ECN 206E PS#6.pdf - ECN 206E Statistics II Problem Set 6...

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ECN 206E — Statistics II Problem Set 6 Spring ’18 Exercise 12.23 The following are results from a regression model analysis: ˆ y = 1 . 50 + 4 . 8 x 1 (2 . 1) + 6 . 9 x 2 (3 . 7) - 7 . 2 x 3 (2 . 8) R 2 = 0 . 71 n = 24 The numbers below the coefficient estimates are the sample standard errors of the coefficient estimates. (a) Compute two-sided 95% confidence intervals for the three regression slope coefficients. (b) For each of the slope coefficients, test the hypothesis H 0 : β j = 0 Answer: Given the regression results where the numbers in parentheses are the sample standard error of the coefficient estimates (a) The two-sided 95% confidence intervals for the three regression slope coefficients are given by b j ± t n - K - 1 ,α/ 2 s b j 95% CI for x 1 = 4 . 8 ± 2 . 086(2 . 1); 0 . 4194 up to 9 . 1806 95% CI for x 2 = 6 . 9 ± 2 . 086(3 . 7); 0 - . 8182 up to 14 . 6182 95% CI for x 3 = - 7 . 2 ± 2 . 086(2 . 8); - 13 . 0408 up to - 1 . 3592 (b) Testing the hypothesis H 0 : β j = 0, H 1 : β j > 0 For x 1 : t = 4 . 8 2 . 1 = 2 . 286; t 20 , 0 . 005 / 0 . 01 = 1 . 725, 2 . 528 Therefore, reject H 0 at the 5% level but not at the 1% level For x 2 : t = 6 . 9 3 . 7 = 1 . 865; t 20 , 0 . 005 / 0 . 01 = 1 . 725, 2 . 528 Therefore, reject H 0 at the 5% level but not at the 1% level For x 3 : t = - 7 . 2 2 . 8 = - 2 . 571; t 20 , 0 . 005 / 0 . 01 = 1 . 725, 2 . 528 Therefore, do not reject H 0 at either level 1/7 T.A. Bu˘ gra Aky¨uz
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ECN 206E — Statistics II Problem Set 6 Spring ’18 Exercise 12.28 The following model was fitted to a sample of 30 families in order to explain household milk consumption: y = β 0 + β 1 x 1 + β 2 x i + ε where y i = milk consumption, in quarts per week x 1 i = weekly income, in hundreds of dollars x 2 i = family size The least squares estimates of the regression parameters were as follows: b 0 = - 0 . 025 b 1 = 0 . 052 b 2 = 1 . 14 The estimated standard errors were as follows: s b 1 = 0 . 023 s b 2 = 0 . 35 (a) Test, against the appropriate one-sided alternative, the null hypothesis that, for fixed family size, milk consumption does not depend linearly on income. (b) Find 90%, 95%, and 99% confidence intervals for β 2 . Answer: (a) H 0 : β 1 = 0; H 1 : β 1 > 0 t = 0 . 052 0 . 023 = 2 . 26 t 27 , 0 . 025 / 0 . 01 = 2 . 052, 2 . 473 Therefore, reject H 0 at the 2.5% level but not at the 1% level (b) t 27 , 0 . 05 / 0 . 025 / 0 . 005 = 1 . 703, 2 . 052, 2 . 771 90% CI: 1 . 14 ± 1 . 703(0 . 35); 0 . 5439 up to 1 . 7361 95% CI: 1 . 14 ± 2 . 052(0 . 35); 0 . 4218 up to 1 . 8582 99% CI: 1 . 14 ± 2 . 771(0 . 35); 0 . 1701 up to 2 . 1099 2/7 T.A. Bu˘ gra Aky¨uz
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ECN 206E — Statistics II Problem Set 6 Spring ’18 Exercise 12.34 In a study of differences in levels of community demand for firefighters, the following sample regression was obtained, based on data from 39 towns in Maryland: y = - 0 . 00232 - 0 . 00024 x 1 (0 . 00010) - 0 . 00002 x 2 (0 . 000018) + 0 . 00034 x 3 (0 . 00012) +0 . 48122 x 4 (0 . 77954) + 0 . 04950 x 5 (0 . 01172) - 0 . 00010 x 6 (0 . 00005) + 0 . 00645 x 7 (0 . 00306) ¯ R 2 = 0 . 3572 where y = number of full-time firefighters per capita x 1 = maximum base salary of firefighters, in thousands of dollars x 2 = percentage of population x 3 =
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