806 APPENDIX D 8. Two-way analysis of variance includes a test for an interaction between gender and age bracket, but the two separate tests fail to include that test for an interaction. 9. a. Because the effect from an interaction has a P-value of 0.3973, conclude that there is not sufficient evidence to warrant rejection of the null hypothesis of no interaction. Pulse rates do not appear to be affected by an interaction between gender and age bracket. b. If there does appear to be an interaction, we should not proceed to test for an effect from the row factor of age bracket and we should not proceed to test for an effect from the column factor of gender. If there is an interaction, we should not consider the effect of either factor without also considering the effect of the other factor. 10. Pulse rates do not appear to be affected by an interaction between gender and age bracket, they do not appear to be affected by age bracket, but they do appear to be affected by gender. Chapter 12: Review Exercises 1. With test statistic F = 2.1436 and P-value 0.096, fail to reject H0: m1 = m2 = m3 = m4. Conclude that there is not sufficient evidence to warrant rejection of the claim that subjects in the four age brackets have the same mean cholesterol level. It appears that for the given age brackets, age does not have a significant effect on cholesterol. 2. Test statistic: F = 2.3831. P-value: 0.092. Fail to reject H0: m1 = m2 = m3 = m4. There is not sufficient evidence to warrant rejection of the claim that subjects in the four different age brackets have the same mean cholesterol level. It appears that for those four age brackets, age does not have an effect on LDL cholesterol. 3. For interaction, the test statistic is F = 0.8476 and the P-value is 0.6413, so there is not sufficient evidence to warrant rejection of the null hypothesis of no interaction effect. For the row variable of hospital, the test statistic is F = 0.3549 and the P-value is 0.7857, so there does not appear to be an effect from hospital. For the column variable of day of the week, the test statistic is F = 0.8386 and the P-value is 0.5426, so there does not appear to be an effect from day of the week. It appears that birth weights are not affected by an interaction between hospital and day of the week, they are not affected by the hospital, and they are not affected by the day of the week. 4. For interaction, the test statistic is F = 1.0605 and the P-value is 0.3795, so there is not sufficient evidence to warrant rejection of the null hypothesis of no interaction effect. For the row variable of hospital, the test statistic is F = 0.6423 and the P-value is 0.5935, so there does not appear to be an effect from hospital. For the column variable of day of weekday>weekend, the test statistic is F = 0.4097 and the P-value is 0.5267, so there does not appear to be an effect from whether the day is a weekday or weekend day. It appears that birth weights are not affected by an interaction between hospital and weekday>weekend, they are not affected by the hospital, and they are not affected by whether the day is a weekday or weekend day. the femur side (left>right) and a test for an effect from vehicle size category. It appears that an interaction between the femur side and vehicle size category has an effect on the force measurements. (Remember, these results are based on fabricated data used in one of the cells, so this conclusion does not necessarily apply to real data.) 7. For interaction, the test statistic is F = 3.6296 and the P-value is 0.0749, so there is not sufficient evidence to warrant rejection of the null hypothesis of no interaction effect. For the row variable of gender, the test statistic is F = 5.3519 and the P-value is 0.0343, so there does appear to be an effect from gender. For the column variable of handedness, the test statistic is F = 2.2407 and the P-value is 0.1539, so there does not appear to be an effect from handedness. It appears that the distance between pupils is affected by gender. For distances between pupils, we expect the following: (1) No interaction between genders; (2) No effect from handedness (right, left); (3) There could be differences based on gender. The results are as expected. 9. For interaction, the test statistic is F = 2.1727 and the P-value is 0.1599, so there is not sufficient evidence to warrant rejection of the null hypothesis of no interaction effect. For the row variable of gender, the test statistic is F = 27.8569 and the P-value is 0.0001, so there does appear to be an effect from gender. For the column variable of handedness, the test statistic is F = 1.4991 and the P-value is 0.2385, so there does not appear to be an effect from handedness. It appears that sitting height is affected by gender. For sitting heights, it is reasonable to not expect an interaction between genders and handedness, it is reasonable to not expect that handedness (right, left) would affect sitting heights, and it is reasonable to expect that there could be differences based on gender, so the results are as expected. 11. a. Test statistics and P-values do not change. b. Test statistics and P-values do not change. c. Test statistics and P-values do not change. d. An outlier can dramatically affect and change test statistics and P-values. Chapter 12: Quick Quiz 1. H0: m1 = m2 = m3 = m4. Because the displayed P-value of 0.000 is small, reject H0. There is sufficient evidence to warrant rejection of the claim that the four samples have the same mean weight. 2. No. It appears that mean weights of Diet Coke and Diet Pepsi are lower than the mean weights of regular Coke and regular Pepsi, but the method of analysis of variance does not justify a conclusion that any particular means are significantly different from the others. 3. Right-tailed. 4. Test statistic: F = 503.06. Larger test statistics result in smaller P-values. 5. The four samples are categorized using only one factor: the type of cola (regular Coke, Diet Coke, regular Pepsi, Diet Pepsi). 6. One-way analysis of variance is used to test a null hypothesis that three or more samples are from populations with equal means. 7. With one-way analysis of variance, data from the different samples are categorized using only one factor, but with two-way analysis of variance, the sample data are categorized into different cells determined by two different factors.
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