MAST20005/MAST90058: Week 6 Problems
1. To analyse the data from a brain study, we use the regression model: Yi = α + βxi + εi, εi ∼ N(0,σ2), i = 1,…,n. The response is the brain weight (on a log-scale) for n = 62 terrestrial mammals, while the predictor is the body weight (also on a log-scale). Consider the following (partial) R output:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.13479
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x 0.75169
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
Residual standard error: 0.6943 on 60 degrees of freedom
Multiple R-squared: 0.9208, Adjusted R-squared: 0.9195
F-statistic: 697.4 on 1 and 60 DF, p-value: < 2.2e-16
Recall that var(βˆ) = σ2/K, where K = i(xi − x ̄)2. Find a 95% confidence interval for β. You may use the following information:
[1] 3.123128
> qt(c(0.999, 0.99, 0.975, 0.95), df = 60)
[1] 3.231 2.39 2.00 1.67
2. Consider random variables X1,X2,X3 having joint density f(x1,x2,x3). Suppose that E(X3 |X1 =x1,X2 =x2)=α+β1(x1 −μ1)+β2(x2 −μ2)
where μi = E(Xi). Show that:
(a) α = μ3,
(b) both of:
β1 = σ13σ2 − σ12σ23 , σ12σ2 − σ122
where σi2 = var(Xi) and σij = cov(Xi,Xj).
3. Consider the simple linear model Y = α0 + β(xi − x ̄) + ε where ε ∼ N(0, σ2).
(a) Show that
[Yi − α0 − β (xi − x ̄)]2 = n(αˆ0−α0)2+(βˆ−β)2 (xi−x ̄)2+[Yi−αˆ0−βˆ(xi−x ̄)]2
i=1 i=1 i=1
(b) For an appropriate value of c (which one?), show that the endpoints for a 100·(1−γ)%
confidence interval for α0 are:
αˆ 0 ± c √σˆ . n
(c) Letting F−1 be the inverse cdf of χ2n−2, show that a 100·(1−γ)% confidence interval
for σ2 is:
(n−2)σˆ2 (n−2)σˆ2 F−1(1−γ/2),F−1(γ/2) .
β2 = σ23σ12 − σ12σ13 σ12σ2 − σ122
4. Explain why the model μ(x) = β1eβ2x is not a linear model.
5. To fit the quadratic curve y = β1 + β2x + β3×2 to a set of points, we minimise
h ( β 1 , β 2 , β 3 ) = ( y i − β 1 − β 2 x i − β 3 x 2i ) 2 .
By setting the three first partial derivatives of h with respect to β1, β2 and β3 to zero,
6. (a) Show that:
show that β1, β2 and β3 satisfy the normal equations:
y = β n + β x + β x2
x y = β x + β x2 + β x3
ii1i2i3i x2y =β x2 +β x3 +β x4
(xi −x ̄)yi =(xi −x ̄)(yi −y ̄)=xi(yi −y ̄)
i=1 i=1 i=1 nn1nn
=xy−nx ̄y ̄=xy− xy ii iinii
i=1 i=1 i=1 i=1 (b) Prove the following identity for the sum of squared residuals:
n [n (xi − x ̄)(yi − y ̄)]2 2 2i=1
(yi−y ̄) − ni=1(xi−x ̄)2 8 11 17 20 23 26
7. The following table gives the leaf area, y, of a particular type of tree at age x years.
Some useful statistics:
xi =230 yi = 313.2
14.8 17.3 20.8 24.4 9.0 23.7 28.9 11.0 27.8
xiyi =6282.3 (xi x2i =4648 (xi yi2 = 8546.0 (yi
29.3 35.0 33.4 37.8
−x ̄)(yi −y ̄)=741.1 −x ̄)2 =578.8
− y ̄)2 = 1000.2
Using a simple linear regression model, calculate the following:
(a) Estimates of all of the parameters
(b) Standard errors for all of the regression coefficients
(c) A 95% confidence interval for the expectation of Y when x = 18 (d) A 95% prediction interval for Y when x = 18
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