COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman

1. COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick Chapter : 10 Independent Samples t Test

2. Key Terms: Don’t Forget Notecards • Hypothesis Test (p. 233) • Null Hypothesis (p. 236) • Alternative Hypothesis (p. 236) • Alpha Level (level of significance) (pp. 238 & 245) • Critical Region (p. 238) • Estimated Standard Error (p. 286) • t statistic (p. 286) • Degrees of Freedom (p. 287) • t distribution (p. 287) • Confidence Interval (p. 300) • Directional (one-tailed) Hypothesis Test (p. 304) • Independent-measures Research Design (p.318)

3. Formulas • Estimated Standard Error: • Estimated Standard Error: • Pooled Variance: • t-Score Formula: • Degrees of Freedom: n1 = n2 n1 ≠ n2

4. More Formulas • Cohen’s d: • Confidence Interval: • Hartley’s F-max Test:

5. Identifying the Independent-Measures Design • Question 1: What is the defining characteristic of an independent-measures research study?

6. Identifying the Independent-Measures Design • Question 1 Answer: • An independent-measures study uses a separate group of participants to represent each of the populations or treatment conditions being compared.

7. Pooled Variance and Estimated Standard Error • Question 2: One sample from an independent-measures study has n = 4 with SS = 100. The other sample has n = 8 and SS = 140. • Compute the pooled variance for the sample. • Compute the estimated standard error for the mean difference.

8. Pooled Variance and Estimated Standard Error • Question 2 Answer:

9. t Test for Two Independent Samples -- Two-tailed Example • Question 3: A researcher would like to determine whether access to computers has an effect on grades for high school students. One group of n = 16 students has home room each day in a computer classroom in which each student has a computer. A comparison group of n = 16 students has home room in a traditional classroom. At the end of the school year, the average grade is recorded for each student The data are as follows:

10. t Test for Two Independent Samples -- Two-tailed Example • Question 3: • Is there a significant difference between the two groups? Use a two-tailed test with α = 0.05. • Compute Cohen’s d to measure the size of the difference. • Compute the 90% confidence interval for the population mean difference between a computer classroom and a regular classroom.

11. t Test for Two Independent Samples -- Two-tailed Example • Question 3a Answer: • Step 1: State hypotheses • H0: Treatment has no effect. ( = 0) • H1: Treatment has an effect. (≠ 0) • Step 2: Set Criteria for Decision (α = 0.05) df = 16 + 16 – 2 = 30 Critical t = ± 2.042 • If -2.042 ≤ tsample≤ 2.042, fail to reject H0 • If tsample < -2.042 or tsample > 2.042, reject H0

12. t Test for Two Independent Samples -- Two-tailed Example df= 30 t Distribution with α = 0.05 Critical region Critical region t = - 2.042 t = + 2.042

13. t Test for Two Independent Samples -- Two-tailed Example • Question 3a Answer: • Step 3: Compute sample statistic

14. Two-Tailed Hypothesis Test Using the t Statistic df= 30 t Distribution with α = 0.05 Critical region Critical region t = - 2.042 t = 1.17 t = + 2.042

15. t Test for Two Independent Samples -- Two-tailed Example • Question 3a Answer: • Step 4: Make a decision • For a Two-tailed Test: • tsample (1.17) <tcritical (2.042) • Thus, we fail to reject the null and cannot conclude that access to computers has an effect on grades. • If -2.042 ≤ tsample≤ 2.042, fail to reject H0 • If tsample < -2.042 or tsample > 2.042, reject H0

16. t Test for Two Independent Samples -- Two-tailed Example • Question 3b Answer: • Cohen’s d: • This is a small, or small-to-medium, effect.

17. t Test for Two Independent Samples -- Two-tailed Example • Question 3c Answer: • df= 30 • Critical t = ± 1.697 • Thus, the population mean difference is estimated to be between – 1.591 and 8.591. The fact that zero is an acceptable value (inside the interval) is consistent with the decision that there is no significant difference between the two population means. (Fail to reject the null) Our α = 0.10 because our confidence interval leaves 10% split between the 2-tails.

18. t Test for Two Independent Samples -- One-tailed Example • A researcher is using an independent-measures design to evaluate the difference between two treatment conditions with n = 8 in each treatment. The first treatment produces M = 63 with a variance of s2 = 18, and the second treatment has M = 58 with s2= 14. • Use a one-tailed test with α = 0.05 to determine whether the scores in the first treatment are significantly greater than scores in the second. • Measure the effect size with r2.

19. t Test for Two Independent Samples -- One-tailed Example • Question 4 Answer: • Step 1: State hypotheses • H0: No difference between treatments. () • H1: Treatment 1 has a greater effect. (> 0) • Step 2: Set Criteria for Decision (α = 0.05) df = 8 + 8 – 2 = 14 Critical t = 1.761 • If tsample≤ 1.761, fail to reject H0 • If tsample > 1.761, reject H0

20. t Test for Two Independent Samples -- One-tailed Example df= 14 t Distribution with α = 0.05 Critical region Because this is a one-tailed test‚ there is only one critical region. t = + 1.761

21. t Test for Two Independent Samples -- One-tailed Example • Question 4 Answer: • Step 3: Compute sample statistic

22. t Test for Two Independent Samples -- One-tailed Example df= 14 t Distribution with α = 0.05 Critical region Because this is a one-tailed test‚ there is only one critical region. t = + 1.761 t = 2.50

23. t Test for Two Independent Samples -- One-tailed Example • Question 4 Answer: • Step 4: Make a decision • For a One-tailed Test: • tsample(2.50) > tcritical(1.761) • Thus, we reject the null and conclude that the treatment 1 has a significantly greater effect. • If tsample≤ 1.761, fail to reject H0 • If tsample > 1.761, reject H0

24. t Test for Two Independent Samples -- One-tailed Example • Question 4 Answer: • This is a large effect.

25. Assumptions Underlying the Independent-Measures t Test • Question 5: What three assumptions must be satisfied before you use the independent-measures t formula for hypothesis testing?

26. Assumptions Underlying the Independent-Measures t Test • Question 5 Answer: • The observations within each sample must be independent. • The two populations from which the samples are selected must be normal. • The two populations from which the samples are selected must have equal variances (homogeneity of variance).

27. Testing for Homogeneity of Variance • Question 6:Suppose that two independent samples each have n = 10 with sample variances of 12.34 and 9.15. Do these samples violate the homogeneity of variance assumption? (Use Hartley’s F-Max Test with α = 0.05)

28. Testing for Homogeneity of Variance • Question 6 Answer: • F-max = • df = 10 – 1 = 9 • k = 2 Critical levels for α = 0.05 are in regular type; critical levels for α = 0.01 are in bold type. Critical F-max = 4.03

29. Testing for Homogeneity of Variance • Question 6 Answer: • Because the obtained F-max value (1.35) is smaller than the critical value (4.03), we can conclude that the data do not provide evidence that the homogeneity of variance assumption has been violated.

30. Frequently Asked Questions FAQs • What’s the difference between and ? • There seems to be some confusion as to how to work the formula for pooled variance. In the first example (the correct one), the numerator is the sum of SS1 and SS2. The denominator is the sum of df1 and df2. • In the second example (the incorrect method), we’re dividing, then adding. This yields a completely different, and ultimately wrong, result. SS = 10; df= 15 SS = 5; df = 20