Statistical Analysis of Temperature and Weight Gain Data, Exams of Data Analysis & Statistical Methods

Download Statistical Analysis of Temperature and Weight Gain Data and more Exams Data Analysis & Statistical Methods in PDF only on Docsity! Stat/For/Hort 572 Zhu March 7, 2006 Midterm I NAME: Instructions: 1. For hypothesis testing, follow the standard procedures. That is, state the H0 and the HA, select and compute the test statistic, state the null distribution, report the p-value, draw a conclusion at the α = 0.05 level, and interpret the testing result. 2. This exam is open book. You may use textbooks, notebooks, and a calculator. 3. Do all your work in the spaces provided. If you need additional space, use the back of the preceding page, indicating clearly that you have done so. 4. To get full credit, you must show your work. Partial credit will be awarded. 5. Do not dwell too long on any one question. Answer as many questions as you can. 6. Note that some questions have multiple parts. For some questions, these parts are independent, and so, for example, you can work on parts (b) or (c) separately from part (a). For grader’s use: Question Possible Points Score 1 35 2 35 3 30 Total 100 1 Stat/For/Hort 572 Zhu March 7, 2006 1. In a study of air temperature pattern over time, a researcher obtained the weekly average temperature in two different study sites from the 16th week to the 29th week of a given year. The data set (temp.dat) consists of three variables: weekly average temperature in oC (temp), week of the year (week), and the square of the week (weeksq). The following (edited) R code and output may be used to answer Questions 1(a)–(d). > temp.dat; temp week weeksq 1 2.87 16 256 2 3.31 17 289 3 8.57 18 324 4 7.91 19 361 5 10.48 20 400 6 8.74 21 441 7 12.63 22 484 8 15.94 23 529 9 18.56 24 576 10 18.14 25 625 11 20.12 26 676 12 15.21 27 729 13 19.56 28 784 14 16.44 29 841 15 2.51 16 256 16 3.56 17 289 17 9.35 18 324 18 6.63 19 361 19 10.94 20 400 20 8.06 21 441 21 9.56 22 484 22 16.17 23 529 23 17.20 24 576 24 17.43 25 625 25 19.70 26 676 26 17.21 27 729 27 23.55 28 784 28 17.91 29 841 > temp.lm1 = lm(temp~week, data=temp.dat); > summary(temp.lm1); Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -17.3763 2.6803 -6.483 7.17e-07 *** week 1.3409 0.1173 11.436 1.21e-11 *** --- Residual standard error: 2.501 on 26 degrees of freedom Multiple R-Squared: 0.8342,Adjusted R-squared: 0.8278 F-statistic: 130.8 on 1 and 26 DF, p-value: 1.209e-11 2 Stat/For/Hort 572 Zhu March 7, 2006 w1 0.9360 0.1070 8.745 0.000124 *** w2 2.2800 0.4665 4.887 0.002747 ** w3 -0.0920 0.1514 -0.608 0.565604 --- Residual standard error: 0.1692 on 6 degrees of freedom Multiple R-Squared: 0.9879, Adjusted R-squared: 0.9818 F-statistic: 163.1 on 3 and 6 DF, p-value: 3.873e-06 > anova(turkey.lm); Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) w1 1 3.9605 3.9605 138.2856 2.283e-05 *** w2 1 10.0400 10.0400 350.5601 1.499e-06 *** w3 1 0.0106 0.0106 0.3694 0.5656 Residuals 6 0.1718 0.0286 --- (a) Consider the model y = b0 + b1w1 + b2w2 + b3w3 + e. Briefly interpret b3 and perform a test for H0 : [b3 = 0|b0, b1, b2]. (b) Consider the model y = b0 + b1w1 + b2w2 + b3w3 + e. Briefly interpret b0 and construct a 90% confidence interval for b0. (c) Perform a test to determine whether the relation between weight gain and diet supplement level is the same on the two farms. (d) Perform a test for H0 : [b2 = 0|b0, b1] and briefly interpret the result under the context of this experiment. 3. The data study consists of a response variable y and three explanatory variables x1, x2, x3. In addition, an explanatory variable x4 is created such that it has the value of 1 for the third observation and 0 for all other observations. Use the following (edited) R code and output to answer Questions 3(a)–(c). > cor(study); x1 x2 x3 x4 y x1 1.0000000 0.7971536 -0.9561373 -0.4294967 -0.4332871 x2 0.7971536 1.0000000 -0.7317592 -0.1929603 0.1068198 x3 -0.9561373 -0.7317592 1.0000000 0.3117618 0.5926742 x4 -0.4294967 -0.1929603 0.3117618 1.0000000 0.2514859 y -0.4332871 0.1068198 0.5926742 0.2514859 1.0000000 > study.lm23 = lm(y~x2+x3, data=study); > summary(study.lm23); Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 11.5130 1.3892 8.288 1.67e-05 *** x2 3.1730 0.1877 16.905 3.98e-08 *** x3 3.3085 0.1577 20.981 5.95e-09 *** --- 5 Stat/For/Hort 572 Zhu March 7, 2006 Residual standard error: 0.7802 on 9 degrees of freedom Multiple R-Squared: 0.9802, Adjusted R-squared: 0.9758 F-statistic: 222.7 on 2 and 9 DF, p-value: 2.166e-08 > anova(study.lm23); Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x2 1 3.156 3.156 5.1847 0.0488 * x3 1 267.924 267.924 440.1964 5.95e-09 *** Residuals 9 5.478 0.609 --- > study.lm123 = lm(y~x1+x2+x3, data=study); > summary(study.lm123); Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 9.4114 1.6571 5.679 0.000465 *** x1 1.8841 0.9994 1.885 0.096138 . x2 2.9982 0.1898 15.794 2.58e-07 *** x3 3.8700 0.3287 11.772 2.48e-06 *** --- Residual standard error: 0.6886 on 8 degrees of freedom Multiple R-Squared: 0.9863, Adjusted R-squared: 0.9811 F-statistic: 191.8 on 3 and 8 DF, p-value: 8.659e-08 > anova(study.lm123); Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x1 1 51.920 51.920 109.51 6.041e-06 *** x2 1 155.141 155.141 327.22 8.955e-08 *** x3 1 65.704 65.704 138.58 2.481e-06 *** Residuals 8 3.793 0.474 > study.lm1234 = lm(y~x1+x2+x3+x4, data=study); > summary(study.lm1234); Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.8393 1.4579 4.691 0.00223 ** x1 3.3599 0.8645 3.886 0.00600 ** x2 2.8381 0.1450 19.577 2.26e-07 *** x3 4.2151 0.2603 16.192 8.34e-07 *** x4 1.9560 0.6557 2.983 0.02043 * --- 6 Stat/For/Hort 572 Zhu March 7, 2006 Residual standard error: 0.4884 on 7 degrees of freedom Multiple R-Squared: 0.994, Adjusted R-squared: 0.9905 F-statistic: 288.1 on 4 and 7 DF, p-value: 7.664e-08 > anova(study.lm1234); Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x1 1 51.920 51.920 217.6265 1.574e-06 *** x2 1 155.141 155.141 650.2793 3.643e-08 *** x3 1 65.704 65.704 275.4013 7.046e-07 *** x4 1 2.123 2.123 8.8981 0.02043 * Residuals 7 1.670 0.239 --- (a) Consider the three simple linear regression models: y = b0 + b1x1 + e, y = b0 + b1x2 + e, and y = b0 + b1x3 + e. Which model has the smallest MSE? Briefly explain why. (b) For the model y = b0 + b1x1 + b2x2 + b3x3 + e, the third observation has the largest studentized residual. Perform an outlier test on this observation. (c) Perform a backward elimination for model selection, starting with the full model y = b0 + b1x1 + b2x2 + b3x3 + e. 7