Sales\tPOS\tLocal\tNat 12.85\t5.6\t5.6\t3.8 11.55\t4.1\t4.8\t4.8 12.78\t3.7\t3.5\t3.6 11.19\t4.8\t4.5\t5.2 9\t3.4\t3.7\t2.9 9.34\t6.1\t5.8\t3.4 13.8\t7.7\t7.2\t3.8 8.79\t4\t4\t3.8 8.54\t2.8\t2.3\t2.9 6.23\t3.2\t3\t2.8 11.77\t4.2\t4.5\t5.1 8.04\t2.7\t2.1\t4.3 5.8\t1.8\t2.5\t2.3 11.57\t5\t4.6\t3.6 7.03\t2.9\t3.2\t4 0.27\t0\t0.2\t2.7 5.1\t1.4\t2.2\t3.8 9.91\t4.2\t4.3\t4.3 6.56\t2.4\t2.2\t3.7 14.17\t4.7\t4.7\t3.4 8.32\t4.5\t4.4\t2.7 7.32\t3.6\t2.9\t2.8 3.45\t0.6\t0.8\t3.4 13.73\t5.6\t4.7\t5.3 8.06\t3.2\t3.3\t3.6 9.94\t3.7\t3.5\t4.3 11.54\t5.5\t4.9\t3.2 10.8\t3\t3.6\t4.6 12.33\t5.8\t5\t4.5 2.96\t3.5\t3.1\t3 7.38\t2.3\t2\t2.2 8.68\t2\t1.8\t2.5 11.51\t4.9\t5.3\t3.8 1.6\t0.1\t0.3\t2.7 10.93\t3.6\t3.8\t3.8 11.61\t4.9\t4.4\t2.5 17.99\t8.4\t8.2\t3.9 9.58\t2.1\t2.3\t3.9 7.05\t1.9\t1.8\t3.8 8.85\t2.4\t2\t2.4 7.53\t3.6\t3.5\t2.4 10.47\t3.6\t3.7\t4.4 11.03\t3.9\t3.6\t2.9 12.31\t5.5\t5\t5.5 ID\tStay\tAge\tInfctRsk\tCulture\tXray\tBeds\tMedSchool\tRegion\tCensus\tNurses\tFacilities 1\t7.13\t55.7\t4.1\t9\t39.6\t279\t2\t4\t207\t241\t60 2\t8.82\t58.2\t1.6\t3.8\t51.7\t80\t2\t2\t51\t52\t40 3\t8.34\t56.9\t2.7\t8.1\t74\t107\t2\t3\t82\t54\t20 4\t8.95\t53.7\t5.6\t18.9\t122.8\t147\t2\t4\t53\t148\t40 5\t11.2\t56.5\t5.7\t34.5\t88.9\t180\t2\t1\t134\t151\t40 6\t9.76\t50.9\t5.1\t21.9\t97\t150\t2\t2\t147\t106\t40 7\t9.68\t57.8\t4.6\t16.7\t79\t186\t2\t3\t151\t129\t40 8\t11.18\t45.7\t5.4\t60.5\t85.8\t640\t1\t2\t399\t360\t60 9\t8.67\t48.2\t4.3\t24.4\t90.8\t182\t2\t3\t130\t118\t40 10\t8.84\t56.3\t6.3\t29.6\t82.6\t85\t2\t1\t59\t66\t40 11\t11.07\t53.2\t4.9\t28.5\t122\t768\t1\t1\t591\t656\t80 12\t8.3\t57.2\t4.3\t6.8\t83.8\t167\t2\t3\t105\t59\t40 13\t12.78\t56.8\t7.7\t46\t116.9\t322\t1\t1\t252\t349\t57.1 14\t7.58\t56.7\t3.7\t20.8\t88\t97\t2\t2\t59\t79\t37.1 15\t9\t56.3\t4.2\t14.6\t76.4\t72\t2\t3\t61\t38\t17.1 16\t11.08\t50.2\t5.5\t18.6\t63.6\t387\t2\t3\t326\t405\t57.1 17\t8.28\t48.1\t4.5\t26\t101.8\t108\t2\t4\t84\t73\t37.1 18\t11.62\t53.9\t6.4\t25.5\t99.2\t133\t2\t1\t113\t101\t37.1 19\t9.06\t52.8\t4.2\t6.9\t75.9\t134\t2\t2\t103\t125\t37.1 20\t9.35\t53.8\t4.1\t15.9\t80.9\t833\t2\t3\t547\t519\t77.1 21\t7.53\t42\t4.2\t23.1\t98.9\t95\t2\t4\t47\t49\t17.1 22\t10.24\t49\t4.8\t36.3\t112.6\t195\t2\t2\t163\t170\t37.1 23\t9.78\t52.3\t5\t17.6\t95.9\t270\t1\t1\t240\t198\t57.1 24\t9.84\t62.2\t4.8\t12\t82.3\t600\t2\t3\t468\t497\t57.1 25\t9.2\t52.2\t4\t17.5\t71.1\t298\t1\t4\t244\t236\t57.1 26\t8.28\t49.5\t3.9\t12\t113.1\t546\t1\t2\t413\t436\t57.1 27\t9.31\t47.2\t4.5\t30.2\t101.3\t170\t2\t1\t124\t173\t37.1 28\t8.19\t52.1\t3.2\t10.8\t59.2\t176\t2\t1\t156\t88\t37.1 29\t11.65\t54.5\t4.4\t18.6\t96.1\t248\t2\t1\t217\t189\t37.1 30\t9.89\t50.5\t4.9\t17.7\t103.6\t167\t2\t2\t113\t106\t37.1 31\t11.03\t49.9\t5\t19.7\t102.1\t318\t2\t1\t270\t335\t57.1 32\t9.84\t53\t5.2\t17.7\t72.6\t210\t2\t2\t200\t239\t54.3 33\t11.77\t54.1\t5.3\t17.3\t56\t196\t2\t1\t164\t165\t34.3 34\t13.59\t54\t6.1\t24.2\t111.7\t312\t2\t1\t258\t169\t54.3 35\t9.74\t54.4\t6.3\t11.4\t76.1\t221\t2\t2\t170\t172\t54.3 36\t10.33\t55.8\t5\t21.2\t104.3\t266\t2\t1\t181\t149\t54.3 37\t9.97\t58.2\t2.8\t16.5\t76.5\t90\t2\t2\t69\t42\t34.3 38\t7.84\t49.1\t4.6\t7.1\t87.9\t60\t2\t3\t50\t45\t34.3 39\t10.47\t53.2\t4.1\t5.7\t69.1\t196\t2\t2\t168\t153\t54.3 40\t8.16\t60.9\t1.3\t1.9\t58\t73\t2\t3\t49\t21\t14.3 41\t8.48\t51.1\t3.7\t12.1\t92.8\t166\t2\t3\t145\t118\t34.3 42\t10.72\t53.8\t4.7\t23.2\t94.1\t113\t2\t3\t90\t107\t34.3 43\t11.2\t45\t3\t7\t78.9\t130\t2\t3\t95\t56\t34.3 44\t10.12\t51.7\t5.6\t14.9\t79.1\t362\t1\t3\t313\t264\t54.3 45\t8.37\t50.7\t5.5\t15.1\t84.8\t115\t2\t2\t96\t88\t34.3 46\t10.16\t54.2\t4.6\t8.4\t51.5\t831\t1\t4\t581\t629\t74.3 47\t19.56\t59.9\t6.5\t17.2\t113.7\t306\t2\t1\t273\t172\t51.4 48\t10.9\t57.2\t5.5\t10.6\t71.9\t593\t2\t2\t446\t211\t51.4 49\t7.67\t51.7\t1.8\t2.5\t40.4\t106\t2\t3\t93\t35\t11.4 50\t8.88\t51.5\t4.2\t10.1\t86.9\t305\t2\t3\t238\t197\t51.4 51\t11.48\t57.6\t5.6\t20.3\t82\t252\t2\t1\t207\t251\t51.4 52\t9.23\t51.6\t4.3\t11.6\t42.6\t620\t2\t2\t413\t420\t71.4 53\t11.41\t61.1\t7.6\t16.6\t97.9\t535\t2\t3\t330\t273\t51.4 54\t12.07\t43.7\t7.8\t52.4\t105.3\t157\t2\t2\t115\t76\t31.4 55\t8.63\t54\t3.1\t8.4\t56.2\t76\t2\t1\t39\t44\t31.4 56\t11.15\t56.5\t3.9\t7.7\t73.9\t281\t2\t1\t217\t199\t51.4 57\t7.14\t59\t3.7\t2.6\t75.8\t70\t2\t4\t37\t35\t31.4 58\t7.65\t47.1\t4.3\t16.4\t65.7\t318\t2\t4\t265\t314\t51.4 59\t10.73\t50.6\t3.9\t19.3\t101\t445\t1\t2\t374\t345\t51.4 60\t11.46\t56.9\t4.5\t15.6\t97.7\t191\t2\t3\t153\t132\t31.4 61\t10.42\t58\t3.4\t8\t59\t119\t2\t1\t67\t64\t31.4 62\t11.18\t51\t5.7\t18.8\t55.9\t595\t1\t2\t546\t392\t68.6 63\t7.93\t64.1\t5.4\t7.5\t98.1\t68\t2\t4\t42\t49\t28.6 64\t9.66\t52.1\t4.4\t9.9\t98.3\t83\t2\t2\t66\t95\t28.6 65\t7.78\t45.5\t5\t20.9\t71.6\t489\t2\t3\t391\t329\t48.6 66\t9.42\t50.6\t4.3\t24.8\t62.8\t508\t2\t1\t421\t528\t48.6 67\t10.02\t49.5\t4.4\t8.3\t93\t265\t2\t2\t191\t202\t48.6 68\t8.58\t55\t3.7\t7.4\t95.9\t304\t2\t3\t248\t218\t48.6 69\t9.61\t52.4\t4.5\t6.9\t87.2\t487\t2\t3\t404\t220\t48.6 70\t8.03\t54.2\t3.5\t24.3\t87.3\t97\t2\t1\t65\t55\t28.6 71\t7.39\t51\t4.2\t14.6\t88.4\t72\t2\t2\t38\t67\t28.6 72\t7.08\t52\t2\t12.3\t56.4\t87\t2\t3\t52\t57\t28.6 73\t9.53\t51.5\t5.2\t15\t65.7\t298\t2\t3\t241\t193\t48.6 74\t10.05\t52\t4.5\t36.7\t87.5\t184\t1\t1\t144\t151\t68.6 75\t8.45\t38.8\t3.4\t12.9\t85\t235\t2\t2\t143\t124\t48.6 76\t6.7\t48.6\t4.5\t13\t80.8\t76\t2\t4\t51\t79\t28.6 77\t8.9\t49.7\t2.9\t12.7\t86.9\t52\t2\t1\t37\t35\t28.6 78\t10.23\t53.2\t4.9\t9.9\t77.9\t752\t1\t2\t595\t446\t68.6 79\t8.88\t55.8\t4.4\t14.1\t76.8\t237\t2\t2\t165\t182\t48.6 80\t10.3\t59.6\t5.1\t27.8\t88.9\t175\t2\t2\t113\t73\t45.7 81\t10.79\t44.2\t2.9\t2.6\t56.6\t461\t1\t2\t320\t196\t65.7 82\t7.94\t49.5\t3.5\t6.2\t92.3\t195\t2\t2\t139\t116\t45.7 83\t7.63\t52.1\t5.5\t11.6\t61.1\t197\t2\t4\t109\t110\t45.7 84\t8.77\t54.5\t4.7\t5.2\t47\t143\t2\t4\t85\t87\t25.7 85\t8.09\t56.9\t1.7\t7.6\t56.9\t92\t2\t3\t61\t61\t45.7 86\t9.05\t51.2\t4.1\t20.5\t79.8\t195\t2\t3\t127\t112\t45.7 87\t7.91\t52.8\t2.9\t11.9\t79.5\t477\t2\t3\t349\t188\t65.7 88\t10.39\t54.6\t4.3\t14\t88.3\t353\t2\t2\t223\t200\t65.7 89\t9.36\t54.1\t4.8\t18.3\t90.6\t165\t2\t1\t127\t158\t45.7 90\t11.41\t50.4\t5.8\t23.8\t73\t424\t1\t3\t359\t335\t45.7 91\t8.86\t51.3\t2.9\t9.5\t87.5\t100\t2\t3\t65\t53\t25.7 92\t8.93\t56\t2\t6.2\t72.5\t95\t2\t3\t59\t56\t25.7 93\t8.92\t53.9\t1.3\t2.2\t79.5\t56\t2\t2\t40\t14\t5.7 94\t8.15\t54.9\t5.3\t12.3\t79.8\t99\t2\t4\t55\t71\t25.7 95\t9.77\t50.2\t5.3\t15.7\t89.7\t154\t2\t2\t123\t148\t25.7 96\t8.54\t56.1\t2.5\t27\t82.5\t98\t2\t1\t57\t75\t45.7 97\t8.66\t52.8\t3.8\t6.8\t69.5\t246\t2\t3\t178\t177\t45.7 98\t12.01\t52.8\t4.8\t10.8\t96.9\t298\t2\t1\t237\t115\t45.7 99\t7.95\t51.8\t2.3\t4.6\t54.9\t163\t2\t3\t128\t93\t42.9 100\t10.15\t51.9\t6.2\t16.4\t59.2\t568\t1\t3\t452\t371\t62.9 101\t9.76\t53.2\t2.6\t6.9\t80.1\t64\t2\t4\t47\t55\t22.9 102\t9.89\t45.2\t4.3\t11.8\t108.7\t190\t2\t1\t141\t112\t42.9 103\t7.14\t57.6\t2.7\t13.1\t92.6\t92\t2\t4\t40\t50\t22.9 104\t13.95\t65.9\t6.6\t15.6\t133.5\t356\t2\t1\t308\t182\t62.9 105\t9.44\t52.5\t4.5\t10.9\t58.5\t297\t2\t3\t230\t263\t42.9 106\t10.8\t63.9\t2.9\t1.6\t57.4\t130\t2\t3\t69\t62\t22.9 107\t7.14\t51.7\t1.4\t4.1\t45.7\t115\t2\t3\t90\t19\t22.9 108\t8.02\t55\t2.1\t3.8\t46.5\t91\t2\t2\t44\t32\t22.9 109\t11.8\t53.8\t5.7\t9.1\t116.9\t571\t1\t2\t441\t469\t62.9 110\t9.5\t49.3\t5.8\t42\t70.9\t98\t2\t3\t68\t46\t22.9 111\t7.7\t56.9\t4.4\t12.2\t67.9\t129\t2\t4\t85\t136\t62.9 112\t17.94\t56.2\t5.9\t26.4\t91.8\t835\t1\t1\t791\t407\t62.9 113\t9.41\t59.5\t3.1\t20.6\t91.7\t29\t2\t3\t20\t22\t22.9 Homework 4 Please bring a hard copy of your homework to class and submit it at the beginning of the class. You may either type and/or write down your answers. For the questions that require software, please paste your output under the question or append it to the end of the homework and label it with the appropriate homework problem number. Make sure to include your name in the homework submission and staple all the pages together. 1. Provide the calculation of each of the following extra sums of squares in terms of the difference of sum of squared regression values. a. (3 pts) SSR(X6 | X1) b. (3 pts) SSR(X3, X4 | X1) c. (3 pts) SSR(X4 | X1, X2, X3) 2. For a multiple regression model with five predictor variables (X1, ..., X5), what is the form of the general linear F-statistic for each of the following scenarios in terms of sequential SS? In each case, you may assume that the full model contains all 5 predictors. Use SS notation. a. (4 pts) H0: 5 = 0 b. (4 pts) H0: 2 = 4 = 0 c. (4 pts) H0: 1 = 2 = 3 = 4 = 5 = 0 3. Data from 55 college students are used to estimate a multiple regression model with Y = left forearm length, X1= left foot length, and X2 = right foot length. All variables were measured in centimeters. Answer the following questions based on the Minitab output given below. The regression equation is LeftArm = 11.7 + 0.352 LeftFoot + 0.185 RtFoot Predictor Constant LeftFoot RtFoot Coef 11.710 0.3519 0.1850 S = 1.79628 SE Coef 2.518 0.2961 0.2816 R-Sq = 36.9% T 4.65 1.19 0.66 P 0.000 0.240 0.514 R-Sq(adj) = 34.4% Analysis of Variance Source Regression Residual Error Total Source LeftFoot RtFoot DF 1 1 DF 2 52 54 SS 98.019 167.785 265.804 MS 49.010 3.227 F 15.19 P 0.000 Seq SS 96.627 1.393 a. (3 pts) Test the hypothesis that the model has at least one significant predictor. Report the test statistic and p-value, and make a conclusion, b. (5 pts) Interpret the results of the individual t-tests for each of the two predictors. Then, explain why these results do not contradict the result of the F-test done in part a. c. (5 pts) Calculate the partial R2 of X2 (RtFoot) given X1 (LeftFoot), and interpret in context. d. (5 pts) Using the Mintab output below, calculate the VIF for \"RtFoot\" (X2). Then, comment on what this means in the model. Correlation: LeftFoot, RtFoot Pearson correlation of LeftFoot and RtFoot = 0.944 P-Value = 0.000 e. (3 pts) Are your answers from parts c and d consistent with each other? Why or why not? 4. The following are output from the full data set from the previous question. Suppose that we are interested in determining if we can simultaneously remove the following predictors from the model: X5 = LeftHand, X6 = RtHand, X7 = HeadCirc, and X8 = Nose. The regression equation is Height = 17.9 + 0.769 LeftArm - 0.024 RtArm + 0.563 LeftFoot + 0.429 RtFoot + 0.306 LeftHand - 0.047 RtHand + 0.081 HeadCirc - 0.161 Nose Predictor Constant LeftArm RtArm LeftFoot RtFoot LeftHand RtHand HeadCirc Nose Coef 17.950 0.7685 -0.0242 0.5631 0.4285 0.3064 -0.0473 0.0814 -0.1611 S = 2.22129 SE Coef 8.207 0.3283 0.3588 0.4173 0.4098 0.7174 0.7107 0.1557 0.5050 R-Sq = 78.5% T 2.19 2.34 -0.07 1.35 1.05 0.43 -0.07 0.52 -0.32 P 0.034 0.024 0.947 0.184 0.301 0.671 0.947 0.604 0.751 VIF 5.806 5.996 11.891 12.679 8.937 9.362 1.341 1.157 R-Sq(adj) = 74.7% Analysis of Variance Source Regression Residual Error Total Source LeftArm RtArm LeftFoot RtFoot LeftHand RtHand HeadCirc Nose DF 1 1 1 1 1 1 1 1 DF 8 46 54 SS 827.78 226.97 1054.75 MS 103.47 4.93 F 20.97 P 0.000 Seq SS 590.21 5.26 220.47 5.65 4.28 0.04 1.36 0.50 a. (3 pts) What are the null and alternative hypotheses for this test? b. (2 pts) The full model is shown in the output. What predictors would be included in the reduced model? c. (6 pts) Calculate the test statistic needed to conduct this test. (Hint: You'll need to use the sequential SS table at the bottom of the output.) d. (2 pts) What distribution does your test statistic follow? e. (4 pts) Note that the critical value for this test, for a significance level of = .05, is F* = 2.57. Based on this, what would you conclude? 5. (Software needed) The \"hospital_infct.txt\" file contains data on several hospitals in the United States collected to try to understand what affects the risk of infection. The dataset contains several variables, but for this exercise we are interested in the following: Y = InfctRsk, the risk of infection at a hospital X1 = Stay, average length of stay at the hospital X2 = Age, average age of patients at hospital X3 = Culture, average number of bacterial cultures per day at the hospital X4= Beds, the number of beds in the hospital X5 = Census, the average daily number of patients a. (5 pts) Fit a multiple linear regression model predicting Y = infection risk from the 5 predictors outlined above and calculate the VIFs (if not already included in the output). Provide the output and comment on the VIFs. b. Refer to the model in part a. There should be three predictors that each are statistically insignificant in the model, based on their p-values in the Coefficient table. We are now interested in testing whether we can drop all three of these from the model simultaneously. i. (4 pts) What are the null and alternative hypotheses of this test? What are the \"full\" and \"reduced\" models? Be sure to define your notation. ii. (5 pts) Fit the reduced model you specified and then calculate the test statistic using the sums-of-squares values from the ANOVA tables of the full and reduced models. What distribution does this test statistic follow? iii. (5 pts) Using either the p-value or critical value method, make a conclusion based on the p-value at the = .05 significance level. Is it reasonable to use the reduced, simpler model over the full model? 6. (Software needed) (Problem 10.13 from text) An assistant in the district sales office of a national cosmetics firm obtained data on advertising expenditures and sales last year in the district's 44 territories. The variables include Y = Sales (in thousands of cases), X1 = POS (thousands of dollars spent on point-of-sale displays), X2 = Local (local media advertising expenditures), and X3 = Nat (prorated share of national advertising). Data are stored in \"Cosmetics.txt\". a. (5 pts) Create a scatterplot matrix and calculate the bivariate correlations (correlation matrix) for each pair of variables and comment on the relationships observed. b. Fit a multiple linear regression model with the given response and three predictors, and print the model summary. i. (4 pts) Based on the output, is Local a significant predictor of Sales in this model? Does this agree with the relationship between Local and Sales that you observed in part a? If not, what might be one explanation for this? ii. (3 pts) Locate or calculate the VIFs and note whether or not these suggest problems of multicollinearity. c. Refit the model but remove the Local predictor. i. (3 pts) Locate or calculate the VIFs again. Comment. ii. (4 pts) Calculate the partial R2 of Local given the other two predictors. (Hint: There is a command that can do this in R, but in Minitab you'll have to calculate using sequential SS). d. (3 pts) Explain why we would remove Local from the model based on your findings in part c. Software instructions you may need for this homework: Minitab: Matrix scatterplot: Graph > Matrix Plot > Simple Correlation matrix: Stat > Basic Statistics > Correlation Fit regression model: Stat > Regression > Regression > Fit Regression Model F distribution: Calc > Probability Distributions > F R: Install an R package: install.packages(\"package_name\") Load an R package: library(package_name) Read in data: read.table(\"path\