x9 - 0.00518 x10 30 cases used, 2 cases contain missing values Predictor Constant x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Coef 19.35 -0.07613 -0.07425 0.12098 1.317 5.976 0.289 -3.398 0.1853 -0.4094 -0.005184 S = 3.14565 SE Coef 28.99 0.05623 0.08663 0.08907 3.021 3.076 1.251 2.859 0.1259 0.3124 0.005742 R-Sq = 83.5% T 0.67 -1.35 -0.86 1.36 0.44 1.94 0.23 -1.19 1.47 -1.31 -0.90 P 0.512 0.192 0.402 0.190 0.668 0.067 0.820 0.249 0.158 0.206 0.378 R-Sq(adj) = 74.8% Analysis of Variance Source Regression Residual Error Total Source x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 DF 1 1 1 1 1 1 1 1 1 1 DF 10 19 29 SS 951.098 188.007 1139.105 MS 95.110 9.895 F 9.61 P 0.000 Seq SS 866.227 5.436 4.441 14.489 2.876 9.074 6.644 0.347 33.502 8.063 Unusual Observations Obs 22 31 x1 360 360 y 21.470 13.770 Fit 16.170 19.049 SE Fit 2.212 2.080 Residual 5.300 -5.279 St Resid 2.37R -2.24R R denotes an observation with a large standardized residual. y x1 18.9 17 20 18.25 20.07 11.2 22.12 21.47 34.7 30.4 16.5 36.5 21.5 19.7 20.3 17.8 14.39 14.89 17.8 16.41 23.54 21.47 16.59 31.9 29.4 13.27 23.9 19.73 13.9 13.27 13.77 16.5 x2 350 350 250 351 225 440 231 262 89.7 96.9 350 85.3 171 258 140 302 500 440 350 318 231 360 400 96.9 140 460 133.6 318 351 351 360 360 x3 165 170 105 143 95 215 110 110 70 75 155 80 109 110 83 129 190 215 155 145 110 180 185 75 86 223 96 140 148 148 195 165 x4 260 275 185 255 170 330 175 200 81 83 250 83 146 195 109 220 360 330 250 255 175 290 83 366 120 255 243 243 295 255 x5 8 8.5 8.25 8 8.4 8.2 8 8.5 8.2 9 8.5 8.5 8.2 8 8.4 8 8.5 8.2 8.5 8.5 8 8.4 7.6 9 8 8 8.4 8.5 8 8 8.25 8.5 x6 2.56 2.56 2.73 3 2.76 2.88 2.56 2.56 3.9 4.3 3.08 3.89 3.22 3.08 3.4 3 2.73 2.71 3.08 2.45 2.56 2.45 3.08 4.3 2.92 3 3.91 2.71 3.25 3.26 3.15 2.73 x7 4 4 1 2 1 4 2 2 2 2 4 2 2 1 2 2 4 4 4 2 2 2 4 2 2 4 2 2 2 2 4 4 x8 3 3 3 3 3 3 3 3 4 5 3 4 4 3 4 3 3 3 3 3 3 3 3 5 4 3 5 3 3 3 3 3 x9 200.3 199.6 196.7 199.9 194.1 184.5 179.3 179.3 155.7 165.2 195.4 160.6 170.4 171.5 168.8 199.9 224.1 231 196.7 197.6 179.3 214.2 196 165.2 176.4 228 171.5 215.3 215.5 216.1 209.3 185.2 x10 69.9 72.9 72.2 74 71.8 69 65.4 65.4 64 65 74.4 62.2 66.9 77 69.4 74 79.8 79.7 72.2 71 65.4 76.3 73 61.8 65.4 79.8 63.4 76.3 78.5 78.5 77.4 69 x11 3910 3860 3510 3890 3365 4215 3020 3180 1905 2320 3885 2009 2655 3375 2700 3890 5290 5185 3910 3660 3050 4250 3850 2275 2150 5430 2535 4370 4540 4715 4215 3660 1 1 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1