Mini 11 One-Way ANOVA

Mini 11 One-Way ANOVA Does a baseball player's salary depend on the position that he plays? The following data depict the 2005 season salaries (in millions of dollars) for players randomly selected from all players in their respective positions in the National Baseball League. The data is below. The statistical results are provided for you. 1. Complete the other steps of hypothesis testing, including the conclusion given the results below. a. State: i. H0: μ pitcher = μcatcher = μoutfielder = μ shortstop ii. H1: μ pitcher ≠ μcatcher ≠ μoutfielder≠ μshortstop b. Plan i. 1 nominal IV 3 ≤levels and 1 DV scale (ratio or interval) ii. Random selection: All positions and players were selected at random with different skill levels and experience playing baseball. iii. Normal distribution: all positions and players came from a normally distributed population with equal variances iv. Cut off 1. DF between = 4 − 1 = 3 2. DF within =( 20 − 1 ) , ( 19 − 1 ) , ( 18 − 1 ) , ( 17 − 1 )= 16 3. N = 20 ,K = 4 ,Cut off F ( 4,20 )= 3.10 c. Do: Salary Sum of Squares df Mean Square F Sig. Between Groups 5.458 3 1.819 .272 .845 Within Groups 107.063 16 6.691 Total 112.521 19 d. Conclude: i. Fail to reject null hypothesis, not statistically significant enough ii. F = 0.272 < 3.10 , P > 0.05 2. Given the results, is it appropriate to conduct a post-hoc test, and why or why not? What results in particular would you look for in the post-hoc analyses? a. No it wouldn’t because F = 0.272 is not statistically significant enough to conduct a post-hoc test, it wouldn’t be necessary to. Table: Baseball Salaries and Positions

Pitcher Catcher Outfielder Shortstop 0.600 0.650 1.350 0.322 6.050 3.000 7.750 8.250 3.000 0.750 0.575 0.445 0.750 3.133 3.100 3.400 1.600 0.324 2.325 0.318 Stat 95 Assignment: Comparing Three Means using One-Way ANOVA and Post- Hoc Tests Using SPSS Scenario : Dr. Olson is interested in the effects of three different methods for losing weight in long-distance truck-drivers. He tested a sample of 60 truck drivers. Participants were randomly assigned to one of three groups of 20 people each. Participants in the Diet-Only group followed a low-calorie diet but did not exercise. Participants in the Exercise-only group followed an exercise regimen but did not change their diet. Participants in the Diet + Exercise group followed both a low-calorie diet and an exercise regimen. Dr. Olson measured the number of pounds each individual lost after 6 months. Diet-Only Group Exercise-Only Group Diet+Exercise Group 5 9 15 7 10 18 10 8 20 10 11 22

LbsLost Sum of Squares df Mean Square F Sig. Between Groups 1210.033 2 605.017 35.711 .000 Within Groups 965.700 57 16.942 Total 2175.733 59 Dependent Variable: Scheffe (I) 1-Diet,2- Excer,3-Both Mean Difference (I-J) Std. Error Sig. 95% Confidence Interval Lower Bound Upper Bound 1.00 2.00 2.25000 1.30162 0.233 -1.0216 5.5216 3.00 -8.20000 * 1.30162 0.000 -11.4716 -4.9284 2.00 1.00 -2.25000 1.30162 0.233 -5.5216 1.0216 3.00 -10.45000 * 1.30162 0.000 -13.7216 -7.1784 3.00 1.00 8.20000 * 1.30162 0.000 4.9284 11.4716 2.00 10.45000 * 1.30162 0.000 7.1784 13.7216 Questions ( Be sure to use complete sentences) :

1. In this study, what was the independent and dependent variable? a. Independent variable would be Diet, Exercise, or both and the Dependent variable would be Weight 2. Describe the center of each distribution. a. 3. Describe the variability in each distribution: What is the standard deviation for each group? What does this measure tell us about the dispersion of scores in each group? a. Group 1 = 3.94, Group 2 = 3.39, Group 3 = 4.88 b. Variance i. Group 1 = 15.52 ii. Group 2 = 11.49 iii. Group 3 = 23.81 c. Tells that group 3 has a larger dispersion of scores and therefore a significant difference compared to the rest. 4. What would the overall null hypothesis state (in everyday language)? What would the overall alternative hypothesis state (in everyday language)? a. The null hypothesis would be that there wouldn’t be a difference between diet, exercising, or doing both for losing weight. The alternative hypothesis would be that there would be a difference between diet, exercising, or doing both for losing weight. 5. Looking at the source table provided, were the results of the ANOVA statistically significant? Explain how you arrived at this decision. a. Yes the results were statistically significant because it’s seen that group 3/dieting and exercising has a significant change compared to the two other groups with either diet or exercise. 6. Were the differences between each group statistically significant, as revealed by the post-hoc tests? Explain how you arrived at your decisions. Which of the three weight-loss techniques was most effective; which one was least effective? Explain your answers. a. Result between group 3 and the other two were statistically significant because of the large difference mean difference and significance (zero for both). Most effective was group 3 and least effective was both diet and exercise. 7. From all of this information, can you conclude that the different weight-loss techniques caused the truck drivers to lose weight? Explain your answer. a. From this information, I can conclude that different weight loss methods caused truck drivers to lose weight. In particular, choosing both methods of dieting and exercising resulted in much more weight loss compared to doing one or the other. I conclude this information from the ANOVA table showing statistical significance between the three different groups. Based on materials developed by Frank Payne, Ph.D. and Nancy Da Silva, Ph.D.Modified 08/2008 by Sean Laraway, Ph.D. and Mike Abrams, Ph.D.

Within 300.857 18 16.714 Total 1020.950 20 Table: Beer and Attractiveness, Group Means Group M 0 beer 21.5714 1 beer 26.5714 3 beers 35.7143 Is it appropriate to run a post-hoc analysis for this experiment? Why or why not? It is appropriate to run a post-hoc analysis because there’s a significant ANOVA and we have three groups. Answer Each of the Multiple Choice Questions Below. 6. The assumptions of the ANOVA are that samples: A) are selected randomly and that the samples come from populations with unequal variances. B) are selected randomly, the population distribution is normal, and the samples come from populations with equal variances. C) are selected randomly and the population distribution is normal. D) come from populations that are heteroscedastic and they are normally distributed.

7. The grand mean is the: A) mean of all the scores in the study, regardless of the condition. B) average of each of your group means. C) mean of all the scores in the study squared. D) average difference of each of your group means. 8. If you have unequal sample sizes, you can test for homoscedasticity A) by making sure that the within-groups variance is no more than two times the between-groups variance. B) by making sure that your largest sample variance is no more than two times the smallest variance. C) with an ANOVA. D) by making sure that your largest sample variance is no more than five times the smallest sample variance. 9. Running post-hoc tests is different from running multiple t tests because: A) once you have calculated the ANOVA, you already have performed all the calculations necessary for the post-hoc tests. B) post-hoc tests control for the increase in Type I error associated with making multiple-group comparisons. C) post-hoc tests can only be conducted if we have a theory regarding

B) dividing SS Between by SS Within . C) dividing df Between by SS Between . D) dividing SS Within by SS Within .