Chapter 12: One-Way Analysis of Variance

SECTION 12.1 EXERCISES

For Exercises 12.1 and 12.2, see page 651 for Exercises 12.3 and 12.4, see page 653 for Exercise 12.5, Exercise 12.6, Exercise 12.7 and Exercise 12.8, see pages 655–656 and for Exercises 12.9 and 12.10, see page 663.

Question 12.11

12.11 A one-way ANOVA example. A study compared five groups with six observations per group. An F statistic of 4.81 was reported.

(a) Give the degrees of freedom for this statistic and the entries from Table E that correspond to this distribution.
(b) Sketch a picture of this F distribution with the information from the table included.
(c) Based on the table information, how would you report the P-value?
(d) Can you conclude that all pairs of group means are different? Explain your answer.

12.11 (a) 4 and 25. In Table E, 4.18 < F < 6.49. (c) 0.001 < P-value < 0.01. (d) Because the P-value is small we reject H₀; however, this does not say that all pairs of group means are different, only that at least one mean is different.

Question 12.12

12.12 Visualizing the ANOVA model. For each of the following settings, draw a picture of the ANOVA model similar to Figure 12.6 (page 652). To sketch the Normal curves, you may want to review the 68–95–99.7 rule on page 57.

(a) μ₁ = 17, μ₂ = 13, μ₃ = 12, and σ = 2.
(b) μ₁ = 17, μ₂ = 13, μ₃ = 12, and σ = 4.
(c) μ₁ = 20, μ₂ = 12, μ₃ = 10, and σ = 3.

Question 12.13

12.13 Visualizing the ANOVA model, continued. Refer to the previous exercise. If SRSs of size n = 5 were obtained from each of the three populations, under which setting would you most likely obtain a significant ANOVA F test? Explain your answer.

12.13 Generally, the one with the largest difference between means and the smallest standard deviation will be the most significant. If this is not clear a ratio between the two can be used. So part (b) can be ruled out because it has a larger sigma than part (a) with the same means. Of the remaining two we can see that part (c) has a bigger max difference relative to its sigma (10 to 3) than part (a) does (5 to 2), so part (c) will be the most significant.

Question 12.14

12.14 Calculating the ANOVA F test P-value. For each of the following situations, find the degrees of freedom for the F statistic and then use Table E to approximate the P-value.

(a) Six groups are being compared with five observations per group. The value of the F statistic is 2.47.
(b) Four groups are being compared with 11 observations per group. The value of the F statistic is 5.03.
(c) Five groups are being compared using 65 total observations. The value of the F statistic is 3.11.

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Question 12.15

12.15 Calculating the ANOVA F test P-value, continued. For each of the following situations, find the F statistic and the degrees of freedom. Then draw a sketch of the distribution under the null hypothesis and shade in the portion corresponding to the P-value. State how you would report the P-value.

(a) Compare three groups with 21 observations per group, MSE = 50, and MSG = 340.
(b) Compare eight groups with six observations per group, SSG = 77, and SSE = 190.

12.15 (a) F = 6.8, DFG = 2, DFE = 60. 0.001 < P-value < 0.01. (b) DFG = 7, DFE = 40. MSG = 11, MSE = 4.75; F = 2.32, 0.025 < P-value < 0.05.

Question 12.16

12.16 Calculating the pooled standard deviation. An experiment was run to compare three groups. The sample sizes were 28, 33, and 102, and the corresponding estimated standard deviations were 2.7, 2.6, and 4.8.

(a) Is it reasonable to use the assumption of equal standard deviations when we analyze these data? Give a reason for your answer.
(b) Give the values of the variances for the three groups.
(c) Find the pooled variance.
(d) What is the value of the pooled standard deviation?
(e) Explain why your answer in part (d) is much closer to the standard deviation for the third group than to either of the other two standard deviations.

Question 12.17

12.17 Describing the ANOVA model. For each of the following situations, identify the response variable and the populations to be compared, and give I, n_i, and N.

(a) A poultry farmer is interested in reducing the cholesterol level in his marketable eggs. He wants to compare two different cholesterol-lowering drugs added to the hens’ standard diet as well as an all-vegetarian diet. He assigns 25 of his hens to each of the three treatments.
(b) A researcher is interested in students’ opinions regarding an additional annual fee to support non-income-producing varsity sports. Students were asked to rate their acceptance of this fee on a seven-point scale. She received 94 responses, of which 31 were from students who attend varsity football or basketball games only, 18 were from students who also attend other varsity competitions, and 45 were from students who did not attend any varsity games.
(c) A professor wants to evaluate the effectiveness of his teaching assistants. In one class period, the 42 students were randomly divided into three equal-sized groups, and each group was taught power calculations from one of the assistants. At the beginning of the next class, each student took a quiz on power calculations, and these scores were compared.

12.17 (a) Response: egg cholesterol level. Populations: chickens with different diets or drugs. I = 3, n₁= n₂= n₃= 25, N = 75. (b) Response: rating on seven-point scale. Populations: the three groups of students. I = 3, n₁= 31, n₂= 18, n₃= 45, N = 94. (c) Response: quiz score. Populations: students in each TA group. I = 3, n₁= n₂= n₃= 14, N = 42.

Question 12.18

12.18 Describing the ANOVA model, continued. For each of the following situations, identify the response variable and the populations to be compared, and give I, n_i, and N.

(a) A developer of a virtual-reality (VR) teaching tool for the deaf wants to compare the effectiveness of different navigation methods. A total of 40 children were available for the experiment, of which equal numbers were randomly assigned to use a joystick, wand, dancemat, or gesture-based pinch gloves. The time (in seconds) to complete a designed VR path is recorded for each child.
(b) To study the effects of pesticides on birds, an experimenter randomly (and equally) allocated 65 chicks to five diets (a control and four with a different pesticide included). After a month, the calcium content (milligrams) in a 1-centimeter length of bone from each chick was measured.
(c) A university sandwich shop wants to compare the effects of providing free food with a sandwich order on sales. The experiment will be conducted from 11:00 A.M. to 2:00 P.M. for the next 20 weekdays. On each day, customers will be offered one of the following: a free drink, free chips, a free cookie, or nothing. Each option will be offered five times.

Question 12.19

12.19 Determining the degrees of freedom. Refer to Exercise 12.17. For each situation, give the following:

(a) Degrees of freedom for group, for error, and for the total.
(b) Null and alternative hypotheses.
(c) Numerator and denominator degrees of freedom for the F statistic.

12.19 For all three situations, we have H₀: $μ_{1}$ = $μ_{2}$ = $μ$ ₃. $H_{α}$ : not all of the $μ$ _i are equal. DFG = I – 1 = 2, DFE = N − I, and DFT = N − 1. The degrees of freedom for the F test are DFG and DFE. (a) DFG 2, DFE 72, DFT 74; F(2,72). (b) DFG 2, DFE 91, DFT 93; F(2,91). (c) DFG 2, DFE 39, DFT 41; F(2,39).

Question 12.20

12.20 Determining the degrees of freedom, continued. Refer to Exercise 12.18. For each situation, give the following:

(a) Degrees of freedom for group, for error, and for the total.
(b) Null and alternative hypotheses.
(c) Numerator and denominator degrees of freedom for the F statistic.

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Question 12.21

12.21 Data collection and the interpretation of results. Refer to Exercise 12.17. For each situation, discuss the method of obtaining the data and how this will affect the extent to which the results can be generalized.

12.21 (a) This sounds like a fairly well-designed experiment, so the results should at least apply to this farmer’s breed of chicken. (b) It would be good to know what proportion of the total student body falls in each of these groups—that is, is anyone overrepresented in this sample? (c) How well a TA teaches one topic (power calculations) might not reflect that TA’s overall effectiveness.

Question 12.22

12.22 Data collection, continued. Refer to Exercise 12.18. For each situation, discuss the method of obtaining the data and how this will affect the extent to which the results can be generalized.

Question 12.23

12.23 The effects of two stimulant drugs. An experimenter was interested in investigating the effects of two stimulant drugs (labeled A and B). She divided 25 rats equally into five groups (placebo, Drug A low, Drug A high, Drug B low, and Drug B high) and, 20 minutes after injection of the drug, recorded each rat’s activity level (higher score is more active). The following table summarizes the results:

Treatment	$\bar{x}$	s²
Placebo	11.80	17.20
Low A	15.25	13.10
High A	18.55	10.25
Low B	16.15	7.75
High B	17.10	12.50

(a) Plot the means versus the type of treatment. Does there appear to be a difference in the activity level? Explain.
(b) Is it reasonable to assume that the variances are equal? Explain your answer, and if reasonable, compute s_p.
(c) Give the degrees of freedom for the F statistic.
(d) The F statistic is 2.64. Find the associated P-value and state your conclusions.

12.23 (a) Both drugs cause an increase in activity level; Drug B appears to have a greater effect. (b) Yes; $\sqrt{17.2}$ < 2 $\sqrt{7.75}$ , s_p = 3.487. (c) DFG = 4, DFE = 20. (d) 2.25 < F < 2.87, 0.05 < P-value < 0.10.

Question 12.24

12.24 Perceptions of social media. It is estimated that more than 90% of North American students use social media. This has prompted much research on the mental health impacts of these technologies. In one study, researchers investigated how mental health workers perceive the association between social media and mental disorders. A sample of psychiatrists from Canada completed a questionnaire, from which a perception score was obtained (a higher score indicating a stronger perceived association). The following ANOVA table summarizes a comparison of these scores across three age groups (generations).

Source	DF	SS	MS	F
Age	2	137.78	68.89	0.45
Error	45	6899.54	153.32
Total	47	7037.32

(a) How many psychiatrists completed the questionnaire?
(b) What is the estimated common standard deviation?
(c) What is the P-value? Make sure to specify the degrees of freedom of the F statistic.
(d) State your conclusion using the P-value from part (c) and a 5% significance level.

Question 12.25

12.25 Pain tolerance among sports teams. Many have argued that sports such as football require the ability to withstand pain from injury for extended periods of time. To see if there is greater pain tolerance among certain sports teams, a group of researchers assessed 183 male Division II athletes from five sports.⁶ Each athlete was asked to put his dominant hand and forearm in a 3°C water bath and keep it in there until the pain became intolerable. The total amount of time (in seconds) that each athlete maintained his hand and forearm in the bath was recorded. Following this procedure, each athlete completed a series of surveys on aggression and competitiveness. In their report, the researchers state:

A univariate between subjects (sports team) ANOVA was performed on the total amount of time athletes were able to keep their hand and forearm in the water bath, and found it to be statistically significant, F(4,146) = 4.96, p < .001.

Further analysis revealed that the lacrosse and soccer players tolerated the pain for a significantly longer period of time and swimmers tolerated the pain for a significantly shorter period of time than athletes from the other teams.

(a) Based on the description of the experiment, what should the degrees of freedom be for this analysis?
(b) Assuming that the degrees of freedom reported are correct, data from how many athletes were used in this analysis?
(c) The researchers do not comment on the missing data in their report. List two reasons these data may not have been used, and for each, explain how the omission could impact or bias the results.

12.25 (a) 4 and 178. (b) 5 + 146 = 151 athletes actually participated. (c) For example, the individuals could have been outliers in terms of their ability to withstand the water-bath pain. In either case of low or high outliers, their removal would lessen the standard deviation for their sport and move that sports mean.

Question 12.26

12.26 Constructing an ANOVA table Refer to Exercise 12.5 (page 655). Using the table of group means and standard deviations, construct an ANOVA table similar to that on page 662. Based on the F statistic and degrees of freedom, compute the P-value. What do you conclude?