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

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

Formulas n 1 = n 2 n 1 ≠ n 2

More Formulas

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

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

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. Question 2: One sample from an independent-measures study has n = 4 with SS = 100. The other sample has n = 8 and SS = 140. a) Compute the pooled variance for the sample. b) Compute the estimated standard error for the mean difference.

Pooled Variance and Estimated Standard Error

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: 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: ComputerTraditional M = 86M = 82.5 SS = 1005SS = 1155

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

t Test for Two Independent Samples -- Two-tailed Example Critical t = ± df = – 2 = 30 If ≤ t sample ≤ 2.042, fail to reject H 0 If t sample 2.042, reject H 0

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

t Test for Two Independent Samples -- Two-tailed Example

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

t Test for Two Independent Samples -- Two-tailed Example Question 3a Answer: Question 3a Answer: Step 4: Make a decision Step 4: Make a decision For a Two-tailed Test: For a Two-tailed Test: t sample (1.17) < t critical (2.042) t sample (1.17) < t critical (2.042) Thus, we fail to reject the null and cannot conclude that access to computers has an effect on grades. Thus, we fail to reject the null and cannot conclude that access to computers has an effect on grades. If ≤ t sample ≤ 2.042, fail to reject H 0 If t sample 2.042, reject H 0

t Test for Two Independent Samples -- Two-tailed Example Magnitude of dEvaluation of Effect Size d = 0.2Small effect (mean difference around 0.2 standard deviations) d = 0.5Medium effect (mean difference around 0.5 standard deviations) d = 0.8Large effect (mean difference around 0.8 standard deviations)

t Test for Two Independent Samples -- Two-tailed Example Our α = 0.10 because our confidence interval leaves 10% split between the 2-tails.

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 s 2 = 18, and the second treatment has M = 58 with s 2 = 14. 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 s 2 = 18, and the second treatment has M = 58 with s 2 = 14. a) 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. b) Measure the effect size with r 2.

t Test for Two Independent Samples -- One-tailed Example Critical t = df = – 2 = 14 If t sample ≤ 1.761, fail to reject H 0 If t sample > 1.761, reject H 0

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

t Test for Two Independent Samples -- One-tailed Example

t Distribution with α = 0.05 t = Critical region df = 14 t = 2.50 Because this is a one-tailed test‚ there is only one critical region.

t Test for Two Independent Samples -- One-tailed Example Question 4 Answer: Question 4 Answer: Step 4: Make a decision Step 4: Make a decision For a One-tailed Test: For a One-tailed Test: t sample (2.50) > t critical (1.761) t sample (2.50) > t critical (1.761) Thus, we reject the null and conclude that the treatment 1 has a significantly greater effect. Thus, we reject the null and conclude that the treatment 1 has a significantly greater effect. If t sample ≤ 1.761, fail to reject H 0 If t sample > 1.761, reject H 0

t Test for Two Independent Samples -- One-tailed Example Percent of Variance Explained as Measured by r 2 Evaluation of Effect Size r 2 = 0.01 (0.01*100 = 1%)Small effect r 2 = 0.09 (0.09*100 = 9%)Medium effect r 2 = 0.25 (0.25*100 = 25%)Large effect

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? Question 5: What three assumptions must be satisfied before you use the independent-measures t formula for hypothesis testing?

Assumptions Underlying the Independent-Measures t Test Question 5 Answer: Question 5 Answer: 1) The observations within each sample must be independent. 2) The two populations from which the samples are selected must be normal. 3) The two populations from which the samples are selected must have equal variances (homogeneity of variance).

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

Testing for Homogeneity of Variance 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

Testing for Homogeneity of Variance Question 6 Answer: 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. 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.

Frequently Asked Questions FAQs SS = 10; df = 15 SS = 5; df = 20