1. The ANOVA table below summarizes an experiment with 4 groups, each with 8 observations. Complete the table, and give the appropriate conclusion at the .05 significance level.

Source df SS MS F

Between __ 480 ___ ___

Within __ ___ ___

Total __ 2160

2. Suppose the USGA wants to compare the mean distances reached of four different brands of golf balls struck with a driver. A completely randomized design is employed, with Iron Byron, the USGA's robotic golfer, using a driver to hit a random sample of 10 balls of each brand in a random sequence. The distance is recorded for each hit , and the results are shown below, organized by brand. We save the 4 samples of distance of each brand in R as 4 vectors.

distance.A=c( 251.2, 245.1, 248.0, 251.1, 260.5, 250.0, 253.9, 244.6, 254.6, 248.8)

distance.B=c( 263.2, 262.9, 265.0, 254.5, 264.3, 257.0, 262.8, 264.4, 260.6, 255.9)

distance.C=c( 269.7, 263.2, 277.5, 267.4, 270.5, 265.5, 270.7, 272.9, 275.6, 266.5)

distance.D=c( 251.6, 248.6, 249.4, 242.0, 246.5, 251.3, 261.8 , 249.0, 247.1, 245.9)

distance.total=c(distance.A, distance.B, distance.C, distance.D)

(note: in R, we use functions sum(), mean() and var() to calculate sum, mean and variance, respectively. )

(a) Find the total variation (TSS) of the whole data set. Note: you can find the variance of the whole data set and then multiply it by (the total sample size - 1)

(b) Find the variance for each sample.

(c) Find SSW.

(d) Find SSB.

(e) Find df_B, df_W.

(f) Find MSW and MSB.

(g) conduct the F test to answer if these four brands have the same mean distances.

3. What assumptions are required for the ANOVA F test?

4. You are comparing mean hardness for epoxy resins cured under one of four different conditions:

#1: catalyst A for 1 hour

#2: catalyst A for 2 hours

#3: catalyst B for 1 hour

#4: catalyst B for 2 hours

Denote the mean hardness under condition #1, #2, #3, #4 as u1, u2, u3, u4, respectively.

(a) Specify the contrast corresponding to the statement "The average of the means for catalyst A equals that for Catalyst B."

(b) Specify the contrast corresponding to the statement "the difference between the means at 1 and 2 hours is the same for Catalyst A as it is for catalyst B."

(c) Show that the two contrasts for part (a) and part (b) are orthogonal.