MAST20005/MAST90058: Week 9 Problems
Some useful information for many of the problems is shown at end of this problem sheet.
1. In a one-way ANOVA with I treatments and J observations per treatment, let μ = I−1 μi.
(a) Express E(X ̄··) in terms of μ. (Hint: X ̄·· = I−1 X ̄i·)
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(b) Compute E(X ̄2)
i· (c) Compute E(X ̄2)
(d) Compute E(SS(T)) and then show that
E(MS(T))=σ2+ J (μi−μ)2
(e) Using the result of (d), what is E(MS(T)) when H0 is true?
When H0 is false, how does E(MS(T)) compare with σ2?
2. In an experiment to compare the tensile strengths of five different types of copper wire, four samples of each type were used. In an ANOVA, the between-groups and within- groups mean squares statistics were computed as MS(T) = 2573.3 and MS(E) = 1394.2 respectively. Use the F-test at a 5% significance level to test H0 : μ1 = μ2 = μ3 = μ4 = μ5 against the alternative H1 : H ̄0, where μi is the mean tensile strength of copper wire of type i.
3. Consider the following partial output from a regression of the average brain and body weights for 62 species of mammals (variables are transformed on the log-scale).
lm(formula = brain ~ body)
Estimate Std. Error
(Intercept) 2.13479 0.09604
Body 0.75169 0.02846
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
(a) Test the null hypothesis of no association between body and brain weights at the α = 0.01 level of significance.
(b) Use the following approximate distribution to obtain a test of size α for the null hy- pothesis H0 : ρ = 0 against H1 : ρ ̸= 0 based on R, the sample correlation coefficient.
1 1+R 1 1+ρ 1
2ln 1−R ≈N 2ln 1−ρ , n−3
(c) What is the sample correlation coefficient for these data?
(d) Apply the procedure in (b) to the mammals data using the significance level α = 0.01.
(e) Based on the above results, state your conclusion about the relationship between body and brain weight of mammals.
4. Let X1,...,Xn be a random sample from N(μ,σ2), where μ and σ2 are unknown. We wishtotestH0:σ2 =σ02 againstH1:σ2 ̸=σ02.
(a) Find L0 and L1, the maximised likelihoods under H0 and H1, that are required in order to write a likelihood ratio.
(b) Show that the likelihood ratio test rejects H0 if w > c1 or w < c2 (for some constants c1 and c2), where w = i(xi − x ̄)2/σ02.
Some potentially helpful R output:
> p <- c(0.95, 0.975, 0.99, 0.995)
> qnorm(p)
[1] 1.644854 1.959964 2.326348 2.575829
> qt(p, 60)
[1] 1.670649 2.000298 2.390119 2.660283
> qchisq(p, 60)
[1] 79.08194 83.29767 88.37942 91.95170
> qf(p, 4, 15)
[1] 3.055568 3.804271 4.893210 5.802907
> qf(p, 5, 20)
[1] 2.710890 3.289056 4.102685 4.761574
> qf(p, 15, 4)
[1] 5.857805 8.656541 14.198202 20.438268
> qf(p, 20, 5)
[1] 4.558131 6.328555 9.552646 12.903488
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