hw4.dvi
Statistics 106
Winter 2020
Homework 4
Due date: March 13 (Friday)
A study aims to test the effect of price discount and package size on the sales of saltine
crackers. Three levels of discount (10%, 20% and 30%) and three package sizes (Small/Medium/Large)
are tested. Five stores are randomly assigned to each combination of discount and package
size. The sample means of sales in thousand dollars observed over an entire month for each
discount-size combination are reported in Table 1.
Package size (Factor B)
Small Medium Large Y i··
10% XXXXXXXXXX
Discount 20% XXXXXXXXXX
(Factor A) 30% XXXXXXXXXX
Y
·j· Y ··· =
Table 1: Sample means of saltine cracker sales in thousand dollars across the levels of discount
and package size.
1. Identify factors, their type and nature, treatments, experimental units and the response
variable. (4p)
2. Identify the individual sample size n, the number of levels of discount and package
size, a and b, the number of treatments r, the overall sample size nT . (4p)
3. Which kind of experimental design is this? Why? (4p)
4. Produce the interaction plot of Factor A (discount) and Factor B (package size), and
viceversa, commenting about the main effects of both factors and their interactions.
(8p)
5. Complete Table 1. Then, calculate the sum of treatment squares for factor A (SSA),
factor B (SSB) and their interactions (SSAB). Finally, complete Table 2. (8p)
6. Test for the overall presence of main effects for factors A and B and of their interactions.
(6p)
7. Can the simulatenous effect µ11 be assumed equal to 12? Perform the individual-wise
test at a significance level of 5% and comment. (4p)
8. Derive a confidence interval at 95% for µ1·, both individual-wise and family-wise ove
the family composed by all the row-wise means µ1i, i = 1, . . . , a. (8p)
1
Source of Sum of Degrees of MS
Variation Squares (SS) Freedom (df) = SS
df
Factor SSA = MSA =
A
Factor SSB = MSB =
B
Interaction SSAB = MSAB =
AB
Between SSTR = MSTR =
treatments
Within SSE = 1000 MSE =
treatments
Total SSTO =
Table 2: Saltine crackers sales by discount and package size: two-factor analysis of variance
9. Test the null hypothesis µ23 − µ22 = 0 at a significance level of 5%, both individual-
wise and family-wise (over all pairwise differences involving the simultaenous effects).
Comment. (8p)
10. Derive the confidence interval at 95% for the contrast −1
4
µ31 +
1
2
µ32 −
1
4
µ33, both
individual-wise and family-wise (over the infinite-dimensional family of contrasts in-
volving all simultaneous effects). Comment. (8p)
Suppose now that Table 1 is refe
ed to only one store for each discount-size combina-
tion instead of five.
11. Which is the effect of this change on the underlying model we allow for the data?
Explain. (8p)
12. Produce a new ANOVA table with the respective degrees of freedom. (6p)
13. Test for the overall presence of main effects for factors A and B. Are these tests reliable?
(6p)
14. Estimate an individual-wise confidence interval at 95% for Y 2·. (4p)
15. Estimate the treatment mean for stores with a 20% discount and a medium package
size. Comment. (6p)
16. Perform Tukey’s additivity test with a significance level of 5% and 1%. Comment.
(8p)
2