Mini 10_ Effect Size, Power, Signa Detection Theory

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San Jose State University *

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May 21, 2024

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Mini #10 Effect Size & Power & Signal Detection Theory 1. The Secretary of Labor wants to know if productivity increases when you give employees 2 vacations a year instead of one. Employees who receive 1 and 2 vacations a year are selected at random from nearby companies to fill out a survey which rates their productivity as an index on a scale 1-10. The table below provides the productivity Mean and Standard Deviation, and Sample Size for both groups of employees with 1 Vacation per year and those with 2 Vacations per year. Mean1Vac 6 Mean2Vac 8.2 S 1Vac 1.3 S 2Vac 1.2 N1Vac 16 N2Vac 9 IV: Type of Employee DV: Productivity rating Ho: Mean Productivity rating 2 Vacation = Mean Productivity 1 Vacation H1: Mean Productivity rating 2 Vacation ≠ Mean Productivity 1 Vacation pooled var: 1.603043 sdiff var: 0.278306 sdiff: 0.527547 t-test: -4.17024 Since the difference in productivity index means between employees who take one annual vacation and those who take two exceeds what we would expect from pure chance, we reject the null hypothesis. Our sample size is 25, yielding a t-value of -4.17 with 23 degrees of freedom, indicating significance at p<.05. a. Using the information provided conduct all steps for the appropriate hypothesis test. b. If appropriate, calculate and interpret (small medium, large) the effect size using Cohen’s d , and calculate and interpret the explained variance, r 2 . d = 1.7376 and r 2 = .430565 c. Without repeating all the steps for hypothesis testing, change S 1Vac = 1.6 and S 2Vac =1.5. What is the new t-statistic? -3.3718 d. Would it be appropriate to calculate the effect size? If so, please calculate and interpret the new Cohen’s d and the explained variance, r 2 . yes the results are statistically significant e. What happened to the power as a result of increasing the variability (you do not need to calculate power to answer this)? Alpha did not change while the power did. In other words, while the likelihood of rejecting the null hypothesis when
it was false has reduced, the likelihood of rejecting the null hypothesis when it was true has remained unchanged. f. In terms of Signal Detection Theory, what does the change in Power mean in terms of Hits and in terms of False Alarms? While the likelihood of committing a False Alarm has not altered, the likelihood of committing a Hit has. 2. A professor wants to know if students perform better solo or in groups. He compares grades on the final project of the class where there was an option to do the project solo or with a group. The table below provides the Mean, Standard Deviation of final project grades and Sample Size for the number of recorded grades of the randomly selected solo projects and group projects (only one grade was picked from each randomly selected group projects). IV: Solo and Group (Project Type) DV: Grades Ho: MeanGradeSolo = MeanGradeGroup H1: MeanGradeSolo ≠ MeanGradeGroup pooled var: 58.5 sdiffvar: 7.8 sdiff: 2.792848 t-test: -429669 We reject the Null Hypothesis, indicating that there is a significant difference between the average performance of students working individually and those working in groups. With a sample size of 30, the calculated t-value of -4.29669 demonstrates this distinction, with a p-value of less than .05 a. Using the information provided conduct all steps for the appropriate hypothesis test. b. If appropriate, calculate and interpret (small medium, large) the effect size using Cohen’s d , and calculate and interpret the explained variance, r 2 . 1.623996 c. Without repeating all the steps for hypothesis testing, change Nsolo = 10 and NGroup = 10. What is the new t-statistic? -3.5082 d. Would it be appropriate to calculate the effect size? If so, please calculate and interpret the new Cohen’s d and the explained variance, r 2 . Yes, the results are statistically significant, so it is appropriate to report the effect size. MeanSolo 70 MeanGroup 82 S Solo 9 S Group 6 N Solo 15 N Group 15
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