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Unformatted text preview: The Hong Kong University of Science and Technology IS O M 111  Business S tatistics Problem Sheet 6 Solutions 1. (a) = 1.7483 =  8.0068 ∴ =  8.0068 + 1.7483 x 1 ˆ β ˆ β y ˆ When there is a 1% increase in the market return, the average return on Dalton Company’s stock tends to increase by 1.7483 %. (b) H : β 1 = 1 H a : β 1 > 1 Test statistic t = 2.5963; and t α , n – 2 = t 0.05, 8 = 1.86 Since t > t 0.05, 8 , reject H at 0.05 level of significance. We have sufficient evidence that Dalton’s Company’s Stock is sensitive to the market. (c) A 90% P. I. for y given x = 10.5 is (6.6151, 14.0856) (d) r = 0.9063 ⇒ R 2 = 0.8214 or 82.14% (e) A 98% C. I. for E( y  x = 14) is (13.2046, 19.7342) 2. (a) =  364.6 1 ˆ β On average, the Calorie content decreases by 364.6 when there is an unit increase in the Price per ounce. (b) H : β 1 = 0 H a : β 1 ≠ 0 Test statistic t =  3.603; and t 0.025, 7 = 2.365 Since  t  > t 0.025, 7 , reject H at 0.05 level of significance. The price is a good predictor of the calorie content. (c) R 2 = 0.6498 = 64.98% That means 64.98% of the variability of the calorie content can be explained by the price. 3. (a) = 0.564347826 = 7.086956522 1 ˆ β ˆ β (b) 35.30434783 (c) A 95% P. I. is (2.039713895, 45.99506871) 4. (a) y = 0.22072 + 0.47114 x ˆ (b) No. It is not meaningful to interpret (reasons should be given) . (c) For a unit increase in the test score ( x ), the grade point average ( y ) is expected to increase by 0.47114 points. (d) A 95% C. I. for β 1 is 0.47114 ± 2.069(0.0932) = (0.27823, 0.66405) The interval (0.27823, 0.66405) has 95% chance to cover the true slope parameter β 1 . Page 1 of 6 Assignment 6 Suggested Solutions (e) H : β 1 = 0.5 H a : β 1 ≠ 0.5 By (d), 0.5 is included in the 95% C. I. We do not reject Ho at 0.05 level of significance. (f) 95% P. I. is (1.9783, 4.3995) (g) 95% C. I. is (2.32196, 2.83088) (h) r = 0.7253 ⇒ x and y have quite high positive linear relationship. 5. (a) = 0.9258 = 73.3856 r = 0.9496 1 ˆ β ˆ β (b) For x = 65 ⇒ = 133.562 For x = 165 ⇒ = 226.142 y ˆ y ˆ The first one is more reliable because the xvalue for the first estimate is inside the range (23 to 100) of the data set, while the second one is outside the range. 6. (a) = 3.235 = 12.353 ⇒ = 12.353 + 3.235 x 1 ˆ β ˆ β y ˆ (b) r = 0.9272 H : ρ = 0 H a : ρ > 0 Test statistic t = 4.9491, and t α , n – 2 = t 0.05, 4 = 2.132 Since t > t 0.05, 4 , reject H at 0.05 level of significance. The ρ is positive. (c) For G, x = 8 ⇒ = 38.2353 For H, x = 15 ⇒ = 60.8824 y ˆ y ˆ The estimation for district G is more reliable because the xvalue (8) for the estimate is inside the range (4 to 11) of the data set, while the second one (15) is outside the range....
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 Fall '08
 HU
 Business

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