2006 with answers.doc

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Department
Economics (Arts)
Course
ECON 227D2
Professor
Kenneth Mac Kenzie
Semester
Winter

Description
1 1. Data have been collected on nutrition for a suitably random sample of twelve popular dry cereals. For a standard serving of each cereal measurements were made of the numbers of units of protein, carbohydrates, fat, vitaminA, and calories. Brand Protein Carbo Fat Calories VitA Life 6 19 1 110 0 Grape Nuts 3 23 0 100 25 Super Sugar Crisp 2 26 0 110 25 Special K 6 21 0 110 25 Rice Krispies 2 25 0 110 25 Raisin Bran 3 28 1 120 25 Product 19 2 24 0 110 100 Wheaties 3 23 1 110 25 Total 3 23 1 110 100 Puffed Rice 1 13 0 50 0 Sugar Corn Pops 1 26 0 110 25 Sugar Smacks 2 25 0 110 25 a) Predict the number of calories of another cereal with 5 units of protein, 23 carbohydrate units, 1 fat unit, and 25 units of vitamin A. Use MODEL I. 111.2875 b) What proportion of the variation in calories is explained by the regression relationship in i) MODEL I ? ii) MODEL II ? MODEL I: 0.777403 MODEL II: 0.956654 2 c) Is the increase in R significant from MODEL I to MODEL II ? Justify your answer numerically from the printouts. The ‘protein’ p-value is listed as 0.001. Since this is less than 0.05, the increase is significant. d) Give a 90% confidence interval for the marginal contribution of a unit of carbohydrates based on the results in MODEL I. t DF = error DF = 8 t = 1.860 answer: 3.6630 ± 1.860 × 0.7889 = 3.663 ±1.467 e) Find the missing numbers in the analysis-of-variance table for MODEL II. 2 Analysis of Variance Source DF SS MS F Regression 4 3348.289 837.07225 38.62 Residual Error 7 151.711 21.673 Total 11 3500.000 f) Find Pearson’s correlation coefficient for the Carbo and Fat data, and determine whether these two variables are significantly correlated. r = 0.04669296 H 0 ρ = 0 H a: ρ ≠ 0 critical t with DF = 10 2.228 test statistic =0.0466296 = 0.1478 2 1 − 0.0466296 10 Do not reject the null hypothesis. Carbo and Fat data are not significantly correlated. MODEL I The regression equation is Calories = 16.71 + 3.663 Carbo + 9.726 Fat + 0.0241 VitA Predictor Coef StDev T P Constant 16.71 17.74 0.94 0.374 Carbo 3.6630 0.7889 4.64 0.002 Fat 9.726 6.071 1.60 0.148 VitA 0.02410 0.09612 0.25 0.808 S = 9.868 R-Sq = 77.7403% Analysis of Variance Source DF SS MS F P Regression 3 2720.91 906.97 9.31 0.005 Residual Error 8 779.09 97.39 Total 11 3500.00 MODEL II 3 The regression equation is Calories= -4.082+ 3.9805Carbo+ 5.215Protein+ 2.208Fat+0.06052VitA Predictor Coef StDev T P Constant -4.082 9.217 -0.44 0.671 Carbo 3.9805 0.3768 10.56 0.000 Protein 5.2150 0.9693 5.38 0.001 Fat 2.208 3.187 0.69 0.511 VitA 0.06052 0.04585 1.32 0.228 S = 4.655 R-Sq = 95.6654% Analysis of Variance Source DF SS MS F P Regression * ******* ****** 38.62 0.000 Residual Error * ****** ***** Total ** ******* ECON 227 2. Regress Calories on VitA either using statistics functions on the calculator or using formulas and the following partial computations: ∑ x = 400 ∑ y = 1260 ∑ x = 25000 ∑ xy = 44000 ∑ y = 135800 . a) Find the OLS line equation y = 99 .2857 + 0.1714 x b) Test whether the number of units of vitamin A is significant for predicting the number of calories in a standard serving. Method I t test H : ρ = 0 0 H a: ρ ≠ 0 critical t with DF = 102.228 0.312984319 test statistic = 2 = 1.042 1 − 0.312984319 10 Method II: ANOVA with the same null and alternative hypotheses. Analysis of Variance 4 Source DF SS MS F Regression 1 342.857 342.857 1.086 Residual Error 10 3157.143 315.714 Total 11 3500.000 Critical 1,10 4.965 Do not reject the null hypothesis. The number of units of vitamin A is not significant for predicting the number of calories in a standard serving. c) Regardless of your conclusion in b) form a 95% prediction interval for the number of calories in standard servings with 25 units of vitamin A. Use the simple-regression model of this question. Note the change in the question. 1 (25 − 33.3) 103.57 ± 2.228 315.714 1+ + 2 = 103.57 ± 41.32 12 400 25000 − 12 d) What is the standard error of the estimate in this model? MSE ≈ 17 .77 e) Show that the point (x , ) satisfies the OLS equation. 3. a) Test whether the variances of Carbo and VitA are significantly different. 2 H : σ1 = 1 0 σ 2 2 σ 2 H a 1 ≠ 1 σ 2 critical F11,11,0.025omewhere between 3.4 and 3.8 2 32.56694736 test statistic = 2 = 67.82945736 3.954284213 Reject the null hypothesis. The variances of Carbo and VitA are significantly different. b) Test whether the mean of VitA is significantly greater than the mean of Carbo. 5 H : μ ≤ μ 0 V C H a μ V μ C 2 2 s2 s 2 32.5669 2 3.9543 2   1 + 2   +  n1 n 2  12 12  df = 2 2 = 2 2  s2  s 2 32.5669 2  3.9543 2   1   2      n1  n 2  12   12  + + n1−1 n 21 11 11 =11.3 Truncate to 11 critical t : 1.796 33.3 − 23 test statistic = =1.09 32.5669 2 3.9543 2 + 12 12 Do not reject the null hypothesis. The mean of VitA is not significantly greater than the mean of Carbo. c) What are the technical requirements for performing the tests you used in parts a) and b) ? Bo
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