lab9 - STAT 350 – Spring 2009 Lab 9 SOLUTION Regression...

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Unformatted text preview: STAT 350 – Spring 2009 Lab 9 SOLUTION Regression For this lab we will use the two data sets (Fitness and Children) as we used in the previous lab. Unless otherwise stated below, the calculations, analyses, and graphs for this lab are to be done in SAS. All SAS code (contents of the editor window) and all SAS output should be given as an appendix. Fitness The first data set is on the worksheet titled “Fitness”. This data set gives measurements made on men involved in a phyical fitness course at N.C. State University (and taken from the SAS documentation). The variables are: • Age (in years) • Weight (in kilograms) • Oxygen Intake Rate (in ml per kg body weight per minute) • Time to Run 1.5 miles (in minutes) • Heart Rate (pulse) while Resting • Heart Rate (pulse) while Running (taken at the same time Oxygen rate was measured) • Maximum heart rate recorded while running Children The second data set is on the worksheet titled “Children”. The data set is from Lewis and Taylor, 1967 (and taken from the SAS documentation.) The variables are: • gender ("f" for females, "m" for males) • Age (in months) • Height (in inches) • Weight (in pounds) 1. For the following problems, use the Fitness data set Note: do NOT sort this data set! a. Give the least squares regression equation for predicting Maximum Running Pulse from Age (that is, y is Maximum Running Pulse and x is Age). ˆ y = 210.06928 − 0.76126 x Lab 9 - SOLUTION Page 1 b. Using the plot option within PROC REG, give a scatter plot of Maximum Running Pulse versus Age with the fitted regression line. M ul se = 210. 07 - 0. 7613 A axP ge 195 N 31 R sq 0. 1874 AR dj sq 0. 1594 190 RS ME 8. 4021 185 180 175 170 165 160 155 37. 5 40. 0 42. 5 45. 0 47. 5 50. 0 52. 5 55. 0 57. 5 A ge c. What percent of the total variation in maximum running pulse can be explained by the linear relationship between maximum running pulse and age in these men? 18.74% - This is just the value of R-Square as a percent d. What is the correlation between maximum running pulse and age? − 0.1874 = -0.432897 0.1874 is R-Square, we know that r is the negative square-root because as age increases, MaxPulse decreases. Lab 9 - SOLUTION Page 2 e. How much, one average, does Maximum Running Pulse increase when a man ages 1 year? Using SAS, obtain a point estimate and a 90% Confidence Interval (give your answers below and also highlight these values in the output). This is asking for a point estimate and 90% CI for β. The point estimate is just the slope of the regression line (b): -0.76126 90% CI: -1.26140 to -0.26112 These values are highlighted in green on the output. f. Based on this data set, is there evidence of a statistically significant linear relationship between a man's age and maximum running pulse? Give the value of the appropriate test statistic, along with its degrees of freedom, and the associated p-value. There are two equally acceptable answers here: Yes. t = -2.59, df = 29, p = 0.0150 OR Yes. F = 6.69, df1 = 1, df2 = 29, p = 0.0150 g. Using SAS, find the predicted maximum running pulse for the 15th subject (give your answers below and also highlight these values in the output). 168.9610 This value is highlighted in cyan on the output h. Using SAS, find the residual for the 15th subject (give your answers below and also highlight these values in the output). 1.0390 This value is highlighted in cyan on the output i. The 15th subject is 54 years old. Using SAS, find the 90% confidence interval for the true mean maximum running pulse for all 54 year old men of this population (give your answers below and also highlight these values in the output). 164.8899 to 173.0321 These values are highlighted in cyan on the output j. Assume you randomly select another 54 year old man from this population (not one of the men in this data set). Using SAS, find the 90% prediction interval for the maximum running pulse of this man (give your answers below and also highlight these values in the output). 154.1158 to 183.8063 These values are highlighted in cyan on the output The prediction interval would be the same for any 54 year old man, so we can take the PI associated with the 15th individual in the output (or any other the other individuals age 54 in the output – check the predicted value and prediction intervals for observations #22 and #26) A prediction interval is always for an un-observed value, so 154.1158 to 183.8063 does not predict the Max Running Pulse of #15 – we KNOW that is 170. A prediction interval for that subject is moot. Lab 9 - SOLUTION Page 3 2. For the following problems, use the Children data set a. Regression analysis: Predicting Height from Age i. Give the least squares regression equation to predict a child's height (in inches) from the child's age (in months) ˆ y = 38.51927 + 0.13894x ii. Using the plot option within PROC REG, give a scatter plot of Height versus Age with the fitted regression line. hei ght = 38. 519 + 1389 age 0. 75 N 237 R sq 0. 4210 AR dj sq 0. 4186 RS ME 3. 0085 70 65 60 55 50 130 140 150 160 170 180 190 200 210 220 230 240 250 age Lab 9 - SOLUTION Page 4 iii. Based on this analysis, how much, on average, does a child's height increase in one month? Use SAS to obtain a point estimate and a 99% confidence interval (give your answers below and also highlight these values in the output). This is asking for a point estimate and 99% CI for β. The point estimate is just the slope of the regression line (b): 0.13894 99% CI: 0.11134 to 0.16654 These values are highlighted in magenta on the output. iv. What percent of the total variation in heights of these children can be explained by the linear relationship between height and age? 42.10% (this is R-Square) v. Based on this data set, is there evidence of a statistically significant linear relationship between a child's age and height? Give the value of the appropriate test statistic, along with its degrees of freedom, and the associated p-value. There are two equally acceptable answers here: Yes. t = 13.07, df = 235, p <0.0001 OR Yes. F = 170.88, df1 = 1, df2 = 235, p <0.0001 b. Does scale matter in simple linear regression? In SAS, make two new variables: • age_years - the age of each child in years • height_cm – the height of each child in centimeters. Note – these variables must be made in SAS to get full credit. Making these variables in Excel (or another program) then bringing into SAS will NOT get full credit! Let Model 1 denote the original analysis from part (a), with height (in inches) as the response and age (in months) as the explanatory variable. Run 2 additional regression analyses: Model 2: height (in inches) is the response, age (in years) is the explanatory variable Model 3: height (in cm) is the response, age (in month) is the explanatory variable i. Summarize your findings by completing the following table. Model #: units of Height (y): units of Age (x): estimate of y-intercept (a): estimate of slope (b): Coefficient of Determination (R2): test statistic (t) for H0: β = 0 Lab 9 - SOLUTION 1 inches months 38.51927 0.13894 0.4210 13.07 2 inches years 38.51927 1.66723 0.4210 13.07 3 cm months 97.83895 0.35290 0.4210 13.07 Page 5 ii. What did you learn from this analysis? You should address how multiplying x by a constant (c) affects the estimate of slope and y-intercept; how multiplying y by a constant (c) affects the estimate of slope and y-intercept; how altering the scale affects the coefficient of determination and the test of H0: β = 0. Changing the units (scale) of the x axis, affected only the estimate of slope. You should notice that b2 = 12×b1 It makes sense that if a child grows, on average, 0.13894 inches in 1 month, a child would grow 12×0.13894 = 1.66723 inches in 12 months (1 year) Slope is not affected because a 0 month-old = 0 year-old child is predicted to be 38.51927 inches. Changing the units (scales) of the y axis, affects both the slope and y-intercept. You should notice that b3 = 2.54×b1 and a3 = 2.54×a1. A 0 month-old child is predicted to be 38.51927 inches which is 2.54×38.51927 = 97.83895 cm. For every increase of 1 month, a child is predicted to grow 0.13894 inches, which is 2.54×0.13894 = 0.35290 cm The coefficient of determination (r2) is not affected by scale, just like correlation (r). The value of the test statistic t is also not affected. It would be problematic if the relationship between two variables (such as age and height) was either significant or not depending on the units used! Recall that t = b/sb. When the scale of slope is changed (e.g., multiplied by a constant, c), both the numerator and denominator are multiplied by that constant so the constant cancels out. Lab 9 - SOLUTION Page 6 c. Regression Diagnostics i. Give a Residual Plot for the analysis in part (a). Discuss what you determine regarding the key assumptions of simple linear regression from this plot. hei ght = 38. 519 + 1389 age 0. 10. 0 N 237 R sq 0. 4210 AR dj sq 0. 4186 7. 5 RS ME 3. 0085 5. 0 2. 5 0. 0 - 2. 5 - 5. 0 - 7. 5 - 10. 0 130 140 150 160 170 180 190 200 210 220 230 240 250 age The residual plot is used to test 2 key assumptions: (1) is the relationship between x and y linear and (2) is the variance of the residuals (errors) constant across x? Based on this plot, I am not concerned about violating either of these assumptions Lab 9 - SOLUTION Page 7 ii. Give a QQ-plot of the residuals for the analysis in part (a). Discuss what you determine regarding the key assumptions of simple linear regression from this plot. 10. 0 7. 5 5. 0 2. 5 R e s i d u a l 0 - 2. 5 - 5. 0 - 7. 5 - 10. 0 -3 -2 -1 0 Nm or al 1 2 3 Q uant i l es The QQ-plot of the residuals is used to test the assumption that the residuals (or errors) are normally distributed. Based on this plot, I am not concerned about violating this assumption Appendix Do not forget to include your SAS code (contents of the Editor window) and all SAS output (contents of the Output window, NOT the Log window) as an appendix. Note: You may truncate(cut/shorten) the data set part(s) of the code to save paper. You may also truncate unnecessary output. However, any output used to answer the questions above must be given in the appendix. Lab 9 - SOLUTION Page 8 data fitness; input Age Weight Oxygen RunTime datalines; 44 89.47 44.609 11.37 44 85.84 54.297 8.65 38 89.02 49.874 9.22 40 75.98 45.681 11.95 44 81.42 39.442 13.08 44 73.03 50.541 10.13 45 66.45 44.754 11.12 54 83.12 51.855 10.33 51 69.63 40.836 10.95 48 91.63 46.774 10.25 57 73.37 39.407 12.63 52 76.32 45.441 9.63 51 67.25 45.118 11.08 51 73.71 45.790 10.47 49 76.32 48.673 9.40 52 82.78 47.467 10.50 ; RestPulse RunPulse MaxPulse @@; 62 45 55 70 63 45 51 50 57 48 58 48 48 59 56 53 178 156 178 176 174 168 176 166 168 162 174 164 172 186 186 170 data children; input sex $ age height weight @@; age_years = age/12; height_cm = height*2.54; datalines; f 143 56.3 85.0 f 155 62.3 105.0 f 191 62.5 112.5 f 171 62.5 112.0 f 160 62.0 94.5 f 140 53.8 68.5 f 157 64.5 123.5 f 149 58.3 93.0 f 191 65.3 107.0 f 150 59.5 78.5 f 141 61.8 85.0 f 140 53.5 81.0 f 185 63.3 101.0 f 166 61.5 103.5 f 210 65.5 140.0 f 146 56.3 83.5 f 149 64.3 110.5 f 139 57.5 96.0 f 169 62.3 99.5 f 177 61.8 142.5 f 173 62.8 102.5 f 166 59.3 89.5 f 150 61.3 94.0 f 184 62.3 108.0 f 144 59.5 93.5 f 177 61.3 112.0 f 146 60.0 109.0 f 145 59.0 91.5 f 155 61.3 107.0 f 167 62.3 92.5 f 183 64.5 102.5 f 185 60.0 106.0 f 154 60.0 114.0 f 156 54.5 75.0 f 152 60.5 105.0 f 191 63.3 113.5 f 148 60.5 84.5 f 189 64.3 113.5 f 164 65.3 98.0 f 157 60.5 112.0 f 177 61.3 81.0 f 171 61.5 91.0 f 183 66.5 112.0 f 143 61.5 116.5 f 182 65.5 133.0 f 182 62.0 91.5 f 165 55.5 67.0 f 154 61.0 122.5 f 163 56.5 84.0 f 141 56.0 72.5 f 171 63.0 84.0 f 167 61.0 93.5 f 193 59.8 115.0 f 141 61.3 85.0 f 169 61.5 85.0 f 175 60.3 86.0 m 157 60.5 105.0 m 144 57.3 76.5 m 139 60.5 87.0 m 189 67.0 128.0 m 146 57.5 90.0 m 160 60.5 84.0 m 151 66.3 117.0 m 141 53.3 84.0 m 153 60.0 84.0 m 206 68.3 134.0 m 176 65.0 118.5 m 140 59.5 94.5 m 146 57.3 83.0 m 183 66.0 105.5 m 151 61.0 81.0 m 144 62.8 94.0 m 193 66.3 133.0 m 162 64.5 119.0 m 143 57.5 75.0 m 175 64.0 92.0 m 173 69.0 112.5 m 170 63.8 112.5 m 144 59.5 88.0 m 156 66.3 106.0 m 147 57.0 84.0 m 188 67.3 112.0 m 150 59.5 84.0 m 193 67.8 127.5 m 140 58.5 86.5 m 156 58.3 92.5 m 184 66.5 112.0 m 156 68.5 114.0 m 168 66.5 111.5 m 149 52.5 81.0 m 203 66.5 117.0 m 142 58.8 84.0 m 200 71.0 147.0 m 152 59.5 105.0 m 145 56.5 91.0 m 143 57.5 101.0 Lab 9 - SOLUTION f f f f f f f f f f f f f f f f f f f f f f f f f f f f m m m m m m m m m m m m m m m m m m m m 182 168 180 180 176 168 176 170 172 164 176 166 172 188 188 172 153 185 139 143 147 164 175 170 186 185 168 139 178 147 183 148 144 190 143 147 172 179 142 150 147 182 164 180 150 183 156 150 250 185 140 160 164 175 174 149 169 157 156 144 142 189 174 163 40 42 47 43 38 45 47 49 51 49 54 50 54 57 48 63.3 59.0 61.5 51.3 61.3 58.0 60.8 64.3 57.8 65.3 61.5 52.8 63.5 55.8 64.3 56.3 55.8 66.8 58.3 59.5 64.8 63.0 56.0 54.5 51.5 64.0 63.3 61.3 59.5 64.8 61.8 59.0 67.5 66.0 56.5 59.3 60.5 68.0 66.0 57.0 62.0 58.0 61.5 57.0 55.0 66.3 69.8 65.3 75.07 68.15 77.45 81.19 81.87 87.66 79.15 81.42 77.91 73.37 79.38 70.87 91.63 59.08 61.24 108.0 104.0 104.0 50.5 115.0 83.5 93.5 90.0 95.0 118.0 95.0 63.5 148.5 75.0 109.5 77.0 73.5 140.0 77.5 101.0 142.0 98.5 72.5 74.0 64.0 111.5 108.0 110.5 84.0 111.0 112.0 99.5 171.5 105.0 84.0 78.5 95.0 112.0 108.0 92.0 100.0 80.5 108.5 84.0 70.0 112.0 119.5 117.5 45.313 59.571 44.811 49.091 60.055 37.388 47.273 49.156 46.672 50.388 46.080 54.625 39.203 50.545 47.920 f f f f f f f f f f f f f f f f f f f f f f f f f f f m m m m m m m m m m m m m m m m m m m m m 161 142 178 145 180 176 180 162 197 182 169 147 197 145 143 147 154 140 178 148 190 186 165 155 210 144 186 165 150 147 173 164 176 180 151 178 186 175 164 144 172 168 158 176 188 188 166 166 10.07 8.17 11.63 10.85 8.63 14.03 10.60 8.95 10.00 10.08 11.17 8.92 12.88 9.93 11.50 59.0 56.5 61.5 58.8 63.3 61.3 59.0 58.0 61.5 58.3 62.0 59.8 64.8 57.8 55.5 58.3 62.8 60.0 66.5 59.0 56.8 57.0 61.3 66.0 62.0 61.0 63.5 64.8 60.8 50.5 61.3 57.8 63.8 61.8 58.3 67.3 66.0 63.5 63.5 60.0 65.0 60.0 65.0 61.5 71.0 65.8 62.5 67.3 62 40 58 64 48 56 47 44 48 67 62 48 44 49 52 185 166 176 162 170 186 162 180 162 168 156 146 168 148 170 185 172 176 170 186 192 164 185 168 168 165 155 172 155 176 92.0 69.0 103.5 89.0 114.0 112.0 112.0 84.0 121.0 104.5 98.5 84.5 112.0 84.0 84.0 111.5 93.5 77.0 117.5 95.0 98.5 83.5 106.5 144.5 116.0 92.0 108.0 98.0 128.0 79.0 93.0 95.0 98.5 104.0 86.0 119.5 112.0 98.5 108.0 117.5 112.0 93.5 121.0 81.0 140.0 150.5 84.0 121.0 Page 9 m m m m m m m m m m m 182 177 150 171 142 144 193 139 196 153 164 67.0 63.0 59.0 61.8 56.0 60.0 72.0 55.0 64.5 57.8 66.5 133.0 111.0 98.0 112.0 87.5 89.0 150.0 73.5 98.0 79.5 112.0 m m m m m m m m m m m 173 177 150 162 148 206 194 186 164 155 189 66.0 60.5 61.8 63.0 60.5 69.5 65.3 66.5 58.0 57.3 65.0 112.0 112.0 118.0 91.0 118.0 171.5 134.5 112.0 84.0 80.5 114.0 m m m m m m m m m m m 155 175 188 141 140 159 152 161 159 178 164 61.8 65.5 63.3 57.5 56.8 63.3 60.8 56.8 62.8 63.5 61.5 91.5 114.0 115.5 85.0 83.5 112.0 97.0 75.0 99.0 102.5 140.0 m m m m m m m m m m m 162 166 163 174 160 149 146 153 178 142 167 60.0 62.0 66.0 63.0 64.0 56.3 55.0 64.8 63.8 55.0 62.0 105.0 91.0 112.0 112.0 116.0 72.0 71.5 128.0 112.0 76.0 107.5 m 151 59.3 87.0 ; options nodate pageno=1; /* Problem #1 */ title 'Fitness Data: Problem #1'; proc reg data=fitness alpha=0.10; model MaxPulse = Age/clb clm cli; plot MaxPulse*Age; run; /* Problem #2a */ title 'Children Data: Problem #2a'; proc reg data=children alpha=0.01; model height=age/clb; plot height*age; run; /* Problem #2b */ title 'Children Data: Problem #2b'; proc reg data=children; Model_2: model height=age_years; Model_3: model height_cm=age; run; /* Problem #2c */ title 'Children Data: Problem #2c'; proc reg data=children; model height=age; plot r.*age; output out=child_out r=resid; run; proc univariate data=child_out noprint; var resid; qqplot resid / normal (mu=est sigma=est); run; Lab 9 - SOLUTION Page 10 Fitness Data: Problem #1 1 The REG Procedure Model: MODEL1 Dependent Variable: MaxPulse Number of Observations Read Number of Observations Used 31 31 Analysis of Variance Source DF Sum of Squares Mean Square Model Error Corrected Total 1 29 30 472.17999 2047.23937 2519.41935 472.17999 70.59446 Root MSE Dependent Mean Coeff Var 8.40205 173.77419 4.83504 R-Square Adj R-Sq F Value Pr > F 6.69 0.0150 0.1874 0.1594 Parameter Estimates Variable Intercept Age DF Parameter Estimate Standard Error t Value Pr > |t| 1 1 210.06928 -0.76126 14.11483 0.29435 14.88 -2.59 <.0001 0.0150 90% Confidence Limits 186.08639 -1.26140 234.05218 -0.26112 Fitness Data: Problem #1 Answers for #1(e) 2 The REG Procedure Model: MODEL1 Dependent Variable: MaxPulse Output Statistics Obs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Dependent Predicted Std Error Variable Value Mean Predict 182.0000 185.0000 168.0000 172.0000 180.0000 176.0000 180.0000 170.0000 176.0000 186.0000 168.0000 192.0000 176.0000 164.0000 170.0000 185.0000 172.0000 168.0000 164.0000 168.0000 176.0000 165.0000 166.0000 155.0000 172.0000 172.0000 188.0000 155.0000 188.0000 176.0000 172.0000 176.5737 179.6187 176.5737 178.0962 181.1413 174.2899 179.6187 177.3349 176.5737 181.1413 176.5737 175.8124 175.8124 174.2899 168.9610 172.7674 171.2448 171.2448 173.5286 172.7674 166.6773 168.9610 170.4836 172.0061 171.2448 168.9610 171.2448 166.6773 172.7674 173.5286 170.4836 1.8571 2.7174 1.8571 2.2517 3.2236 1.5222 2.7174 2.0428 1.8571 3.2236 1.8571 1.7025 1.7025 1.5222 2.3960 1.5585 1.7983 1.7983 1.5120 1.5585 3.1317 2.3960 1.9739 1.6567 1.7983 2.3960 1.7983 3.1317 1.5585 1.5120 1.9739 90% CL Mean 173.4182 175.0015 173.4182 174.2703 175.6640 171.7035 175.0015 173.8640 173.4182 175.6640 173.4182 172.9197 172.9197 171.7035 164.8899 170.1193 168.1894 168.1894 170.9595 170.1193 161.3561 164.8899 167.1297 169.1912 168.1894 164.8899 168.1894 161.3561 170.1193 170.9595 167.1297 Sum of Residuals Sum of Squared Residuals Predicted Residual SS (PRESS) Lab 9 - SOLUTION 179.7292 184.2359 179.7292 181.9221 186.6186 176.8762 184.2359 180.8058 179.7292 186.6186 179.7292 178.7051 178.7051 176.8762 173.0321 175.4154 174.3003 174.3003 176.0978 175.4154 171.9984 173.0321 173.8374 174.8210 174.3003 173.0321 174.3003 171.9984 175.4154 176.0978 173.8374 90% CL Predict 161.9529 164.6145 161.9529 163.3163 165.8504 159.7813 164.6145 162.6429 161.9529 165.8504 161.9529 161.2461 161.2461 159.7813 154.1158 158.2477 156.6454 156.6454 159.0231 158.2477 151.4417 154.1158 155.8188 157.4551 156.6454 154.1158 156.6454 151.4417 158.2477 159.0231 155.8188 191.1944 194.6230 191.1944 192.8761 196.4321 188.7984 194.6230 192.0270 191.1944 196.4321 191.1944 190.3787 190.3787 188.7984 183.8063 187.2870 185.8443 185.8443 188.0341 187.2870 181.9128 183.8063 185.1484 186.5571 185.8443 183.8063 185.8443 181.9128 187.2870 188.0341 185.1484 Residual 5.4263 5.3813 -8.5737 -6.0962 -1.1413 1.7101 0.3813 -7.3349 -0.5737 4.8587 -8.5737 16.1876 0.1876 -10.2899 1.0390 12.2326 0.7552 -3.2448 -9.5286 -4.7674 9.3227 -3.9610 -4.4836 -17.0061 0.7552 3.0390 16.7552 -11.6773 15.2326 2.4714 1.5164 Answers for #1(g), (h), (i), and (j) 0 2047.23937 2299.68804 Page 11 Children Data: Problem #2a 3 The REG Procedure Model: MODEL1 Dependent Variable: height Number of Observations Read Number of Observations Used 237 237 Analysis of Variance DF Sum of Squares Mean Square 1 235 236 1546.65059 2126.97169 3673.62228 1546.65059 9.05094 Root MSE Dependent Mean Coeff Var 3.00848 61.36456 4.90263 Source Model Error Corrected Total R-Square Adj R-Sq F Value Pr > F 170.88 <.0001 0.4210 0.4186 Parameter Estimates Variable DF Parameter Estimate Standard Error t Value Pr > |t| 1 1 38.51927 0.13894 1.75851 0.01063 21.90 13.07 <.0001 <.0001 Intercept age 99% Confidence Limits 33.95257 0.11134 43.08598 0.16654 Children Data: Problem #2b 4 The REG Procedure Model: Model_2 Dependent Variable: height Number of Observations Read Number of Observations Used 237 237 Analysis of Variance DF Sum of Squares Mean Square 1 235 236 1546.65059 2126.97169 3673.62228 1546.65059 9.05094 Root MSE Dependent Mean Coeff Var 3.00848 61.36456 4.90263 Source Model Error Corrected Total R-Square Adj R-Sq F Value Pr > F 170.88 <.0001 0.4210 0.4186 Parameter Estimates Variable DF Parameter Estimate Standard Error t Value Pr > |t| Intercept age_years 1 1 38.51927 1.66723 1.75851 0.12754 21.90 13.07 <.0001 <.0001 Lab 9 - SOLUTION Page 12 Children Data: Problem #2b 5 The REG Procedure Model: Model_3 Dependent Variable: height_cm Number of Observations Read Number of Observations Used 237 237 Analysis of Variance DF Sum of Squares Mean Square 1 235 236 9978.37095 13722 23701 9978.37095 58.39307 Root MSE Dependent Mean Coeff Var 7.64154 155.86597 4.90263 Source Model Error Corrected Total R-Square Adj R-Sq F Value Pr > F 170.88 <.0001 0.4210 0.4186 Parameter Estimates Variable Intercept age DF Parameter Estimate Standard Error t Value Pr > |t| 1 1 97.83895 0.35290 4.46663 0.02700 21.90 13.07 <.0001 <.0001 Children Data: Problem #2c 6 The REG Procedure Model: MODEL1 Dependent Variable: height Number of Observations Read Number of Observations Used 237 237 Analysis of Variance DF Sum of Squares Mean Square 1 235 236 1546.65059 2126.97169 3673.62228 1546.65059 9.05094 Root MSE Dependent Mean Coeff Var 3.00848 61.36456 4.90263 Source Model Error Corrected Total R-Square Adj R-Sq F Value Pr > F 170.88 <.0001 0.4210 0.4186 Parameter Estimates Variable Intercept age Lab 9 - SOLUTION DF Parameter Estimate Standard Error t Value Pr > |t| 1 1 38.51927 0.13894 1.75851 0.01063 21.90 13.07 <.0001 <.0001 Page 13 ...
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