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Ch. 10: The Independent Measures t - test

Ch. 10: The Independent Measures t - test. Chapter Overview. Overview: The independent measures vs. repeated measures designs The t- statistic for an independent measures design An example of the two-sample, independent measures t-test The role of sample variability

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Ch. 10: The Independent Measures t - test

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  1. Ch. 10: The Independent Measures t - test

  2. Chapter Overview • Overview: The independent measures vs. repeated measures designs • The t- statistic for an independent measures design • An example of the two-sample, independent measures t-test • The role of sample variability • Other factors affecting the value of t • Assumptions • F max test

  3. The independent measures vs. repeated measures design • Independent measures = 2 separate samples (See Figure 10.2, p. 303) • . • Repeated measures = same sample tested twice (Chap. 11) • . • . • Fig. 10.2 presented in class

  4. New notation for the independent measures design • Sample 1 Sample 2 n1 n2 df1 df2 M1 M2 SS1 SS2 • Difference betweens means: M1 – M2 • Standard error of the difference between means: • .

  5. Hypotheses for an independent measures t test • Evaluating the significance of the difference between 2 population means (u1 – u2) • . • . • If there is no treatment effect then: • . • If there is a treatment effect then: • .

  6. The t- statistic for an independent measures design • The overall formula: • t = M1 - M2 - (µ1 - µ2) s M1 – M2 • s M1 – M2 = • Standard error measures ……………..

  7. Formula for independent measures t-test (cont’d) • Final full formula for independent measures t (10.4 on p. 309). t = (M1 - M2) - (:1 - :2) s (M1 - M2) Simplified form: • Comparison with single sample t-test formulas in Table 10.1 (p. 310). • .

  8. Estimated standard error(of the difference between means) • In general, measures how accurately a statistic represents a parameter (e.g. population mean) • E.g. in 1 sample t-test how accurately …. • Or average difference between ……………….. • In 2 sample, independent measures t-test, how accurately the sample mean difference (M1 – M2) represents .. • Or average difference between …. • 2 samples from the same population will have different means

  9. Developing formula for estimated standard error • 2 sources of error • M1 approximates ……………. • Sm1 = √(S12/n1) • M2 approximates …………… • Sm2 = √(S22/n2) • Combine these 2 sources of error to get standard error of the difference between means • formula • Only works when equal n in the 2 samples. Otherwise unbiased • Biased because ……………… • Should give more weight to ………

  10. Pooled variance (s2p) • More general formula for calculating estimated standard error • ……………. • Combine 2 sample variances to get pooled variance but ……… • One sample case formula • Two sample case formula

  11. Estimated standard error: Final formula • Estimated standard error of the difference between means • How accurately the sample mean difference (M1 – M2) represents …. • _________ • Formula for s M1 – M2 = • Note that under Ho, the population mean difference (µ1 - µ2) = 0 • Can redefine standard error of the difference between means as ……….. • Thus, t =

  12. Formula for independent measures t-test (review) • Final full formula for independent measures t t = (M1 - M2) - (:1 - :2) s (M1 - M2) • Under Ho, (:1 - :2) = 0 Simplified form: t = t = ……………………… OR …………………………

  13. A new example: Does Sex Sell? • The effect (if any) of subliminal embeds in advertising. • Ps randomly assigned to receive either: • Subliminal embed ad • No subliminal embed ad

  14. Does Sex Sell (continued) • Ps in each treatment condition (group) rate how likely it is they will buy the product (e.g. coke) on a 5 point scale: • 1 2 3 4 5 • Definitely not Definitely yes

  15. Does sex sell (cont’d) • Results • Embed group No embed group 1 1 4 2 3 1 4 4 • Analysis • Calculate descriptives (and values needed for t) • Calculate t

  16. Reporting the results of an independent measures t-test • Same general form as for one sample t-test. • Descriptive statistics then hypothesis testing statistics are reported. • Example provided in class for subliminal embeds experiment. • Careful consideration of the descriptive statistics (M and SD), can provide a good indication of what the hypothesis test may conclude.

  17. Hypothesis tests with the independent measures t-test • Same general steps in H: testing as before. • Example 10.1 from a memory experiment comparing an experimental group that uses imagery to a control group (no imagery) illustrates the procedure. • See Demo 10.1 on p. 326 also (more complete example). • Descriptive statistics then hypothesis testing statistics are reported in Results section. • Careful consideration of the descriptive statistics (M and SD), can provide a good indication of what the hypothesis test may conclude. • See Fig 10.5a (p. 315) for distribution of scores for the 2 samples.

  18. Measuring Effect Size • Cohen’s d = mean difference standard deviation (of ……..) • r2 = …………….. • Bigger the r2 the ……………..

  19. The one-tailed independent measures t-test • Nothing new. • We use 2 tailed tests all the time.

  20. The role of sample variance • Example 10.4. 2 Figures from text shown for below • Data with low sample variance for each group is shown on next slide …………………….. t (16) = …….., p < .05 is significant. Effect size is ………….. • Data with high sample variance for each group is shown 2 slides down ……………………….. t (16) = ……., p > .05 is not significant. Effect size is …………….

  21. Factors influencing the value of the independent measures t

  22. Assumptions underlying the independent measures t-test • The observations within each sample ……….. • The 2 populations from which the samples are selected must be ……………….. • The 2 populations from which the sample are selected must have equal ……………….(tested by F Max test- next slide). • Interval or ratio level of measurement for dependent variable.

  23. F- Max test • F max = largest sample variance smallest sample variance Compare to required value in Table B.3 Depends on: k = ………. df = …………………….. Alpha level = (usually .05)

  24. Demo 10.1 • Consider SPSS output?

  25. In the literature: Reporting the results of an independent measures t-test • Same general form as for one sample t-test. • Descriptive statistics then hypothesis testing statistics are reported. • Example given for Demo 10.1 • Careful consideration of the descriptive statistics (M and SD), can provide a good indication of what the hypothesis test may conclude.

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