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Suppose in a completely randomized designs experiment there are 3 crossed factors A, B and C.

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1. An experiment was run to determine whether different colored candles (red, white, blue, yellow) burn at different speeds. Each experimenter collected four observations on each color in a random order, and "experimenter" was used as a blocking factor. The design was a general complete block design with v = 4, k = 16, b = 4, and s = 4. The resulting burning times (in seconds) are shown in Table 17.21 in the book and can be downloaded at http://deanvossdraguljic.ietsandbox.net/DeanVoss Draguljic/SAS- data/candle.sas.

(a) Analyze the experiment as though the experimenters represent a random sample from a large population of people who might use these candles in practice. Use a two-way mixed model with interaction where the experimenters are blocks with random effects and the interactions of block and color are random as well. The color has fixed effects. Provide the 95% simultaneous confidence intervals for all pairwise comparisons of color using Tukey's method.

b) Provide the SAS code for 1(a).

(c) In 1(a), suppose we do not treat these experimenters as a random sample from a large population and consequently the model does not have random effects. Provide the 95% simultaneous confidence intervals for all pairwise comparisons of color using Tukey's method.

d) Provide the SAS code for 1(c)

(e) Which model provides shorter confidence intervals? Give a justification if you could.

(f) If a two-way main-effects model was used in 1(a) and 1(c), do you think the two models (one with random effects and one without) would produce the simultaneous confidence intervals of different lengths?

2. Suppose in a completely randomized designs experiment there are 3 crossed factors A, B and C. Their levels are a = 4, b = 3 and c = 3. Each treatment is replicated r = 2 times. The following model is applied to the experiment:

Yijkt = μ+ai + Bj + k + (BY)jk + Eijkt, i = 1, 2, 3, 4; j, k = 1, 2, 3,

where Eijkt's are i.i.d. N(0,02), a;'s represent the fixed effects of A, B,'s and 's are the random effects of Factors B and C, respectively, and (B),k's are the random interaction effects. Furthermore, assume all random terms are independent and

B~ N(0,0),~ N(0, o†), (87)jk ~ N(0,σc).

Partial SAS output is provided below

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