1. Recall Question 5 from the Week 8 lab. In a manufacturing plant, filters are used to remove pollutants. We are interested in comparing the lifespan of 5 different types of filters. Six filters of each type are tested, and the time to failure in hours is given in the dataset filters (on the website, in csv format).
(a) Is µ − τ1 + τ5 estimable?
(b) Is estimable?
(c) In the week 8 lab you were asked to find two solutions to the normal equations. Verify thatthey produce the same estimate of τ4− τ5.
(d) Do your two solutions produce the same estimate of 2µ + τ1?
(e) Write down the quantities corresponding to: (i) the lifespan of type 1 filters; (ii) the differencebetween the lifespans of type 2 and type 3 filters; (iii) the amount by which type 4 filters outlive the average filter; (iv) the expected total time to failure of a set of filters containing one of each type.
Verify directly that all of these quantities are estimable, and estimate them.
(f) Fit a lm model using contr.treatment contrasts (the default). This gives estimates of µ1,µ2− µ1,...,µ5− µ1. Use these to estimate ¯µ,µ1− µ,...,µ¯ 5− µ¯. Check your answers by fitting a contr.sum model.
2. According to the Gauss-Markov theorem, the estimator for tTβ with the lowest variance is tTb. Assuming that tTβ is estimable, show that this variance is σ2tT(XTX)ct. 3. For the one-way classification model, with ni observations in group i, show that
k
SSReg := yˆTyˆ = yTX(XTX)cXTy = X(¯yi)2ni.
i=1
4. Consider the one-way classification model with 3 levels (k = 3). Find all estimable quantities of the form .
5. Consider the two-way classification model
yij = µ + τi + βj + εij.
Suppose that you have at least one sample from each combination of factor levels.
Treatment contrasts for the first factor are defined here as Pi aiτi, where Pi ai = 0. Show that these treatment contrasts are estimable.
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