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1 Homework #8 Business Statistics with Computer Applications II Problems in Book: 14.33, 14.35, 14.36, 14.37, 14.38 Problems not in book: 1. Growth of pine trees. The Department of Biology at Kenyon College conducted an experiment to study the growth of pine trees. In April 1990, volunteers planted 1000 white pine (Pinus strobus) seedlings at the Brown Family Environmental Center. The seedlings were planted in two grids, distinguished by 10-and 15-foot spacings between the seedlings. Table 28.12 (page 28-67) shows the first 10 rows of a subset of the data collected by students at Kenyon College.1 a. Use tree height at the time of planting (Hgt90) and the indicator variable for fertilizer (Fert) to fit a multiple regression model for predicting Hgt97. Specify the estimated regression model and the regression standard error. Are you happy with the fit of this model? Comment on the value of R2 and the plot of the residuals against the predicted values. b. Construct a correlation matrix with Hgt90, Hgt96, Diam96, Grow96, Hgt97, Diam97, Spread97, and Needles97. Which variable is most strongly correlated with the response variable of interest (Hgt97)? Does this make sense to you? c. Add tree height in September 1996 (Hgt96) to the model in part (a). Does this model do a better job of predicting tree height in 1997? Explain. d. What happened to the individual t statistic for Hgt90 when Hgt96 was added to the model? Explain why this change occurred. e. Fit a multiple regression model for predicting Hgt97 based on the explanatory variables Diam97, Hgt96, and Fert. Summarize the results of the individual t tests. Does this model provide a better fit than the previous models? Explain by comparing the values of R2 and s for each model. f. Does the parameter estimate for the variable indicating whether a tree was fertilized or not have the sign you expected? Explain. (Experiments can produce surprising results!) g. Do you think that the model in part (e) should be used for predicting growth in other pine seedlings? Think carefully about the conditions for inference. 2. Catalog spending. The file catalog.xls shows catalog-spending data for 200 randomly selected individuals from a very large (over 20,000 households) data base.2 We are interested in developing a model to predict spending ratio. There are no missing values in the data set, but there are some incorrect entries that must be identified and removed before completing the analysis. Income is coded as an ordinal value, ranging from 1 to 12. Age can be regarded as quantitative, and any value less than 18 is invalid. Length of residence (LOR) is a value ranging from zero to someone's age. LOR should not be higher than age. All of the catalog variables are represented by indicator variables; either the consumer bought and the variable is coded as 1 or 1 I thank Ray and Pat Heithaus for providing data on the pine seedlings at the Brown Family Environmental Center. I thank David Cameron for providing the random sample of 200 observations from a large catalog-spending data base. 2 Homework #8 Business Statistics with Computer Applications II the consumer didn't buy and the variable is coded as 0. The other variables can be viewed as indexes for measuring assets, liquidity, and spending. Find a multiple regression model for predicting the amount of money that consumers will spend on catalog shopping, as measured by spending ratio. Your goal is to identify the best model you can. Remember to check the conditions for inference as you evaluate your models. Row Col 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Hgt90 * Hgt96 * 14 17 * Diam96 * 284 387 * 24 22 18 32 22 17 17 * 21 29 29 * * * 18 * 16.5 22.5 * * * * * 110 75 63 90 36 44 51 96 94 72 16 76 * 4.8 * 219 84.5 85 52 57 40 50 86 58 68 70 83 71 66 70 5.6 2 1.9 4.3 3.8 3.1 3.5 3.9 5.3 4.7 1.5 4 368 * 25 * * * 74 78 3.7 4.9 3 2.7 1.2 2.8 2.9 2.7 3.2 4.5 3.9 2.9 3.1 5.2 362 178 174 314 261 257 226 296 336 307 136 262 23 29 * * * 24 20 9 21 15 29 11 10 13 6 20 26 * 5 5.7 271.5 337 230 238 124 238 246 218 222 274 273 252 244 332 18 18 18.5 22 25 18 18 18 26 70 84 96 88 64 83 71 59 77 74 35 86 0.9 * * * * * 298 327 19.5 29 18 32 96 110 3.9 5.6 5.4 5.4 1.3 4.5 3.3 3.4 3.5 2.5 4.7 4.9 167 * * 4.2 7.4 294 310 318 328 157 282 251 201 262 210 265 327 * Grow96 * 100 * 3.1 60 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 13 15 30 8 15 * 247 249 266 324 215 * 17 21 19 20 15 9 22 19 14 20 18 25 21 18.4 22.5 19.8 24 19 19.8 24 19 16 * * * * 18 19 73 98 96 53 77 55 90 56 73 74 72 100 42 40 102 75 60.5 62 71 74 63.3 64 49.5 62.5 * 3.4 3 5.1 2.1 1.8 1.5 3.7 3.5 5.5 2.7 * 244 355 330 314 280 268 325 311 195 324 146 * * * * 12 18 16 24 22 20 14 23 21 26 15 72 84 81 106 65 5.2 5.4 4.2 6 4.1 1.4 5.3 3.6 4.1 4.4 4.7 6.2 3.3 2 6.3 3.4 3 5.9 4.7 6.3 5.3 3 1.6 3.5 226 209.5 325 209 182 181 307 279 380 235 * * * 300 333 324 278 282 151 310 239 250 278 250 347 247 186 381 284 224.5 280 307 317 271.3 236 162.5 241 24 19 24.5 24 19.5 20 21 * 3.8 2.6 3.6 4.7 2.2 5 55.5 66 44 36 39 89 91 100 81 * 2.4 5.3 5.2 4.6 3.3 3.1 4.3 5.9 3.6 4.4 1.5 * 70 81 81 64 101 98 96 78 0 93 17 * 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 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20 23 16 20 20 * * 340 225 157 29 * 3 1.6 5.1 3.8 5.4 54.5 108 * 2.9 3.1 3.8 4.7 2.5 3.2 3.1 2.7 5.1 5.1 5.1 4.7 4.4 3.5 3.6 1.8 6.5 5.7 5.2 5.8 4.7 87 58 64 88 41 76 16 70 85 80 80 81 82 83 79 58 100 73 78 92 91.3 37 37 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 26 27 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 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 19 17.5 21 10 17 * * 19 16 16 18 24 19 17 11 17 23 11 22 16 20 19 20 13 11.5 15 15 16 14 * * * * 29 18 27 24 8 21 19 19 23 21 * * 31 13 15 14 * * 14 18 19 21 23 20 21 353.5 230.1 246 217 120 * * 118 331 249 225 307 352 387 289 301 288 273 279 305 242 326 294 384 306 281 355.5 349 333 * * 220 293 361 315 159 285 277 240 135 351 * 277 288 346 362 244 332 407 336 318 470 363 297 312 4.8 2.8 2.4 1.9 92.5 69.1 77 55 9 * 0.6 3.8 3.4 2.5 3.1 4.6 5.3 4.2 3.5 3.6 3.1 3.6 4.7 4.2 4.8 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103 91 83 28 63 55 95 78 4.1 4.2 3.5 5.1 1.3 4.3 3.1 6 5.7 * * 2.1 3.6 2.3 6.4 6.7 6.5 4.8 2.4 4.4 5.2 5.6 6.9 6.1 3.5 5.2 2.8 3.5 3.2 4.9 3.5 4.6 3.4 2.3 77 92 77 112 97 84 88 68 75 103 94 102 85 71 74 37 73 78 80 59 87 57 60 * 3.3 0.8 5.4 85 40 100 * 5.6 3.9 4 4.8 6.8 110 80 102 85 110 * 5.5 4.1 5.3 4.2 5 5.2 100 98 84 95 85 89 43 43 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 26 27 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 * 12.5 22 19 21 27 10 21 12 31 25 20 20 203 295 192 329 416 435 279 366 355 299 106 350 * 32 32 26 25 24 16 16 25 22 19 21 17 20 11 19 2 4.1 1.6 4.4 5.4 6.4 4.1 5.2 5.1 4.3 0.7 5.5 * 304 236 253 371 329 312 204 322 281 289 213 346 315 319 231 * 73 83 76 89 93 86 74 106 88 74 46 87 * 4.5 3.4 3.1 5.3 5.6 4.5 1.7 3.6 5.6 4.7 2.3 4.6 4.3 4.2 2.4 * 77 51 70 96 84 81 64 96 34 74 71 100 81 91 76 * Hgt97 * Diam97 * 362 442 * 6.6 9.3 * 369 365 356 365 208 329 298 245 317 275 333 366 * 7 6.9 7.6 7.7 2 6.1 3.9 5 5.8 4.5 5.8 6.7 * 199 * 2.05 * 336 367 * 6 8.06 * 331 371 301 313 194 321 315 271 277 341 349 334 303 382 * * 5.39 8.86 4.75 5.77 2.35 5.25 4.89 4.85 5.14 7.81 6.18 4.46 6.66 7.09 * * 267 248 385 320 325 278 349 392 352 158 327 * 3.6 3.6 5.9 5.5 4.6 5.3 5.8 7.2 6.6 2.3 5.9 * 400 * 6.5 * 279 4.9 Spread97 * 162 250 * 176 215 238 219 127 209 154 167 180 190 140 180 * 84 * 163 200 * 182 200 170 175 115 200 140 137 185 210 169 160 130 196 * * 162 145 254 168 169 171 217 274 255 90 228 * 215 * 147 Needles97 * Deer95 * 66 77 * Deer97 * 0 0 * 72 76 74.5 60.5 56 79.5 70.5 77 67 65 77 58 * * 0 0 0 0 1 0 0 0 0 0 0 0 * 80 * 63 72.5 * * 75 70 45 65 61 84 66 81 80 60 80 * * * * 0 0 0 0 0 0 0 0 0 0 0 * 1 * 83 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 84 * 1 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 * * * 0 * 0 0 80.5 66.7 85.8 80 70.5 89.1 75.3 83 75.8 70 71.5 62.4 70 63.6 * * 0 0 0 0 1 1 0 0 0 0 0 0 1 * * 1 0 0 * 0 0 312 333 327 386 310 * 5 3.8 5.3 6.8 4.3 * 361 388 380 353 340 212 373 293 333 333 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1 0 0 0 0 0 0 0 0 0 0 0 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 0 1 0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 2 3 3 1 3 3 3 3 3 3 3 1 0 3 3 3 3 3 3 2 3 1 3 2 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 3 1 1 3 3 0 3 2 3 0 3 3 2 0 2 2 3 2 2 2 3 1 3 3 1 0 0 2 0 0 0 3 0 2 0 1 2 3 3 1 0 0 2 2 1 1 1 1 1 1 1 1 2 2 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 2 1 3 3 2 2 3 1 3 2 1 2 1 3 2 3 3 3 3 0 2 2 2 1 2 1 2 1 1 0 0 0 0 1 3 3 3 2 2 3 3 3 1 2 3 3 2 2 0 2 3 3 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 0 1 2 1 3 2 1 3 1 3 2 1 0 0 2 0 1 2 2 1 1 1 2 3 2 1 1 2 2 0 1 3 3 3 1 3 3 0 2 3 3 0 2 2 2 2 2 0 2 1 1 2 3 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 1 3 0 2 3 2 2 2 0 3 3 2 3 3 2 2 1 2 1 2 1 0 3 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 602 C h a p t e r 1 4 B u ild in g M u lt ip le R e g r e s s io n M o d e ls products for 10 years during a 20-year period. Use these data and a stepwise regression analysis to predict corn production by the production of soybeans and wheat. Comment on the results. or conversions of commercial ships. Graph each of these predictors separately with the response variable and use Tukey's four-quadrant approach to explore possible recoding schemes for nonlinear relationships. Include any of these in the regression model. Comment on the regression model and its strengths and its weaknesses. Employment Naval Vessels Commercial Ship Repairs or Conversions 133.4 177.3 143.0 142.0 130.3 120.6 120.4 108 99 105 111 100 85 79 431 1335 1419 1631 852 847 806 14.33 The U.S. Bureau of Labor Statistics produces consumer price indexes for several different categories. Shown here are the percentage changes in consumer price indexes over a period of 20 years for food, shelter, apparel, and fuel oil. Also displayed are the percentage changes in consumer price indexes for all commodities. Use these data and a stepwise regression procedure to develop a model that attempts to predict all commodities by the other four variables. Construct scatter plots of each of these variables with all commodities. Examine the graphs in light of Tukey's four-quadrant approach. Develop any other appropriate predictor variables by recoding data and include them in the analysis. Comment on the result of this analysis. All Commodities Food Shelter Apparel Fuel Oil .9 .6 .9 .9 1.2 1.1 2.6 1.9 3.5 4.7 4.5 3.6 3.0 7.4 11.9 8.8 4.3 5.8 7.2 11.3 1.0 1.3 .7 1.6 1.3 2.2 5.0 .9 3.5 5.1 5.7 3.1 4.2 14.5 14.3 8.5 3.0 6.3 9.9 11.0 2.0 .8 1.6 1.2 1.5 1.9 3.0 3.6 4.5 8.3 8.9 4.2 4.6 4.7 9.6 9.9 5.5 6.6 10.2 13.9 1.6 .9 .4 1.3 .9 1.1 2.5 4.1 5.3 5.8 4.2 3.2 2.0 3.7 7.4 4.5 3.7 4.5 3.6 4.3 3.7 2.7 2.6 2.6 2.1 2.4 4.4 7.2 6.0 6.7 6.6 6.2 3.3 4.0 9.3 12.0 9.5 9.6 8.4 9.2 14.34 The U.S. Department of Agriculture publishes data annually on various selected farm products. Shown here are the unit production figures for three farm Corn Soybeans Wheat (million bushels) (million bushels) (million bushels) 4152 6639 4175 7672 8876 8226 7131 4929 7525 7933 1127 1798 1636 1861 2099 1940 1938 1549 1924 1922 1352 2381 2420 2595 2424 2091 2108 1812 2037 2739 14.35 The American Chamber of Commerce Researchers Association compiles cost-of-living indexes for selected metropolitan areas. Shown here are cost-of-living indexes for 25 different cities on five different items for a recent year. Use the data to develop a regression model to predict the grocery cost-of-living index by the indexes of housing, utilities, transportation, and healthcare. Discuss the results, highlighting both the significant and nonsignificant predictors. City Albany Albuquerque Augusta, GA Austin Baltimore Buffalo Colorado Springs Dallas Denver Des Moines El Paso Indianapolis Jacksonville Kansas City Knoxville Los Angeles Louisville Memphis Miami Minneapolis Mobile Nashville New Orleans Oklahoma City Phoenix Grocer y TransporItems Housing Utilities tation Healthcare 108.3 96.3 96.2 98.0 106.0 103.1 94.5 106.8 105.2 88.8 83.9 114.1 117.3 88.5 127.4 98.8 115.6 87.7 108.1 127.6 74.6 89.1 100.9 102.3 97.4 112.8 107.8 93.3 107.5 102.1 94.0 94.9 111.5 100.8 102.4 105.4 91.5 94.3 102.9 96.0 96.1 89.8 93.2 103.3 94.6 99.1 100.3 92.8 99.9 95.8 104.0 98.2 98.9 108.3 95.1 94.6 99.7 90.4 92.4 88.0 211.3 91.0 86.2 123.0 112.3 81.1 107.7 83.4 79.4 108.9 97.2 111.4 90.9 92.1 96.0 96.3 91.7 75.6 79.4 91.1 125.6 105.2 104.9 91.6 122.2 103.4 110.0 105.9 105.7 104.2 102.7 106.0 95.6 91.6 102.1 102.4 101.1 104.3 106.0 102.8 98.1 98.2 97.3 106.8 114.3 96.2 91.4 97.4 96.1 93.6 82.3 128.5 88.4 85.5 137.8 107.5 92.2 90.9 87.0 97.1 95.7 98.7 96.3 104.6 115.2 S u p p le m e n t a r y P r o b le m s INTERPRETING THE OUTPUT 14.36 A stepwise regression procedure was used to analyze a set of 20 observations taken on four predictor variables to predict a dependent variable. The results of this procedure are given next. Discuss the results. STEPWISE REGRESSION OF Y ON 4 PREDICTORS, WITH N = 20 STEP 1 2 CONSTANT 152.2 124.5 - 50.6 7.42 X1 T-RATIO 15.2 75.39 13.9 80.59 14.37 Shown here are the data for y and three predictors, x1, x2, and x3. A stepwise regression procedure has been done on these data; the results are also given. Comment on the outcome of the stepwise analysis in light of the data. y x1 x2 x3 94 97 93 95 90 91 91 94 98 99 90 92 96 21 25 22 27 29 20 18 25 26 24 28 23 25 1 0 1 0 1 1 1 0 0 0 1 1 0 204 198 184 200 182 159 147 196 228 242 162 180 219 Regression Residual Total df SS MS F Significance F 4 24 28 289856.08 611955.23 901811.31 72464.02 25498.13 2.84 0.046 Coefficients Standard Error 124.0800 1.7800 13.4700 1.6800 66.2200 336.79 1.65 5.63 0.26 185.50 t Stat P-value 2.71 0.93 0.42 0.16 2.80 0.012 0.363 0.680 0.878 0.010 SECOND M ODEL Regression Statistics M ultiple R R Square Adjusted R Square Standard Error Observations 1 74.81 2 82.18 3 87.89 X3 T-Value P-Value 0.099 6.90 0.000 0.067 3.65 0.004 0.071 5.22 0.001 -2.26 -2.32 0.043 -2.71 -3.71 0.005 X1 T-Value P-Value S 1.37 R-Sq 81.24 R-Sq(adj) 79.53 0.567 0.321 0.208 159.681 29 ANOVA Intercept X1 X2 X3 X4 Step Constant X2 T-Value P-Value FULL M ODEL M ultiple R R Square Adjusted R Square Standard Error Observations 1.36 2.13 S R-SQ 14.38 Shown below is output from two Excel regression analyses on the same problem. The first output was done on a \"full\" model. In the second output, the variable with the smallest absolute t value has been removed, and the regression has been rerun as a second step of a backward elimination process. Examine the two outputs. Explain what happened, what the results mean, and what might happen in a third step. Regression Statistics - 43.4 6.13 X2 T-RATIO 603 -0.256 -3.08 0.013 1.16 87.82 85.38 0.850 94.07 92.09 0.566 0.321 0.239 156.534 29 ANOVA Regression Residual Total df SS MS 3 25 28 289238.10 612573.20 901811.30 96412.70 24502.90 Coefficients Standard Error Intercept X1 X2 X4 342.92 1.83 5.75 181.22 115.34 1.31