The t Test for Two Related Samples


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The t Test for Two Related Samples

  1. 1. Mary Anne Portuguez MP-IP-1 The t Test for Two Related Samples
  2. 2. PRESENTATION OF PROBLEM: THE WHY OF USING T TEST FOR RELATED SAMPLES Independent-measures design is prone to difficulty in handling individual differences (e.g. IQ, age, gender, etc.) which decreases its efficiency (Kantowitz et. al., 1988). The mean difference between groups may be elucidated by person characteristics rather than the effectiveness of a particular treatment (Gravetter & Wallnau, 2012). Through repeated-measures design, the problem is addressed because the two sets of scores come from same group of individuals. Therefore, the individual in one treatment is perfectly matched with the individuals in the other treatment (Gravetter & Wallnau, 2012).
  3. 3. THE RELATED-SAMPLES T TEST The related-samples t-test is a parametric procedure used with two related samples. Related samples happen when researcher pair each scores in one sample with a particular score in the other sample. Researchers generate related samples to have more equivalent, hence, more comparable samples. There are two types of research designs that produce related samples namely: matched- samples designs and repeated-measures designs (Heiman, 2011).
  4. 4. THE TWO RESEARCH DESIGNS THAT PRODUCE RELATED SAMPLES  Matched-samples design refers to assigning subjects to groups in which pairs of subjects are first matched on some characteristic and then individually assigned randomly to groups (Cozby, 2012; McGuigan, 1990; Gravetter & Wallnau, 2012).  Repeated-measures design pertains to one in which a single sample of individuals is measured or tested under all conditions of independent variable (Heiman, 2011; Gravetter & Wallnau, 2012; McGuigan, 1990; Kantowitz et. al., 1988; Cozby, 2012).
  5. 5. THE T STATISTIC FOR REPEATED-MEASURES DESIGN The main distinction of related-samples t is that it is based on difference scores rather than raw scores obtained from a single group of participants. The test involves computing t statistic and then consulting a statistical table to confirm whether the t value obtained is adequate to indicate a significant mean difference (Gravetter & Wallnau, 2012).
  6. 6. DIFFERENCE SCORES: THE DATA FOR REPEATED-MEASURES STUDY difference scores= D = The sign of each D scores determines the direction of change and the sample of difference scores (D values) will serve as the sample data for the hypothesis test.
  7. 7. SAMPLE:
  9. 9. HYPOTHESIS TESTS FOR REPEATED-MEASURES DESIGN Step 1 State the hypotheses, and select the alpha level. Step 2 Locate the critical region. Step 3 Calculate the t statistic. Step 4 Make a decision. If the t value falls or exceed on the critical region, rejection of the null hypothesis must be done.
  10. 10. EFFECT SIZE FOR REPEATED-MEASURES DESIGN Cohen’s d formula Evaluation of Effect size with Cohen’s d
  11. 11. PERCENTAGE OF VARIANCE ACCOUNTED FOR Criteria for interpreting as proposed by Cohen
  12. 12. SITUATIONAL EXERCISE 1.0 One technique to help people deal with phobias is to have them counteract the feared objects by using imagination to move themselves to a place of safety. IN experimental test of this technique, patients sit in front of a screen and are instructed to relax. Then, they are shown an image of the feared object. The patient signals the researcher as soon as feelings of anxiety reach a point at which viewing the image can no longer be tolerated. The researcher records the amount of time that the patient was able to endure looking at the image. The patient then spends 2 minutes imagining a “safe scene” such as a tropical beach or a familiar room before the image is presented again. As before, the patient signals when the level of anxiety is intolerable. If patients can tolerate the feared objects longer after the imagination exercise, it is viewed as a reduction in phobia.
  13. 13. A sample of n =7 patients. Question: Do the data (presented in the table) indicate that the imagination technique effectively alters phobia? Step 1 State the hypotheses, and select the alpha level. (There is no change in the phobia.) (There is a change.) For this test, we will use α= .01
  14. 14. EXERCISE 1.0 PATIENTS TIME BEFORE TIME AFTER D A 15 24 9 81 B 10 23 13 169 C 7 11 4 16 D 18 25 7 49 E 5 14 9 81 F 9 14 5 25 G 12 21 9 81 Total = 56 = 502
  15. 15. Step 2 Locate the critical region. For this sample, n= 7 has a df = n – 1. 7-1 = 6. Thus, df= 6. From the t distribution, you should find the critical value for α= .01 is ± 3.707.
  16. 16. Step 3 Calculate the t statistic. In the sample data, and
  17. 17.  Step 4 Make a decision. If the t value falls or exceed on the critical region, rejection of the null hypothesis must be done. In our example, the researcher rejects the null hypothesis. The t value falls or exceed the critical region, thus, concludes that the imagination technique does affect the onset of anxiety when patients are exposed to feared object.
  18. 18. MEASURING EFFECT SIZE Cohen’s d The treatment effect is statistically significant. Percentage of Variance
  19. 19. REPORTING THE RESULTS Imagining a safe scene increased the amount of time that the patients could tolerate a feared stimulus before feeling anxiety by M= 8. 00 with SD= 3. The treatment effect is statistically significant, t (6) = 7.05, p < .01, = 0.892.
  21. 21. Repeated-measures design Advantages:  Requires only few participants in a study or experiment (Goodwin, 2010; Gravetter & Wallnau, 2012; Cozby, 2012).  It uses the subjects more efficiently for each individual is measured in both treatment conditions (Gravetter & Wallnau, 2012; McGuigan, 1990).  It saves laboratory time or energy (McGuigan, 1990).  It reduces problems caused by individual differences (e.g. IQ, gender, age, personality, etc.) that differ from one person to another (Gravetter & Wallnau, 2012). Therefore, greater ability to detect an effect to independent variable (Cozby, 2012).  It is a reasonable choice when it comes to study concerning learning, development, physiological psychology, sensation and perception, or other changes that happen over time. A researcher can see difference in behaviors that develop or change over time (McGuigan, 1990; Gravetter & Wallnau, 2012).  It reduces error variance since in within-groups design, the experimenter or researcher repeats measure on the same participants. It removes individual differences from error variance (McGuigan, 1990).
  22. 22. Disadvantages:  There may be order effects in which changes in scores of participants are caused by earlier treatments (McGuigan, 1990; Gravetter & Wallnau, 2012).  Since within-subject design involves measuring same individuals at one time and then measuring them again at a later time, outside factors that change over time may be responsible for changes in the participants’ scores (Gravetter & Wallnau, 2012).  The researcher must be on guard on factors related to time such as practice effects and fatigue effects. The former refers to participants performing better in the experimental task simply because of practice while the latter refers to decrease in performance due to tasks that are long, difficult, and boring (Kantowitz et. al., 1988).
  23. 23. Independent-measures design: Advantages:  Each subject enters the study fresh and naive with respect to the procedures to be tested (Goodwin, 2010).  It is conservative. There is no chance that one treatment will contaminate the other, since the same person never receives both treatments (Kantowitz et. al., 1988).  The participants are randomly assigned in the different groups differed in ability, thus, prevent systematic biases, and the groups can be considered equivalent in terms of their characteristics (Kantowitz et. al., 1988; Cozby, 2012).  Researcher or experimenter can minimize the possibility of confounding (Kantowitz et. al., 1988).
  24. 24. Disadvantages:  Requires more participants that need to be recruited, tested, and debriefed (Goodwin, 2010).  It must deal with differences among people and this decreases its efficiency (Kantowitz et. al., 1988).
  25. 25. ASSUMPTIONS OF THE RELATED-SAMPLES T TEST The related-samples t statistics requires two basic assumptions (Gravetter & Wallnau, 2012):  The observations within each treatment condition must be independent. The assumption of independence here pertains to scores within each treatment. Within each treatment, the scores are gathered from different persons and must be independent of one another.  The population distribution of difference scores (D values) must be normal.
  26. 26. EXERCISES 2.0 a. A repeated-measures study with a sample of n= 9 participants produces a mean difference of = 3 with a standard deviation of s= 6. Based on the mean and standard deviation you should be able to visualize (or sketch) the sample distribution. Use a two-tailed hypothesis test with = .05 to determine whether it is likely that this sample came from a population with = 0. b. Now assume that the sample mean difference is = 12, and once again visualize the sample distribution. Use a two- tailed hypothesis test with = .05 to determine whether it is likely that this sample came from a population with = 0. c. Explain how the size of the sample mean difference influences the likelihood of finding a significant mean difference.
  27. 27. SOLUTION:  The estimated standard error is 2 points and t(8) = 1.50. With a critical boundary of 2.306, fail to reject the null hypothesis.  With = 12, t (8) = 6. With a critical boundary of 2.306, reject the null hypothesis.  The larger the mean difference, the greater the likelihood of finding a significant difference.
  28. 28. REFERENCES  McGuigan, F.J. (1990). Experimental psychology (5th ed.). Englewood Cliffs, New Jersey: Prentice Hall.  Kantowitz, B.H. et. al. (1988). Experimental psychology (3rd ed.). New York: West Publishing Company.  Gravetter, F.J. & Wallnau, L.B. (2012). Statistics for behavioral sciences. Philippines: Cengage Learning Asia Pte. Ltd.  Cozby, P.C. & Bates, S.C. (2012). Methods in behavioral research (11th ed.). New York: McGraw- Hill Companies, Inc.  Heiman, G.W. (2011). Basic statistics for the behavioral sciences (6th ed.). Belmont, CA: Wadsworth, Cengage Learning.  Goodwin, C.J. (2010). Research in psychology methods and design (6th ed.). USA: John Wiley & Sons, Inc.