STAT 430 – Assignment 2 - People.stat.sfu.ca

STAT 430 – Assignment 2
Due October 04, 2011 (Tuesday) at the beginning of lecture.
1. Three different washing solutions are being compared to study their effectiveness in retarding bacteria
growth in 5-gallon milk containers. The analysis is done in a laboratory, and only three trials can be
run on any day. Observations are taken for four days ( a potential source of variability), and the data
are shown here.
Days
Solution 1 2 3 4
1 13 22 18 39
2 16 24 17 44
3 5 4 1 22
(a). Write out an appropriate statistical model for this experiment.
(b). State clearly your null and alternative hypothesis. Perform ANOVA. What is your conclusion?
(c). Use the appropriate residual plots to check the assumptions of the model.
(d). Estimate the overall mean, the treatment effects and block effects.
(e). Testiftheaverageresponseofthe1stand3rdsolutionarethesamebyusingasuitablecomparison.
(f). Estimate the mean response of the 1st solution. Report the mean, its standard error, and a 95%
confidence interval (unadjusted for multiple comparisons).
(g). Estimate the difference in mean response between the 1st and 3rd solution and find the a 95%
confidence interval (unadjusted for multiple comparisons).
2. A consumer products company relies on direct mail marketing pieces as a major component of its
advertising campaigns. The company has three different designs for a new brochure and wants to
evaluate their effectiveness, as there are substantial differences in costs between the three designs. The
company decides to test the three designs by mailing 5000 samples of each of potential customers in
four different regions of the country. Since there are known regional differences in the customer base,
regions are considered as blocks. The number of responses to each mailing is as follows.
Region
Design NE NW SE SW
1 250 350 219 375
2 400 525 390 580
3 275 340 200 310
1

(a). Analyze the data from this experiment. Hint: Carry out an ANOVA and a suitable hypothesis
test.
(b). Use the Fisher LSD method to make comparisonamong the three designs to determine specifically
which designs differ in the mean response rate.
(c). Analyze the residuals from this experiment.
3. Find all standard Latin squares of side 5 with the following form
A B C D E
B C D
C
D
E
4. Using the notation in class, show that Var(ˆτ j −ˆτ i ) = (2k/(λa))σ 2 for a BIBD. If the experiment is run
as a complete randomized block design, we can show that Var(ˆτ j − ˆτ i ) = Var(ȳ j. −ȳ i. ) = 2σ 2 /b. Now,
prove that 2k/(λa) > 2/b.
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