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# exsols04 - UNIVERSITY OF TORONTO Faculty of Arts and...

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UNIVERSITY OF TORONTO Faculty of Arts and Science JUNE EXAMINATION 2004 STA 302 H1F / STA 1001 H1F Duration - 3 hours Aids Allowed: Calculator NAME: SOLUTIONS STUDENT NUMBER: There are 17 pages including this page. The last page is a table of formulae that may be useful. Tables of the t distribution can be found on page 15 and tables of the F distribution can be found on page 16. Total marks: 75 1abcde 1fg 2ab 2cd 3 4 5a 5b 5cd 5ef 6ab 6cde 1

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Continued 1. Olympic gold medal performances in track and field improve over time. A regression was run with dependent variable longjump , the winning distance in the long jump (in inches), and independent variable year , the year the Olympics was held after 1900 (counting 1900 as year 0). Data from the Olympics held from 1900 through 1984 were used (some Olympics were missed during the World Wars). Here are the data: year 0 4 8 12 20 24 28 32 36 48 longjump 282.9 289.0 294.5 299.3 281.5 293.1 304.8 300.8 317.3 308.0 year 52 56 60 64 68 72 76 80 84 longjump 298.0 308.3 319.8 317.8 350.5 324.5 328.5 336.3 336.3 Some output from SAS is given below. The REG Procedure Descriptive Statistics Uncorrected Standard Variable Sum Mean SS Variance Deviation Intercept 19.00000 1.00000 19.00000 0 0 year 824.00000 43.36842 49184 747.13450 27.33376 longjump 5890.81250 310.04276 1833077 370.73440 19.25446 The REG Procedure Model: MODEL1 Dependent Variable: longjump Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 5054.88650 5054.88650 53.10 <.0001 Error 17 1618.33267 95.19604 Corrected Total 18 6673.21916 Root MSE 9.75685 R-Square 0.7575 Dependent Mean 310.04276 Adj R-Sq 0.7432 Coeff Var 3.14694 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 283.45427 4.28064 66.22 <.0001 year 1 0.61308 0.08413 7.29 <.0001 Questions based on this output are on the next 2 pages. 2
Continued (a) (2 marks) Estimate the mean change in the winning long jump distance from one Olympics to the next, assuming that the Olympics are held every 4 years. 4(0 . 61308) = 2 . 453 (b) (1 mark) Estimate the variance in winning long jump distance that is not ex- plained by the year the Olympics were held. 95 . 196 (c) (1 mark) Estimate the percent of total variability in winning long jump distance that is explained by the year the Olympics were held. 75 . 75% (d) (1 mark) Estimate the correlation between winning long jump distance and the year the Olympics were held. . 7575 = . 8703 (e) (2 marks) What is the observed value of the test statistic for the test H 0 : β 1 = 0 versus H a : β 1 > 0? What is the p -value for this test? Test statistic: 7.29 p -value: < 0 . 00005 3

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Continued (f) (3 marks) Construct simultaneous 99% confidence intervals for the slope and intercept of the regression line. Using the Bonferroni method, t 17 , 0 . 005 / 2 = 3 . 222 CI for slope: . 61308 ± 3 . 222( . 08413) = ( . 342 , . 884) CI for intercept: 283 . 454 ± 3 . 222(4 . 2806) = (269 . 66 , 297 . 246) (g) (5 marks) It has been suggested that the Mexico City Olympics in 1968 saw un- usually good track and field performances, possibly because of the high altitude. Construct an appropriate 95% interval for the predicted winning long jump dis- tance in 1968. Do the data support these suggestions? Explain.
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