**True/False Questions**
1. ANOVA is the preferred method for finding differences among several population proportions.
Answer: False Type: Concept Difficulty: Easy
2. The sum of squares error and the sum of squares treatment add to the sum of squares total.
Answer: True Type: Concept Difficulty: Easy
3. The degrees of freedom for error in an ANOVA are n - r.
Answer: True Type: Concept Difficulty: Easy
4. In an ANOVA, if the sum of squares for error is 400, the sum of squares for treatment is 180, and the total sample size for the four groups compared is 88, then the null hypothesis should not be rejected.
Answer: False Type: Computation Difficulty: Hard
5. In an ANOVA, if: n = 130, r = 3 groups, SSE = 12,490, SSTR = 13,000, and using a a = 0.05, the decision should be to reject the null hypothesis.
Answer: True Type: Computation Difficulty: Hard
6. The F ratio in an ANOVA is the ratio of the variation between samples to the variation within.
Answer: True Type: Concept Difficulty: Medium
7. One assumption needed for ANOVA is that the r populations involved are normally distributed, with equal variances.
Answer: True Type: Concept Difficulty: Easy
8. The test statistic in an ANOVA is the ratio of the SSTR and the SSE.
Answer: False Type: Concept Difficulty: Easy
9. The total sum of squares is the sum of the sum of squares for error and the sum of squares for treatments.
Answer: True Type: Concept Difficulty: Easy
10. The degrees of freedom associated with the sum of squares treatments is n - 1.
Answer: False Type: Concept Difficulty: Easy
11. In a completely randomized ANOVA design, each element chosen has an equal chance of being assigned to any particular treatment.
Answer: True Type: Concept Difficulty: Easy
12. A Latin square design is a blocking design that uses one and only one blocking variable.
Answer: False Type: Concept Difficulty: Medium
13. In a fixed-effects model the levels of the factor are fixed before the study begins.
Answer: True Type: Concept Difficulty: Medium
14. The denominator degrees of freedom for the test statistic in an ANOVA are n - r.
Answer: True Type: Concept Difficulty: Easy
15. The deviation due to different treatments is called the between treatments deviation.
Answer: True Type: Concept Difficulty: Easy
16. In ANOVA variables are referred to as treatments.
Answer: True Type: Concept Difficulty: Easy
17. Sir Godfrey Fisher developed ANOVA.
Answer: True Type: Concept Difficulty: Easy
**Multiple Choice Questions**
18. The degrees of freedom for error in ANOVA are:
A) n- 1
B) n - 2
C) n - r
D) r - 1
E) equal to the number of treatments
Answer: C Type: Concept Difficulty: Easy
19. In ANOVA with 4 groups and a total sample size of 65, the computed F statistic is 2.53. In this case, the p-value is:
A) exactly 0.05
B) much less than 0.05
C) much greater than 0.05
D) cannot tell - it depends on what the SSE is
E) none of the above
Answer: E Type: Computation Difficulty: Medium
20. In ANOVA, the sample size is 500 and seven groups are compared. The total sum of squares is 10,000. The decision should be:
A) do not reject the null hypothesis
B) reject the null hypothesis immediately
C) not enough information given to make a decision on the null hypothesis
D) mildly reject the null hypothesis
E) decide that the test must be repeated since only seven groups were involved
Answer: C Type: Concept Difficulty: Medium
21. Analysis of Variance (ANOVA) is a test for equality of:
A) variances
B) means
C) proportions
D) only two parameters
E) none of the above
Answer: B Type: Concept Difficulty: Easy
22. In ANOVA, if the sample standard deviations within all four groups under study are
approximately equal, we should:
A) immediately decide not to reject the null hypothesis
B) immediately reject the null hypothesis
C) mildly reject the null hypothesis
D) redo the test, since the standard deviations are equal
E) there is not enough information to make a decision on whether to reject or not reject the null hypothesis
Answer: E Type: Concept Difficulty: Medium
23. Analysis of variance is a statistical method of comparing the ________ of several populations.
A) standard deviations
B) variances
C) means
D) proportions
E) none of the above
Answer: C Type: Concept Difficulty: Easy
24. If the r population means are equal, then MSTR/MSE will be:
A) more than 1.00
B) very close to 1.00
C) very close to 0.00
D) close to -1.00
E) a negative value between 0 and - 1
Answer: B Type: Concept Difficulty: Medium
25. Error deviations measure distances:
A) within groups
B) between groups
C) both (a) and (b)
D) none of the above
E) between each value and the grand mean
Answer: A Type: Concept Difficulty: Medium
26. When the r population means are different, it is likely that the average error:
A) is relatively large compared to the average treatment
B) is relatively small compared to the average treatment
C) is about equal to the average treatment
D) none of the above
E) differs, depending on the populations involved
Answer: B Type: Concept Difficulty: Medium
27. How many degrees of freedom are there in a test comparing five populations with each sample containing 37 observations?
A) 185
B) 4
C) 36
D) 32
E) none of the above
Answer: E Type: Concept Difficulty: Medium
Use the following to answer questions 28-30:
An electronics manufacturer was evaluating whether customers' satisfaction depended on which configuration of personal media player they purchased. In a random sample of customers, 12 had purchased cassette players, 15 had purchased CD players, and 11 had purchased MP3 players. Each customer sampled was measured on a satisfaction index, and in the manufacturer's ANOVA, SSE = 1,490 and SSTR = 760.
28. Compute the F test statistic for this hypothesis test.
A) 0.112
B) 8.926
C) 1.96
D) 9.69
E) none of the above
Answer: B Type: Computation Difficulty: Medium
29. Write the null hypothesis for this test.
A) H_{0}: not all the population means are equal
B) H_{0}: at least one of the population means is not equal to the others
C) H_{0}: m_{1 }= m_{2 }= m_{3 }
D) H_{0}: m_{1 }- m_{2}- m_{3} ³ 0
E) none of the above
Answer: C Type: Concept Difficulty: Easy
30. Find the critical value to test this hypothesis at a = 0.05.
A) 1.96
B) between 19.46 and 19.47
C) 1.645
D) between 3.23 and 3.32
E) none of the above
Answer: D Type: Computation Difficulty: Medium
Use the following to answer questions 31-33:
A survey is conducted to find whether students like a particular type of music more than another: rock, pop, classical, and jazz. Fifth randomly selected students were asked to rate one particular type of music, with SSTR = 28,590 and SST = 40,220.
31. Write the null hypothesis to test whether students like these types of music equally, or whether there is a preference.
A) H_{0}: m_{1}= m_{2 }= m_{3 }= m_{4 }
B) H_{0}: m_{1}= m_{2 }= m_{3 }= m_{4} ¹ 0
C) H_{0}: s_{1}= s_{2 }= s_{3 }= s_{4 }
D) H_{0}: at least one of the population means is different
E) none of the above
Answer: A Type: Concept Difficulty: Easy
32. How many degrees of freedom does the sum of squares for treatments have?
A) 4
B) 3
C) 0
D) 49
E) 46
Answer: B Type: Computation Difficulty: Medium
33. Compute the test statistic for this analysis of variance test.
A) 252.826
B) 11.61
C) 37.69
D) 10.89
E) none of the above
Answer: C Type: Computation Difficulty: Hard
Use the following to answer questions 34-37:
Three companies recently were audited in a study of travel and entertainment expense. Twenty (20) months of expenses were randomly sampled from each company, with observations as follows:
34. What is the SSTR for these observations?
A) 7,322.92
B) 5,715.09
C) 1,607.83
D) 803.91
E) 124.12
Answer: C Type: Computation Difficulty: Hard
35. What is the SSE for these observations?
A) 7,322.92
B) 5,715.09
C) 1,607.83
D) 803.91
E) 124.12
Answer: B Type: Computation Difficulty: Hard
36. What would be a critical value for the F-ratio for an ANOVA on these observations?
A) 2.77
B) 2.89
C) 3.16
D) 6.51
E) 9.96
Answer: C Type: Computation Difficulty: Medium
37. What is the observed F-ratio for the ANOVA on these observations?
A) 0.2813
B) 0.8078
C) 3.1643
D) 5.2510
E) 8.0179
Answer: E Type: Computation Difficulty: Hard
38. Three companies recently were audited in a study of travel and entertainment expense. Twenty (20) months of expenses were randomly sampled from each company, with observations as follows:
The observed F-ratio for the ANOVA on these observations is 8.0179.
Which of the following is the most likely outcome of a Tukey's Pairwise Comparison test?
A) At least one pairwise comparison would be likely to result in a statistically significant difference
B) Every possible pairwise comparison would be likely to result in a statistically significant difference
C) No pairwise comparison would be likely to result in a statistically significant difference
D) Both A and B are correct
E) Neither A, B nor C is correct
Answer: A Type: Concept Difficulty: Easy
Use the following to answer questions 39-40:
Standardized stock price indicators in three different countries over a week are listed below. An analyst is interested in knowing if the stock markets of these three countries are dependent on one another. The ANOVA table is given.
39. What are the degrees of freedom for treatment, error and total respectively?
A) 17, 15, 2
B) 15, 17, 2
C) 2, 17, 15
D) 2, 15, 17
E) none of the above
Answer: D Type: Computation Difficulty: Medium
40. Compute the MSTR and MSE respectively.
A) 168, 374.06
B) 374.06, 168
C) 157, 384.05
D) 361.03, 165.03
Answer: B Type: Computation Difficulty: Medium
41. If the p-value for this test is 0.143, at an 0.05 level of significance, are the means for these three stock markets equal or significantly different?
A) at least one of the three means is different from the other two
B) all these means are significantly different
C) insufficient evidence to indicate that the three means are not the same
D) further information is required to make a decision
E) none of the above
Answer: C Type: Concept Difficulty: Medium
Use the following to answer questions 42-45:
A company is interested in testing customer satisfaction with five department stores that it owns. They are also interested in determining whether or not differences exist in the average consumer satisfaction with the three main departments within each store: sportswear, formalwear and businesswear. Fifteen groups of 60 randomly chosen consumers are assigned to each combination cell. The results are analyzed with a two-way ANOVA, shown below.
42. Compute the mean square for the treatment, store.
A) 54.25
B) 144.50
C) 6.68
D) 43.4
E) none of the above
Answer: A Type: Computation Difficulty: Medium
43. Compute the F ratio for the interaction.
A) 6.68
B) 5.26
C) 17.795
D) 8.12
E) none of the above
Answer: B Type: Computation Difficulty: Medium
44. How many degrees of freedom does SSE have?
A) 15
B) 899
C) 900
D) 885
E) none of the above
Answer: D Type: Computation Difficulty: Medium
45. At a = 0.05, is there interaction?
A) yes
B) no
C) insufficient information to determine
D) some
E) none of the above
Answer: A Type: Computation Difficulty: Medium
Use the following to answer questions 46-48:
A research company is interested in testing whether three medications on the market are about equally effective reducing high blood pressure. As a part of the study, a random sample of 225 people is chosen. Each person is given one of the medications to use for a month, and their blood pressure is recorded. To reduce experimental errors, the sample is divided into 75 groups of three people each. The results are shown below.
46. What are the degrees of freedom for the sum of squares for the blocks?
A) 74
B) 2
C) 148
D) 3
E) none of the above
Answer: A Type: Computation Difficulty: Medium
47. How many treatments were there in this test?
A) 75
B) 225
C) 3
D) 224
E) none of the above
Answer: C Type: Computation Difficulty: Medium
48. Compute the F ratio for this ANOVA.
A) 49.74
B) 0.80
C) 40.97
D) 32.60
E) none of the above
Answer: C Type: Computation Difficulty: Hard
49. In general, the degrees of freedom associated with the sum of squares error term, in an ANOVA using blocking is:
A) (n - r)
B) (r - 1)
C) (n - 1)
D) (n - 1)(r - 1)
E) none of the above
Answer: D Type: Computation Difficulty: Medium
50. A blocking design that uses two blocking variables is called a:
A) one-way ANOVA
B) repeated measures design
C) completely randomized block
D) Latin square design
E) randomized complete block design
Answer: D Type: Concept Difficulty: Medium
51. A manager in a large company rates his workers based on two factors: productivity and independence. Following are the results of an ANOVA testing employee performance.
How many levels of productivity and independence were studied?
A) productivity = 8, independence = 6
B) productivity = 4, independence = 2
C) productivity = 5, independence = 3
D) productivity = 3, independence = 1
E) none of the above
Answer: C Type: Concept Difficulty: Medium
Use the following to answer questions 52-55:
The management of an organization wants to test to see whether the rate of turnover of employees is the same in all the departments. Samples over the last year show the following as the number of employees having left the company:
Production: 5, 6, 3, 4, 6, 6
Marketing: 3, 4, 4, 4, 3, 5
Finance: 3, 3, 3, 2, 4, 5
Accounting: 2, 3, 3, 5, 6, 4
The ANOVA table is shown below:
52. How many degrees of freedom are associated with the sum of squares for treatments?
A) 3
B) 4
C) 5
D) 23
E) none of the above
Answer: A Type: Computation Difficulty: Medium
53. State the null hypothesis for this ANOVA.
A) H_{0}: m_{1} - m_{2} - m_{3 }- m_{4 }³ 0
B) H_{0}: all of the population means are different
C) H_{0}: at least one of the population means is different
D) H_{0}: m_{1} = m_{2} = m_{3 }= m_{4}
E) none of the above
Answer: D Type: Concept Difficulty: Medium
54. If the p-value for this ANOVA is 0.117, what is the conclusion, testing at a = 0.05?
A) reject the null hypothesis
B) do not reject the null hypothesis
C) not enough information to make a decision
D) at least one of the population means is significantly different
E) none of the above
Answer: B Type: Concept Difficulty: Medium
55. Construct a 95% confidence interval on the population mean of the Finance department.
A) 3.333 ± (1.725)(1.033/6)
B) 3.333 ± (2.571)(1.033/2.449)
C) 3.333 ± (1.725)(1.033/2.449)
D) 3.333 ± (2.086)(1.033/2.449)
E) none of the above
Answer: D Type: Computation Difficulty: Hard
56. In an ANOVA, we find that the p-value is 0.003. We therefore conclude that:
A) there is no statistical evidence that any population mean is different from any other
B) no two population means are equal
C) no two variances are equal
D) the null hypothesis should be accepted
E) there is strong statistical evidence that not all the population means are equal
Answer: E Type: Concept Difficulty: Medium
57. In an ANOVA with r = 5 groups and a total sample size of 65, using a = 0.05, the critical point for the distribution to be used is:
A) 1.70
B) 2.53
C) 2.064
D) 1.96
E) 1.645
Answer: B Type: Computation Difficulty: Medium
58. When all members of every block are randomly assigned to all treatments, the design is called:
A) repeated measures design
B) Tukey design
C) Latin square design
D) one-way ANOVA
E) randomized complete block design
Answer: E Type: Concept Difficulty: Medium
59. Given the following ANOVA table, with N = 55:
Compute the F ratio for the interaction.
A) 12.0476
B) 3.608
C) 1.611
D) 3.339
E) none of the above
Answer: B Type: Computation Difficulty: Hard
60. Which of the following is the correct F ratio in the one-way ANOVA?
A) MSA/MSE
B) MSBL/MSE
C) MSTR/MSE
D) MSE/MSTR
E) MSE/MST
Answer: C Type: Concept Difficulty: Medium
Use the following to answer questions 61-65:
You are interested in the effect of pay on the productivity of your employees. To test the effect you setup a study. The results follow:
61. What is your null hypothesis?
A) More pay will lower or not affect productivity.
B) More pay will raise or not affect productivity.
C) Pay will not affect productivity.
D) All of the above
E) None of the above
Answer: C Type: Concept Difficulty: Easy
62. What is your alternate hypothesis?
A) More pay will lower or not affect productivity.
B) More pay will raise or not affect productivity.
C) Pay will not affect productivity.
D) All of the above
E) None of the above
Answer: E Type: Concept Difficulty: Easy
63. How many levels of treatment are in the experiment?
A) 1
B) 2
C) 3
D) All of the above
E) None of the above
Answer: C Type: Computation Difficulty: Easy
64. How many subjects were in the study?
A) 38
B) 39
C) 40
D) All of the above
E) None of the above
Answer: B Type: Computation Difficulty: Easy
65. With an a of .05 what is your decision?
A) Reject the null and accept the alternative hypothesis.
B) Accept the null hypothesis.
C) Reject the alternative and accept the null hypothesis.
D) All of the above
E) None of the above
Answer: A Type: Concept Difficulty: Easy
66. If, in a two-way ANOVA, Factor A has five levels and Factor B has three levels, then the number of degrees of freedom for the interaction term is:
A) 2
B) 4
C) 8
D) 12
E) 15
Answer: C Type: Computation Difficulty: Easy
67. In a _____________ effects model, levels of the factor to be studied are decided upon in advance; in a _____________ effects model, only a subset of all possible factor levels are ultimately considered.
A) Random; Fixed
B) Blocked; Random
C) Fixed; Blocked
D) Fixed; Random
E) Blocked; Fixed
Answer: D Type: Concept Difficulty: Easy
68. One factor with four levels is being studied via ANOVA. The total sample size is 24, with an equal number of observations for each factor level. A significant F-ratio is observed and a Tukey's Pairwise Comparison will be conducted as a follow-up. What is the appropriate critical *q *value for this exercise (assume a = 0.05)?
A) 3.58
B) 3.90
C) 3.96
D) 4.64
E) 5.02
Answer: C Type: Computation Difficulty: Easy
Use the following to answer questions 69-70:
A consulting firm is studying the effects of leadership structure on team performance. Four structures are considered, and seven teams are randomly assigned to each structure type. Results on a proprietary performance index are as follows:
70. As a followup, the consulting firm wishes to know which structure to emphasize. Conduct a Tukey's Pairwise Comparison as an input into this decision
Answer:
Tukey's HSD = 7.922 = *q*_{0.05}*[28.89/7]^{½}
The sample means for the leadership structures are as follows:
A: x-bar = 37.9
B: x-bar = 37.8
C: x-bar = 29.4
D: x-bar = 29.9
A is significantly higher than both C & D
B is significantly higher than both C & D
A is not significantly different from B
C is not significantly different from D
It appears that the consulting firm should choose either structure A or B.
71. In a two-way ANOVA, Factor A has four levels and Factor B has three levels. There were 10 observations for every combination of Factors A and B. Sums of squares were as follows:
What can be concluded in this instance?
Answer: The ANOVA table is as follows:
**p-value < 0.001
Because the interaction term is significant, the insignificant finds for Factors A and B are of little concern. Essentially, these findings indicate that Factors A and B are most important to consider *in combination, *not in isolation.
72. You are interested in the effect of reward vs. punishment on communications with employees. A recent study which analyzes the effect of reward vs. punishment on communication yields the following results.
Summary of the Analysis of Variance of the Effect of Reward and Punishment on Employee Communication
A) What are the dependent and independent variables?
B) What is the null and alternate hypotheses?
C) How many levels were in the treatment?
D) How many subjects were in the study?
E) What is your decision?
Answer:
A) Independent: Reward vs. Punishment, Dependent: Communication
B) Null: Reward has no effect on communication.
Alternative: Reward has an effect on communication.
C) 3
D) 30
E) Reject the null hypothesis, reward and punishment have an effect on communication.
Type: Computation Difficulty: Easy
*Aczel, Complete Business Statistics, Sixth Edition *
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