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Two-Sample t -Tests

Two-Sample t -Tests. Two-Sample t -Tests. Independent versus Related Samples Your two samples are independent if you randomly assign individuals into the two treatment groups. Your samples are related if either

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Two-Sample t -Tests

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  1. Two-Sample t-Tests

  2. Two-Sample t-Tests • Independent versus Related Samples • Your two samples are independent if you randomly assign individuals into the two treatment groups. • Your samples are related if either • Each person in sample A is matched to a partner in sample B (matched samples) OR • Each person in the study is measured under both conditions (repeated measures)

  3. Independent Samples t-Test • The steps to conducting an independent samples t-test are: • State your research question hypotheses • Determine your rejection rule • Calculate the t-statistic • Use your rejection rule to decide whether you • Reject the null hypothesis • Fail to reject the null hypothesis

  4. Independent Samples t-Test Two-Tailed Test Hypotheses • H0: m1 - m2 = 0 • H1: m1 - m2 ≠ 0

  5. Independent Samples t-Test • Two-tailed Rejection Rule • If you are using the t-table, reject H0 • if t(obt) > t(crit, a, n1 + n2 - 2) OR • if t(obt) < -t(crit, a, n1 + n2 - 2) • If you are using the SPSS printout, reject H0 if p < .05

  6. Independent Samples t-Test One-tailed test where you believe the scores in sample 1 will be greater than the scores in sample 2. The hypotheses are • H0: m1 - m2 ≤ 0 • H1: m1 - m2 > 0

  7. Independent Samples t-Test • One-Tailed Rejection Rule where you believe the scores in sample 1 will be greater than the scores in sample 2. • If using the t-table, reject H0 • if t(obt) > t(crit, a, n1 + n2 - 2) • If using the SPSS printout, reject Ho if p< .05.

  8. Independent Samples t-Test One-tailed test where you believe the scores in sample 1 will be less than the scores in sample 2. The hypotheses are • H0: m1 - m2 ≥ 0 • H1: m1 - m2 < 0

  9. Independent Samples t-Test • One-tailed Rejection Rule where you believe the scores in sample 1 will be less than the scores in sample 2. • If using the t-table, reject H0 • if t(obt) < -t(crit, a, n1 + n2 - 2) • If using the SPSS printout, reject Ho if p < .05.

  10. Independent Samples t-Test • Calculating the independent samples t-statistic • Calculate the mean for each of the two samples • Calculate the sum of squares for each of the two samples. What’s a sum of squares? It’s the same as the formula for the variance, except don’t do the final step of dividing by n – 1. SS1 for the first sample and SS2 for the second

  11. Independent Samples t-Test • Calculating the independent samples t-statistic: • Step 1. NOBODY PANIC!

  12. Independent Samples t-Test • Step 2. Find the difference between the group means. Note: It doesn’t matter which group you designate as sample 1 and which as sample 2, AS LONG AS you take into consideration which group you mean when you set up your hypotheses and rejection rules. [Continued on next slide.]

  13. Independent Samples t-Test • Step 3a. Divide the number 1 by the number of observations in the first sample (n1). • Step 3b. Divide the number 1 by the number of observations in the second sample (n2). • Step 3c. Add the answers to Step 3a and Step 3b. • Refer to Step 1!

  14. Independent Samples t-Test • Step 4. Add the sum of squares for the first sample (SS1) to the sum of squares for the second sample (SS2). See slide #8 for information about calculating the sum of squares. • Step 5. Find the degrees of freedom by adding the number of observations in the first sample (n1) to the number of observations in the second sample (n2) and then subtracting the number 2. [Continued on next slide.]

  15. Independent Samples t-Test • Step 6. Divide your answer from Step 4 by the answer from Step 5. • Step 7. Multiply your answer from Step 6 to the answer from Step 3c. • Refer to Step 1.

  16. Independent Samples t-Test Step 8. Take the square root of your answer in Step 7. Step 9. Divide your answer from Step 2 by the answer from Step 8. Compare your obtained t-statistic to the critical t-value from your rejection rule and decide the appropriate action.

  17. Independent Samples t-Test • Find the appropriate t (crit) from the t-table in the back of the book, using the correct bar at the top depending on a one-tailed or a two-tailed test, a, and df = n1 + n2 - 2. • OR use the significance level shown on the SPSS printout. If it is less than .05, reject Ho. • Calculate t (obt) using the two independent samples t-test. • Make your decision based on your rejection rule.

  18. Example of Independent Samples t-Test • Independent Variable is brand of oven (two brands) • Dependent Variable is hours it worked until failure.

  19. Example of Independent Samples t-Test • Stuff we’ll need

  20. Example of Independent Samples t-Test • Now, for it!

  21. Related Samples t-Test Two-Tailed Test • Hypotheses • Rejection Rule--Reject H0 • if t(obt) > t(crit, a, N-1) OR if t(obt) < -t(crit, a, N-1) where N is the number of differences • OR if the significance level on the SPSS printout is less than .05

  22. Related Samples t-Test One-tailed test where you believe the scores in sample 1 will be less than the scores in sample 2 (which means that the differences will tend to be less than 0). • Hypotheses • Rejection Rule--Reject H0 • if t(obt) < -t(crit, a, N-1) • OR if the significance level on the SPSS printout is less than .05

  23. Related Samples t-Test One-tailed test where you believe the scores in sample 1 will be greater than the scores in sample 2 (which means that the differences will tend to be greater than 0). • Hypotheses • Rejection Rule--Reject H0 • if t(obt) > t(crit, a, N-1) • OR if the significance level on the SPSS printout is less than .05.

  24. Related Samples t-Test Calculating the related samples t-statistic Step 1. Find the difference between each pair of scores. Again, it doesn’t matter which sample you designate as 1 and which is 2, AS LONG AS you (a) consistently subtract sample 2 from sample 1, and (b) keep the order in mind as you set up your hypotheses and rejection rule. From this point on, ignore the original scores and use only the difference scores (designed with a subscript D). The test is conducted pretty much the same as if it were a one-sample test. [Continued on next slide.]

  25. Related Samples t-Test • Step 4. Divide your answer from Step 3 by N(N – 1). • Step 5. Take the square root of your answer from Step 4. • Step 6. Divide your answer from Step 2 by the answer from Step 5. • Compare this obtained t-value against the critical t-value from the rejection rule and decide the appropriate action. • Step 2. Find the mean of the difference scores. • Step 3. Find the sum of squares of the difference scores. Be sure to use N as the number of pairs of scores (or the number of difference scores) to do the division.

  26. Related Samples t-Test • Find the appropriate t (crit) from the t-table in the back of the book, using the correct bar at the top depending on a one-tailed or a two-tailed test, a, and df = N - 1. • OR use the significance level on the SPSS printout. • Calculate t (obt) using the related samples t-test. • Make your decision based on your rejection rule.

  27. Example of Related-Measures t-Test • Same data as before

  28. Example of Related-Measures t-Test • Stuff we’ll need

  29. Example of Related-Measures t-Test • And now

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