Old Tests Covering Analysis of Variance
Questions 1-4 are concerned with using One-Way Anova in the following situation. A manager wishes
to try three new training methods to determine which will best increase the output of workers (in parts
per minute). 15 workers are randomly assigned to the three training methods and following the
training their output is measured .
1. The objects being experimented on are...
A. one training method.
B. the three training methods.
C. the workers
D. 15.
E. the output in parts per minute.
2. The type of experimental design being used is a __________.
A. completely randomized design.
B. randomized block design.
C. Two-Way factorial design.
D. observational study
E. Greko-Latin square design.
3. What is the rejection region for testing that the mean output is the same for all training methods?
A. Reject H0 if |t| > t(3, 0.025)
B. Reject H0 if F > F(3, 12, 0.05)
C. Reject H0 if F > F(2, 12, 0.05)
D. Reject H0 if F > F(3, 15, 0.05)
E. Reject H0 if |t| > t(12, 0.025)
4. Given the following ANOVA table what is the test statistic value when testing the null hypothesis of no
difference in mean output among the training methods.
-----------------------------------Source
df
SS
MS
--------Method
__
40
__
Error
12
144
__
Total
14
-----------------------------------A. F = (20 / 12)
B. F = (40 / 144)
C. t = (40 / 12)
D. F = ( 2 / 12)
E. not enough information is available to calculate it
Questions 5-10 are concerned with Randomized Block and Two-Way Anova
5. The experiment was repeated with 15 different workers. In this experiment, workers were placed into
five groups depending on their productivity before the training. Within each group, the three workers
were randomly assigned one to each training method. Given the following ANOVA table, does the
productivity before the training help explain the variation in productivity after the training?
-----------------------------------Source
df
SS
MS
--------Method
2
38
19
Group
4
120
30
Error
8
24
3
Total
14
-----------------------------------A. Yes because the F value of 5 is larger than F(4, 14, 0.5)
B. Yes because the F value of 6.33 is larger than F(2, 8, 0.5)
C. Yes because the F value of 10 is larger than F(4, 8, 0.5)
D. No because the t value of 30 is larger than t(14, 0.025)
E. No because the F value of 1.33 is smaller than F(2, 8, 0.5)
6. If you reject the null hypothesis when testing the training methods, then your managerial conclusion
would be...
A. At alpha = 0.05, we can say the mean training method is different between the productivity in parts
per minute before the experiment.
B. At alpha = 0.05, we can say the mean variance in productivity in parts per minute is different between
at least two training methods.
C. At alpha = 0.05, we can say the mean productivity in parts per minute is the same for all training
methods.
D. At alpha = 0.05, we can not say the variance in productivity in parts per minute is different between at
least two training methods.
E. At alpha = 0.05, we can say the mean productivity in parts per minute is different between at least two
training methods.
Questions 7-10 are concerned with the following experiment. The ability of workers to learn a new
training method may be affected by the time of the day when the training method is taught. The same
training methods were used as before, both in three morning classes and in three afternoon classes. Thirty
workers (who were very similar in productivity before the experiment) were randomly assigned to the six
combinations.
The following ANOVA table was obtained.
-----------------------------------Source
df
SS
MS
F
--------- -Method
2
56
28 28
Class
1
18
18 18
Interaction
2
46
23 23
Error
24
24
1
Total
29
------------------------------------
7. What is the test statistic value for testing if the model is useful for estimated the mean of y?
A. 1
B. 24
C. 27
D. 28
E. 23
8. Does are the factors correlated in this situation?
A. No, any experiment that does not use time-series data can not have correlated
B. No, because there are the same number of workers in each of the six combinations.
C. Yes, because interaction is statistically significant.
D. Yes, because dummy variables are being used instead of real numbers.
E. Yes, because this is not a randomized block design.
9. Does the difference between the mean productivity of morning and afternoon classes depend on the
training method?
A. Yes, because 20 is larger than F(1, 24, 0.05)
B. No, because 23 is larger than F(2, 24, 0.05)
C. Yes, because 27 is larger than F(2, 2, 0.05)
D. No, because 1 is not larger than F(24, 24, 0.05)
E. Yes, because 23 is larger than F(2, 24, 0.05)
10. If interaction is significant, which means should be compared?
A. the six treatment or cell means
B. the 3 training method means and the 2 class time means
ANSWERS to questions 1-10
1. C.
2. A
3. C
4. A
5. C
6. E
7. B
8. B
9. E.
10. A
Questions 11-17 refer to the following situation: A researcher wishes to compare the preference of seven
different colors of characters on a terminal screen. Seventy persons are randomly assigned, ten each, to
the seven colors. After 30-minutes, each person scored the color using a 0-9 preference scale. A statistical
analysis of the data follows:
-----------------------------------Source
df
SS
MS
--------Color
6
420
70
Error
63
441
7
Total
69
861
------------------------------------
11-13 use One-Way Anova
11. The experimental unit (the object on which measurements are being made) is
(A) the terminal screen task
(B) the preference score
(C) color
(D) the student
(E) the characters
12. The factor is
(A) the terminal screen task
(B) the preference score
(C) color
(D) the student
(E) the characters
13. To test the null hypothesis of no difference in mean preference, the test statistic value is
(A) 420/4410 = 0.095
(B) 70/70 = 1
(C) 70/7 = 10
(D) 4410/4830 = 0.913
(E) 420/4830 = 0.087
Questions 14-23 use Randomized Block or Two-Way Anova
14. In a similar experiment, seventy students are divided into 10 groups according to their experience with
using computers. In each group, the colors are randomly assigned to the students. The following
incomplete ANOVA table was obtained:
-----------------------------------Source
df
SS
MS
--------Color
6
420
70
Experience __
360
__
Error
54
81
__
Total
69
861
-----------------------------------The F test for the effect of color is now:
(A) 70/1.5 = 46.7
(B) 70/70 = 1.00
(C) 70/4 = 17.5
(D) 3600/810 = 4.44
(E) 70/40 = 1.75
15. The above design is called
(A) a factorial design
(B) a One-Way analysis of variance
(C) a completely randomized design
(D) a randomized complete block
(E) an independent sampling design
16. How would the experiment be changed so that interaction could be measured while still keep color
and student main effects independent?
(A) You needed 12 groups instead of 10.
(B) One student must score one of the colors again.
(C) Each student will be asked to write their scores down twice.
(D) You can not have interaction and maintain independent main effects.
(E) You need 2 students in each group color combination.
17. Using the results of the two ANOVAs given above, which of the following assumptions has been
violated for the completely randomized
design (CRD)?
(A) linearity
(B) normality
(C) equal variance
(D) all of the above
(E) can not be checked with this information.
Questions 18-23 are concerned with the following situation: You are interested in the productivity of
assembly line workers. Two factors are thought to affect productivity: incoming rate (40, 50, & 60 parts
per minute) and the room temperature (65 or 75 degrees F). Twelve workers were randomly assigned to
the 6 combinations. The statistical analysis is on the attached pages.
18. What is the point estimate of the difference in mean productivity when the rate=50 minus the mean
productivity when the rate=40?
(A) 2.05
(B) 0.9032
(C) 4.339
(D) 1.05
(E) 0.173333
19. Interaction occurs when:
(A) the value of rate is correlated with the value of temperature.
(B) the productivity is not the same for all levels of Rate given the temperature.
(C) the difference in mean productivity between 65 and 75 degrees is not the same for all levels of Rate.
(D) the mean productivity is not the same for all levels of Rate given the temperature.
(E) the mean productivity depends on the levels of Rate given the temperature.
20. The F test value for testing the main effect of temperature is
(A) 0.05
(B) 24.92
(C) 125.33
(D) 8.48
(E) 50.51
21. omit
22. omit
23. This design is a:
(A) balanced factorial
(B) unbalanced factorial
(C) complete randomized block
(D) fractional factorial
(E) observational study
Analysis of Variance -
Main Effects & Interaction
Dependent Variable: Y
Source
Production of units per Minute
Sum of
Mean
DF
Squares
Square
RATE
TEMP
RATE*TEMP
Error
Corrected Total
2
1
2
6
11
2.94000000
4.32000000
43.44666667
1.04000000
51.74666667
1.47000000
4.32000000
21.72333333
0.17333333
===========================================================
Comparing the three Rate Means:
Tukey's Studentized Range (HSD) Test for variable: Y
Alpha= 0.05 df= 6 MSE= 0.173333 Critical Value = 4.339
Grouping
Mean
Different From Group
60
29.2500
50,40
50
26.6500
60,40
40
24.6000
50,60
========================================================
F Value
8.48
24.92
125.33
ANSWERS TO Questions 11-23
QUESTION Answer
11 D
12 C
13 C
14 A
15 D
16 E
17 A
18 A
19 C
20 B
21 C
22 E
23 A
Questions 24 -26 use One-Way Anova
24. Suppose you wish to compare three advising procedures for graduate students. Twenty-seven students
are randomly assigned to the three advising methods. Following their advising, they are asked to rate the
advising on a 1 (great) to 10 (terrible) scale. Which of the following is true?
A. The experimental unit is the advising procedure and the factor is the scale.
B. The response is a treatment and the factor is the student.
C. I haven't the foggiest idea what you are talking about. (Hint, this might be true but will still be counted
wrong.)
D. The response is the student and the factor is their opinion.
E. The response is their opinion score and the experimental unit is the student.
25. omit
26. What is a limitation of a Completely Randomized Design?
A. The dependent variable may vary due to factors that are not being measured.
B. The residuals are positively correlated.
C. There is always interaction.
D. The assignment of observations to factors is complicated compared to a Randomized Block Design.
E. The F test can not be used.
Questions 27-29 use Randomized Block or Two-Way Anova
Questions 27-29 are concerned with the following situation: You wish to compare three advising
procedures for graduate students. Twenty-seven students are divided into 9 groups of three each
according to their feeling about advising before the experiment. The students within each group are
randomly assigned to the three advising methods. Following their advising, they are asked to rate the
advising on a 1 (great) to 10 (terrible) scale.
27. Complete the following: You can use (1) _________ to reduce the noise and (2) _____________ to
increase the signal of an experiment.
A. (1) analysis of variance, (2) regression
B. (1) another factor, (2) randomized block
C. (1) one factor, (2) a dependent variable
D. (1) an experiment , (2) an observational study
E. (1) the randomized block design, (2) increase the sample size
28. A difference in the qualitative independent variables of the Randomized Block Design (RBD) and the
Factorial Design (FD) is that
A. the FD design may have quantitative independent variables while the RBD may not.
B. Tukey's procedure can not be used with the variables in a RBD design.
C. the variables are exactly the same.
D. only one variable is of interest in the RBD while both variables and their interaction are of interest in
the FD.
E. the RBD is an experiment while the FD is an observational study.
29. omit
Questions 30-34 are concerned with the following situation: The percent of water removed from paper as
it moves through a dryer depends on the temperature of the dryer and the length of time spent in the dryer.
An experiment was conducted at three temperatures (100, 120, and 140 degrees F) and at three exposure
times (10, 20, and 30 seconds). Two paper specimens were prepared for each of the 3 X 3 = 9 conditions.
The percent of water remaining in the paper was measured. The data and analyses are attached.
30. Does the change in the mean percent of water between any two temperatures depend on the exposure
time?
A. No, because 3.49 < F(4, 9, 0.05)
B. Yes, because 3.49 > F(4, 9, 0.05)
C. Yes, because the p-value is less than 0.05.
D. No, because there are two observations per cell.
E. Yes because 195.04 > F(8, 9,0.05)
31. omit
32. omit
33. omit
Analysis of Variance - Main Effects and Interactions
Source
DF
TIME
TEMP
TIME*TEMP
Error
Corrected Total
2
2
4
9
17
Answers to Questions 24- 33
24 e
25 a
26 a
27 e
28 d
29 a
30 a
31
32
33.
e
Sum
Squares
4456.77
2244.11
60.55
39.00
6800.44
Mean
Square
2228.38
1122.05
15.13
4.33
F Value
514.24
258.94
3.49
p-value
0.0001
0.0001
0.0551