13.1 xijk=μij+εijk, i=1, 2, j=1, 2, k=1, . . . , nij; εijk~N(0, σ). We have I=2, J=2, n11=27, n12=23, n21=19, and n22=31. The parameters of the model are μ11, μ12, μ21, μ22, and σ.

13.3 Answers will vary.

13.5 DFA=2; DFB=3; DFAB=6.

13.1 Answers will vary.

13.3

1. The factors are sex (I=2) and age (J=3). The response variable is the percent of pretend play. N=66.

2. The factors are weeks after harvest (I=5) and amount of water (J=2). The response variable is the percent of seeds germinating. N=30.

3. The factors are mixture (I=6) and freezing/thawing cycles (J=3). The response variable is the strength of the specimen. N=54.

4. The factors are different colored tags (I=4) and the type of buyer (J=2). The response variable is the total dollar amount spent. N=138.

13.5

2. There seems to be a fairly large difference between the means based on how much the rats were allowed to eat but not very much difference based on the chromium level. There may be an interaction: the NM mean is lower than the LM mean, and the NR mean is higher than the LR mean.

3. L: 4.86, N: 4.871, M: 4.485, R: 5.246; LR minus LM: 0.63; NR minus NM: 0.892. Mean GITH levels are lower for M than for R; there is not much difference for L versus N. The difference between M and R is greater among rats that had normal chromium levels in their diets (N).

13.7

1. This is a repeated-measures design because the same students were tested twice.

2. The plot shows a definite interaction; the control group’s mean score decreased, while the expressive writing group’s mean increased somewhat.

3. No, the largest s is more than twice the smallest s; 14.3>2(5.8)=11.6.

13.9 Answers will vary.

13.11 Answers will vary. The table has three training programs with two options for taking it. Six groups total. With a total of 100 employees, the number in each cell will vary.

13.13

1. DFAB=2×4=8.

2. Sample sizes do not need to be all the same.

3. The error variation is not separated.

4. We compare the means of six groups.

13.15

1. Source	df
   A	2
   B	2
   AB	4
   E	27

2. The degrees of freedom to test for the interaction are DFAB=4, DFE=27; 2.73<F<3.31, 0.025<P-value<0.05.

4. 0.025<P-value<0.05.

5. Interaction is not significant; the interaction plot should have roughly parallel lines.

13.17

1. Using α=0,05, the main effect A is not significant, P-value=0.083 (df=1,24); Main effect B is significant, P-value=0.02 (df=2,24);; the interaction is not significant, P-value=0.259(df=2,24)

2. Means associated with levels of B.

13.19

1. Source	df	SS	MS	F
   A	1	7.13	7.13	1.06
   B	1	22.75	22.75	3.39
   AB	1	29.81	29.81	4.44
   E	32	214.72	6.71
   Total	35	274.41

2. 2.59.

3. 21.75%.

4. H0: μCarbo=μNanovs. Ha: μCarbo≠μNano;) H0: μWith=μwithout. Ha: μwith≠μwithout; H0: ) Ha: versus P-value=0.31(df=1, 32); H0: There is no interaction between copper and resveratrol. versus Ha: There is an interaction between copper and resveratrol.

5. The main effect of copper is not significant, P-value=0.31 (df=1,32); Main effect resveratrol is not significant, P-value=0.07 (df=1,32); the interaction is significant, P-value=0.04 (df=1,32).

6. Answers will vary.

13.21 Answers will vary.

13.23

1. For those with real-time feedback, the average total cost for those not informed was only slightly larger than the average total cost for those informed, while for those without feedback, the not informed average was much larger than for those who were informed. In addition, we can see an interaction effect. For those not informed, the lack of real-time feedback increased their spending, while for those who were informed, the lack of real-time feedback decreased their spending.

2. We assume the samples are independent. Groups have equal sample sizes. The sample sizes are large enough for CLT.

3. F=18.18, df=3,190, P-value<0.0001.

4. Interaction: F=16.18, P-value<0.0001; Informed: F= 37.06, P-value<0.0001; Smartcart: F=1.16, P-value=0.2828.

13.25

1. A large value of the AB F statistic indicates that we should reject the hypothesis of no interaction.

2. The relationship is backward: Mean squares equal the sum of squares divided by degrees of freedom.

3. Under H0, the ANOVA test statistics have an F distribution.

4. If the sample sizes are not the same, the sums of squares may not add for “some methods of analysis.”

13.27

1. Response: willingness to pay ($); Factor A: Saturation; Factor B: Carry-on suitcase size; I=2; J=2; N=156.

2. Response: total amount spent at store ($); Factor A: Scanner; Factor B: Self-control; I=2; J=4; N=139.

3. Response: tensile strength; Factor A: Adhesive; Factor B: Surface roughness; I=6; J=4; N=96.

13.29

1. There is a slight interaction. Factor B level 1 is higher than Factor B level 2, but this difference increases as the level of Factor A increases—hence the interaction.

2. There is no interaction. Factor B level 2 is consistently higher than level 1, regardless of Factor A. In addition, the means increase as Factor A level increases.

3. There is an interaction; for Factor A level 1, Factor B level 2 is higher than level 1; this difference increases for Factor A level 2; finally, there is a difference between levels 2 and 1 for Factor B when Factor A is level 3.

4. There is an interaction; for Factor A levels 1 and 3, Factor B level 1 is higher than level 2; however, this reverses for Factor A level 2, where Factor B level 1 is now much higher than level 2.

13.31

1. The Normality assumption is for the error terms, not the measurements; however, recall from Chapter 12 that ANOVA is robust to reasonable departures from Normality, especially when sample sizes are similar.

2. Yes; the largest s is less than twice the smallest: 1.62<2(0.82)=1.64.

3. Source	DF	SS	MS	F	P-value
   Age	6	31.97	5.328	4.4	0.0003
   Sex	1	44.66	44.66	36.879	0.0000
   Age*Sex	6	13.22	2.203	1.819	0.0962
   Error	232	280.95	1.211

13.33

1. There appears to be an interaction effect; the lines are not parallel.

2. There appears to be a significant Focus main effect, so the marginal means for Focus would be useful in explaining this difference.

3. If each participant looked at a picture of each body type, then their responses likely would be related to each other, which violates the independence assumption.

13.35

1. There appears to be an interaction; a thank you increases repurchase intent for consumers with a short history but not for those with long history.

2. Short: 6.245; Long: 7.45; No: 6.61; Yes: 7.085. Generally, the long history consumers are more likely to repurchase. Thank you is misleading, suggesting that no thanks is lower than yes thanks, but that is true only for the short history group.

13.37

1. The plot suggests a possible interaction.

2. By subjecting the same individual to all four treatments, rather than using four individuals, we reduce the within-groups variability.

13.39

1. We’d expect reaction times to slow with older individuals. If bilingualism helps brain functioning, we would not expect that group to slow as much as the monolingual group. The expected interaction is seen in the plot; mean total reaction time for the older bilingual group is much less than for the older monolingual group; the lines are not parallel.

2. The interaction is just barely not significant (F=3.67, P-value=0.059). Both main effects are very significant (P-value=0.000).

13.41

1. There appears to be an interaction effect. A favorable process increases satisfaction for those with a favorable outcome but not for those with an unfavorable outcome.

2. No interaction effect. Favorable outcome means were only slightly higher than the unfavorable outcomes means. However, both favorable process means were higher than both unfavorable process means.

3. Yes, there appears to be a three-factor interaction because the interactions in parts (a) and (b) are different.

13.43 Will ignore differences in sample size because roughly the same. Here is one pair of main effects and their interaction. Humor slightly increases satisfaction (from 3.58 to 3.96). A favorable process greatly increases satisfaction (from 2.79 to 4.75). With humor, a favorable process increases satisfaction slightly more than 1 point (from 2.96 to 4.20) while without humor, a favorable process increases satisfaction more than 2.5 points (from 2.61 to 5.31).

13.45 sp=1.7746. Yes, the largest s is less than twice the smallest s; 2.024<2(1.601)=3.202.

13.47 Patterns are similar for all four responses. Canada has the highest mean responses, then the United States, then France. Females are higher than males in all cases, except for credibility and purchase intention in France. Sex differences are largest for Canada, then the United States, and very little for France (except for credibility and purchase intention, where U.S. differences are the largest).

13.49

1. There is little evidence of an interaction.

2. sp=0.1278.

3. ψ1=1/2(μnew,city+μnew,highway)−1/2(μold,city+μold,highway), ψ2=μnew,city−μnew,highway,ψ3=μold,highway−μold,city.

4. By subjecting the same individual to all four treatments, rather than subjecting four individuals to one treatment each, we reduce the within-groups variability.

13.51

1. sp=38.14.

2. Yes, the largest s is less than twice the smallest s; 42.4<2(31.2)=62.4.

3. Sender Individual: 70.9; Sender Group: 48.85; Responder Individual: 59.75; Responder Group: 60.

4. There appears to be an interaction effect. Individuals send more to groups; groups send more to individuals.

5. Sender: P-value=0.0033; Responder: P-value=0.9748; Interaction: P-value=0.1522. Only the sender effect is significant.

13.53 Yes, the iron pot means are the highest, and the F statistic for testing the effect of the pot type is very large.

13.55

1. Because the readings vary at the hundreth of a mm, the following table is for 100×(x−25).

   Tool	Time	x¯	s
   1	1	3.067	0.115
   1	2	2.800	0.000
   1	3	2.600	0.000
   2	1	1.667	0.115
   2	2	2.000	0.200
   2	3	1.600	0.000
   3	1	0.633	0.153
   3	2	1.267	0.115
   3	3	0.933	0.115
   4	1	1.200	0.000
   4	2	1.933	0.115
   4	3	1.400	0.400
   5	1	−0.267	0.115
   5	2	0.600	0.000
   5	3	0.033	0.153

2. Tools 3, 4, and 5 are fairly consistent across time differences, with time 2 having the largest diameters and time 1 the smallest diameters. For tool 2, however, time 1 diameters get larger than time 3, and for tool 1, time 1 diameters get the largest, bigger than both time 2 and time 3 diameters.

3. FA=412.94, df=4, 30, P-value<0.0001; FB=43.60, df=2, 30, P-value<0.0001; FAB=7.65, df=8, 30, P-value<0.0001.

4. There are differences in diameter among the five different tools; there are also differences in diameter in the different shifts, although not as large as the tool differences. There also appears to be a small interaction effect.

13.57

1. All three F values have df 1 and 945, and the P-values are P-value<0.0001, <0.001, and 0.1463. Sex and handedness both have significant effects on mean lifetime, but there is no significant interaction.

2. Women live about six years longer than men (on the average), while right-handed people average nine more years of life than left-handed people. “There is no interaction” means that handedness affects both sexes in the same way and vice versa.

13.59

• (a–b) The first three means and standard deviations are x¯11=3.2543, s11=0.2287; x¯12=2.7636, s12=0.0666; x¯13=2.8429, s13=0.2333. Standard deviations range from 0.0666 to 0.3437 for a ratio of 5.16—larger than we like.

3. FS=1301.32, df=3,224, P-value=0.000, FW=9.76, df=6,224, P-value=0.000; FSW=5.97, df=18,224, P-value=0.000.

13.61 The seven F statistics are 184.05, 115.93, 208.87, 218.37, 220.01, 174.14, and 230.17, all with df=3 and 32 and P-value<0.0005. The only nonsignificant differences are between species 1 and 3 for water levels 1, 4, and 7. Therefore, for every water level, species 4 has the lowest nitrogen level, and species 2 is next. For water levels 1, 4, and 7, species 1 and 3 are statistically tied for the highest level; for the other levels, species 3 is the highest, and species 1 comes in second.

13.63

• (a–b) The first three means and standard deviations for Fresh are x¯11=109.095, s11=20.949; x¯12=165.138, s12=29.084; x¯13=168.825, s13=18.866. The first three means and standard deviations for Dry are x¯11=40.565, s11=5.581; x¯12=63.863, s12=7.508; x¯13=71.003, s13=6.032. Standard deviation ratios are quite high for both Fresh and Dry biomass: 108.01/6.79=15.9 and 35.76/3.12=11.5.

3. For Fresh: FS=81.45, df=3, 84, P-value=0.000;, FW=43.71, df=6,84, P-value=0.000; FSW=1.79, df=18,84, P-value=0.0404. For Dry: FS=79.93, df=3,84, P-value=0.000; FW=44.79, df=6,84, P-value=0.000; FSW=2.22, df=18,84, P-value=0.008.

13.65 Fresh: the F statistics are 15.88, 11.81, 62.08, 10.83, 22.62, 8.20, and 10.81; Dry: the F statistics are 8.14, 26.26, 22.58, 11.86, 21.38, 14.77, and 8.66; all with df=3 and 12 and P-value<0.003.

13.67

1. Sex: 1 and 174; Floral: 2 and 174; Interaction: 2 and 174.

2. Damage to males was higher for all characteristics. For males, damage was highest under characteristic level 3, while for females, the highest damage occurred at level 2. Interaction should be significant because the distance between the means increases from floral type 1 to floral type 3.

3. Two of the standard deviations are at least half as large as the means. Because the response variable (leaf damage) must be non-negative, this suggests that these distributions are right-skewed.

13.69

1. As the saturation changes from low to high, the purchase goal for large carry-on increases; the purchase goal for small carry-on decreases as saturation changes from low to high.

2. The graph indicates there is an interaction effect (the lines are not parallel).

13.71

1. For saturation; the P-value=0.29; saturation is not significant. For purchase goal, the P-value=0.04; purchase goal is significant at the 5% level. For the interaction effect between saturation and purchase goal, the P-value=0.09; the interaction effect is not significant at the 5% level.