Homework 03
Question 1 [4] A regression analysis involving an intercept term and two covariate measurements on each of n = 22 subjects produced an estimate βˆ = 1.0 and a 90% confidence interval (—2.36, 4.36) for β1. Find the standard error of βˆ.
Question 2 [4+2+3+2+4=15] (Chapter 5, Section 5.2) The output be- low was obtained from fitting a three - covariate linear model,Yi = βO + β(xi( + ei to observed responses:RSS = 100; n = 30; R2 = 0.9
where df is short for "degrees of freedom", est.coef. denotes the least squares estimates of β(, j = 1, 2, 3, and sig. denotes the p value (two- sided) of the t statistic for testing the hypothesis that the coefficient of the respective covariate is zero.
2.1 Find an unbiased estimate of the error variance, σ2.
2.2 Find the two possible numerical values of βˆ .
2.3 Find a 95% confidence interval for β2, the coefficient of X2 in the model.
2.4 Find the regression sum of squares, SSreg, for these data.
2.5 Test at the 5% level the hypothesis HO : β1 = β2 = β3 = 0.
NOTE: Here you could use either the Analysis of Variance method (page 135) or the Partial F-test method (page 137)
Question 3 [2+1+1+2+2+1+2+1+1=13] (Chapter 5, Section 5.2) The Question h data (attached) show the numerical values of a predictor y and three covariates x1, x2 and x3. Assume that a valid linear regres- sion is in force.
E[y|x1, x2, x3] = βO + β1x1 + β2x2 + β3x3
3.1 Write down the least squares estimates of the four regression coefficients.
ˆ = ; βˆ= ; βˆ= ; βˆ = .
3.2 Test the null hypothesis HO : β1 = β2, β3 = 0 at the 10% level of sig- nificance by writing your answers to (a) through (f) below. Follow the same METHOD as the Partial F-test, but make allowance for the fact that the null hypothesis is compound, not simple. This is equivalent to, but simpler to implement, than Model Reduction Method 2 on slide 45 of the Chapter 5 slides and 46:54 min. into Lecture 14.
(a) RSSƒull =
(b) dfƒull =
(c) RSSnull =
(d) dfnull =
(e) Write an expression for your test statistic in terms of the symbols in parts (a) through (d).
(f) The numerical value of the test statistic is
(g) State the critical value for the hypothesis test OR state the p-value associated with (f).
(h) State your decision regarding the validity of HO and justify it with ref- erence to part (g).
Question 4 [4]
A botanist is interested in the efficacy on predator bugs in reducing pests on garden plants. In particular, two species (A and B) of praying mantis are to be compared to see which devours potato beetles at a higher rate. One hundred grams of potato beetles are released into a cage containing potato foliage, and one praying mantis of each species is introduced into the cage for one week. At the end of the week, the reduction (in grams) of potato beetles is measured. Another identical cage is prepared, but this time, there are two praying mantis of Species A and one of Species B. For the third cage, there are two of Species B and one of Species A, and for the fourth cage of the study, there are two of each species introduced. Let βA be the average grams of potato beetles eaten per week per praying mantis for Species A, and let βB be the average for Species B. Write down a linear model in matrix notation that can be used to estimate βA and βB, giving your design matrix X. Assume that the consumption of each praying mantis is independent of others in the cage.
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