CHAPTER 10: SAMPLE PROBLEMS FOR HOMEWORK, CLASS OR EXAMS
These problems are designed to be done without access to a computer, but they may require a calculator.
1. FOR EACH SCENARIO BELOW, IDENTIFY THE EXPERIMENTAL DESIGN
a. You are comparing three versions of a reading exam to see whether they are of equal difficulty. One hundred children each take all three versions of the exam, in random order.
____ completely randomized design
____ randomized block design
____ randomized block design with sampling
b. You are testing treatments designed to protect boat hulls from marine growth. There are two possible primers and three possible top coats, that is, 2x3 = 6 treatment combinations. You select 10 boat hulls, and randomly assign the 6 combinations to a different spot of each hull (6 observations per hull). You have no reason to suspect interactions between hull and treatment.
____ split plot design
____ factorial design within a randomized block
____ repeated measures design with 2 within-subject factors
c. You are testing treatments designed to protect boat hulls from marine growth. There are two possible primers and three possible top coats, that is, 2x3 = 6 treatment combinations. However, primers can only be applied over large sections of the hull. Therefore, you divide each hull in half, and paint one half with primer A and the other with primer B. Then you randomly assign the three top coats to spots on each hull-half.
____ split plot design
____ factorial design within a randomized block
____ repeated measures design with 2 within-subject factors
d. You are comparing four paints with regard to the way they fade in sunlight. Twenty panels of wood are randomly assigned to one paint each, so that there are 5 panels for each paint.
____ randomized block design
____ completely randomized design
____ randomized block design with sampling
e. Twenty volunteers take a test for cognitive ability on three occasions. The occasions are arranged so that once the test is given in quiet conditions, once in noisy conditions, and once in very noisy conditions, in random order. Some of the volunteers are young, and some are old, and this may be an important influence on cognitive ability.
___ repeated measure with 1 between-subject and 1 within-subject
___ factorial design within a randomized block
___ split plot design
2. To compare the calibration of three instruments designed to measure ozone, five different days are selected. On each day, the machines are set out side-by-side, in random order, and ozone measurements are recorded from each.
a. Identify the experimental design. Which, if any, of the factors are random factors?
b. The table below shows the sums of squares treating the data as if it came from an ordinary two-way ANOVA with Instrument and Day as fixed effects, but no interaction. At a = 5%, do the instruments differ in their mean ozone measurement? Show the construction of the test statistic.
Source Sums of Squares
Instrument 9.730
Day 399.600
Error (Instrument*Day) 7.600
3. In a experiment to test the effect of antibiotics, fifteen pigeons are first trained to recognize which symbol marks the correct cup containing food. The measure of their training is the percentage of pecks made to the correct cup (PCT_CCUP). The pigeons are then randomly assigned to one of three groups, and their initial value (Time 0) of PCT_CCUP is recorded. Then the pigeons are given an injection. Group 1 receives a saline injection, Group 2 receives antibiotic ‘C’, and Group 3 receives antibiotic ‘P’. PCT_CCUP is measured 24 hours later, and again 48 hours later. The experiment is designed to test whether the antibiotics cause the pigeons to forget their training, and whether the effect of the antibiotic is different at 24 and 48 hours post-injection.
a. Identify the experimental design.
b. What purposed does the use of a saline injection serve, when the question concerns antibiotics?
c. The table below shows the relevant sums of squares. Fill in the degrees of freedom, and explain the degrees of freedom for the error term.
Source DF Type III SS
time __ 1195.600000
inject __ 149.733333
time*inject __ 134.666667
pigeon(inject) __ 2970.800000
Error 24 92.400000
d. Use the information in the table to test for a main effect of Injection.
e. Use the information in the table to test for an interaction of Time and Injection.
f. The table below shows the sample means for each combination of Time and Injection. Use this to create a profile plot, and to speculate on the nature of the interactions.
pct_ccup
time inject LSMEAN
0 1 63.4000000
0 2 66.6000000
0 3 66.2000000
24 1 54.8000000
24 2 49.2000000
24 3 56.2000000
48 1 63.2000000
48 2 59.2000000
48 3 66.0000000
4. A hospital is experimenting with different wall-surface materials that might result in quieter rooms. On each of three randomly selected floors of the many floors at the hospital, 12 rooms are selected. These 12 rooms are randomly assigned to one of three wall-surface materials. Then a sound test is conducted in each room, for a total of 36 observations.
a. Identify the experimental design.
b. The sums of squares below were obtained by treating the data as an ordinary two-way ANOVA with Floor and Wall_Surface as the factors. Test the null hypothesis that the mean sound observation does not differ by Wall_surface, using a = 5%.
Source DF Type III SS Mean Square
floor 2 250.4751568 125.2375784
wall_surface 2 413.9184920 206.9592460
floor*wall_surface 4 64.2060927 16.0515232
Error 27 339.604993 12.577963
c. If the hospital only had three floors, how would that affect the analysis and its interpretation?
5. An engineering firm is testing combinations of flooring (two possible types) and wall-surfacing (two possible types) which will produce the quietest workspace. They randomly select five different office buildings. The four combinations of flooring and wall-surfacing are then randomly assigned to four offices within each building.
a. Identify the experimental design.
b. The sums of squares below were obtained by treating the data as a three-way ANOVA. Assuming no interaction between the block effect and the treatment effects, describe the impact of flooring type and wall-surfacing type on noise. Use a = 5% for each test.
Source df SS Mean Square
bldg 4 25.41710336 6.35427584
flooring 1 5.59352908 5.59352908
wall_surface 1 35.29569863 35.29569863
flooring*wall_surfac 1 0.00019904 0.00019904
bldg*flooring 4 47.86657433 11.96664358
bldg*wall_surface 4 90.18758100 22.54689525
Error 4 45.2720588 11.3180147
6. Twenty subjects are randomly selected, and each is asked to taste four different sodas (presented in random order). The subjects score the sodas on a continuous scale from 1 (poor) to 10 (excellent). The sums of squares for this randomized block design are presented below.
Source DF Type III SS Mean Square
soda 3 72.6373976 24.2124659
subj 19 264.6497379 13.9289336
Error 57 141.1444222 2.4762179
a. Is there evidence of that the sodas differ in their mean taste score? Use a = 5%.
b. This experiment could have been carried out by allowing each subject to only taste one soda, that is, as a completely randomized design analyzed by a one-way ANOVA. Calculate the relative efficiency of the randomized block design over the completely randomized design.
7. Twenty different volunteers rate two wines on a continuous scale from 1(poor) to 10(excellent), with each wine presented once at 55°F and once at 65°F. That is, each volunteer will have four scores. Subject effects can be quite strong, and we can not assume that the interactions of subject with either temperature or wine are weak.
a. Identify the experimental design.
b. The table below shows the sums of squares treating this as a three-way ANOVA. Carry out the proper test for the null hypothesis of no main effect for wine, using a = 5%.
Source DF Type III SS Mean Square
volunteer 19 168.3272456 8.8593287
wine 1 3.2362543 3.2362543
temp 1 55.1411527 55.1411527
wine*temp 1 13.4542434 13.4542434
volunteer*wine 19 96.4260179 5.0750536
volunteer*temp 19 60.5814116 3.1884953
Error 19 69.4276658 3.6540877
SOLUTIONS.
1a. Randomized block design
b. factorial design within a randomized block
c. split plot design
d. completely randomized design
e. repeated measure with 1 between-subject and 1 within-subject
2. a. This is a randomized block design, with DAY as the block, which is a random effect.
b.
Source Sums of Squares df MS F
Instrument 9.730 2 4.865 5.12
Day 399.600 4
Error (Instrument*Day) 7.600 8 0.95
There is significant evidence that the instruments differ.
3. a. This is a repeated measures design. Pigeon is the subject, Injection is a between subjects factor, and Time is a within-subjects factor.
b. The use of a control, or saline, group allows us to judge whether the effect is due to the antibiotic, as opposed to the trauma of being handled and receiving an injection.
c. The error is formed using the interaction of the within-subject factor with 2 df, and the subject(between-subject factor) with (5-1)*3 = 12 df. Hence error has 24 df.
Source DF Type III SS
time 2 1195.600000
inject 2 149.733333
time*inject 4 134.666667
pigeon(inject) 12 2970.800000
Error 24 92.400000
d. This would use the mean square for pigeon(inject) in the denominator.
F(2,12) = [149.733/2] / [2970.8/12] = 0.302. There is no significant evidence of a main effect for injection.
e. This would use the Error term as the denominator.
F(4,24) = [134.6667/4] / [92.4/24] = 8.74. There is significant evidence of an interaction of Time and Injection.
f. Apparently, all the groups tend to show a decrease in PCT_CCUP at 24 hours post-injection, but tend to rebound by 48 hours. However, pigeons in group 2 (antibiotic ‘C’) show a greater effect at 24 hours.
4. a. This is a randomized block with sampling. Blocks are floors of the hospital.
b. The experimental error is given by the Floor*wall_surface interaction.
F(2,4) = 125.238/16.052 = 7.80. The critical value from the F table at a = 5% is 6.94, so there is significant evidence that at least one of the wall_surface treatments differs in its mean sound reading.
c. Now floor would be a fixed effect and this would be an ordinary two-way ANOVA.
5. a. This is a factorial experiment in a randomized block. Block is office building.
b. The bldg*flooring and bldg*wall_surface interactions should be pooled with the error (from the three-way interaction) to estimate experimental error.
Denominator: [47.867+90.188+45.272]/12 = 15.28
Flooring: F(1,12) = 5.594/15.28 = 0.37, no significant effect of flooring
Wall_surface: F(1,12) = 35.296/15.28 = 2.31, no significant effect of wall_surface
6. a. F(3,57) = 9.78, there is significant evidence that the means for the sodas are not all equal.
b.
7. a. Repeated measures with two within-subjects factors
b. Denominator will use the volunteer*wine interaction, F(1,19) = 0.64