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Independent-Samples t test

Independent-Samples t test. Used to test for a difference between two groups when using a between-subjects design with independent samples. Randomly selected sample DV normally distributed DV measured using ratio or interval scale mean of the population is known

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Independent-Samples t test

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  1. Independent-Samples t test Used to test for a difference between two groups when using a between-subjects design with independent samples

  2. Randomly selected sample DV normally distributed DV measured using ratio or interval scale mean of the population is known SD of the population is known Randomly selected sample DV normally distributed DV measured using ratio or interval scale mean of the population is known SD of the population is not known and must be estimated Single-Sample z-Test t-Test

  3. Randomly selected sample DV normally distributed DV measured using ratio or interval scale mean of the population is known SD of the population is not known and must be estimated Randomly selected sample DV normally distributed DV measured using ratio or interval scale Homogeneity of variance Single-SampleIndependent-Samplest-Testt-Test

  4. General Model for z-Test and Single-Sample t-Test Original Population Sample H0 Treated Population HA

  5. General Model for Independent-Samples t-Test H0: 1 - 2 = 0 Population A Sample A H0 Population B Sample B

  6. General Model for Independent-Samples t-Test HA: 1 - 2 0 Population A Sample A HA Population B Sample B

  7. Sampling Distribution of the Difference Between the Means f μ

  8. Sampling Distribution of Difference Between the Means To create this sampling distribution: • Select 2 random samples from one population • Each sample is the same size as the N of our groups • Compute the sample mean for each sample • Subtract one sample mean from the other and plot the difference • Do this an infinite # of times

  9. Standard Error of the Difference between the Means The averagedistance between the mean of the sampling distribution (of the difference between the means) and all of the differences between the means plotted in the sampling distribution of the differences between the means. • How much difference should you expect between the sample means even if your treatment has no effect?

  10. t Tests Formulas Single-Sample t-Test Independent-Samples t-Test

  11. Formula Definitional Formulas Single-Sample t-Test Independent-Samples t-Test

  12. Single-Sample t-Test Step 1: Step 2: Estimated variance of the population (definitional formula) Step 3: Estimated standard error of the mean

  13. Single-Sample Independent-Samplest-Test t-Test Step 1: • calculate the estimated variance of the population • calculate the estimated variance of the population for each group

  14. Pooled Variance Step 1a: Calculate the pooled variance

  15. Pooled Variance  Equal Sample Sizes SS1 = 50 SS2 = 30 n1 = 6 n2 = 6 Average of s12 and s22

  16. Pooled Variance  Unequal Sample Sizes SS1 = 20 SS2 = 48 n1 = 3 n2 = 9 Average of s12 and s22

  17. Single-Sample Independent-Samplest Test t Test Step 2: • calculate the estimated standard error of the mean • calculate the estimated standard error of the difference between the means (standard error of the difference)

  18. Single-Sample Independent-Samplest-Test t-Test Step 2: • calculate the estimated standard error of the mean • calculate the estimated standard error of the difference between the means (standard error of the difference)

  19. Single-Sample Independent-Samplest-Test t-Test Step 3: • calculate tobt • calculate tobt

  20. Single-Sample Independent-Samplest-Test t-Test Step 3: • calculate tobt • calculate tobt

  21. Hypothesis Testing with Two Independent Samples

  22. Step 1. State the hypotheses (two-tailed) A. Is it a one-tailed or two-tailed test? • Two-tailed B. Research hypotheses • Alternative hypothesis: • There is a difference between the control group and the experimental group. • Null hypothesis: • There is no difference between the control group and the experimental group. C. Statistical hypotheses: • HA: 1 - 2 0 which is equivalent to 12 • H0: 1 - 2 = 0 which is equivalent to 1 = 2

  23. The HA and H0 Hypotheses • The HA says that there is a difference between the groups, so your difference is NOT zero • The H0 says that there is NOT a difference, so your difference equals zero • You can put the control group or the experimental group as group 1 in your equations, but you HAVE TO BE CONSISTENT • You should substitute abbreviated names based on the conditions instead of 1 and 2 as subscripts

  24. Step 1. State the hypotheses. A. Is it a one-tailed or two-tailed test? • One-tailed

  25. Step 1. State the hypotheses (one-tailed) B. Research hypotheses • Alternative hypothesis: • The experimental group will perform better than the control group. • The experimental group’s scores will be lower than the control group’s score. • Null hypothesis: • The experimental group will perform the same as or worse than the control group. • The experimental group’s scores will be the same as or higher than the control group’s scores. C. Statistical hypotheses: • HA: experimental - control > 0 experimental - control < 0 • H0: experimental - control< 0 experimental - control> 0

  26. Step 1. State the hypotheses. A. Is it a one-tailed or two-tailed test? • One-tailed B. Research hypotheses • Alternative hypothesis: • Participants who eat peppermint will score higher than those who don’t eat peppermint on the digit recall test. • Null hypothesis: • Participants who eat peppermint will score the same as or lower than those who don’t eat peppermint on the digit recall test. C. Statistical hypotheses: • HA: peppermint - no peppermint > 0 • H0: peppermint - no peppermint< 0

  27. Step 2. Set the significance level  = .05. Determine tcrit. Factors that must be known to find tcrit 1. Is it a one-tailed or a two-tailed test? • one-tailed 2. What is the alpha level? • .05 3. What are the degrees of freedom? • df = ?

  28. Degrees of Freedom Independent-Samples t-Test Single-Sample t-Test df = (n1 – 1) + (n2 – 1) = n1 + n2 – 2 df = (n– 1)

  29. Step 3. Select and compute the appropriate statistical test. Step 1: Step 1a: Step 3: Step 2:

  30. Step 4. Make a decision. • Determine whether the value of the test statistic is in the critical region. Draw a picture. tcrit = ???

  31. Step 4. Make a decision. • If +tobt > +tcrit OR if -tobt < -tcrit  Reject Ho • If -tcrit < tobt< +tcrit Retain Ho +tcrit -tcrit REJECT H0 REJECT H0 RETAIN H0

  32. Step 5. Report the statistical results. • Reject H0: t(df) = tobt, p < .05 • Retain H0: t(df) = tobt, p > .05

  33. Step 6: Write a conclusion. • State the relationship between the IV and the DV in words, ending with the statistical results. • General format: Members of the first group (M = xx.xx) did/did not score lower/higher/differently than members of the second group (M = xx.xx), t(df) = tobt, p < > .05.

  34. Hypothesis Testing with Two Independent Samples An Example

  35. An Example • Research Question: Are students who calculate statistics by hand better able to select the appropriate statistical test to use than students who do not calculate statistics by hand (who use SPSS)? • Assume that past research has consistently shown that students who calculate statistics by hand are better, so we decide to generate a directional (one-tailed) hypothesis.

  36. Step 1. State the hypotheses. A. Is it a one-tailed or two-tailed test? • One-tailed B. Research hypotheses • Alternative hypothesis: • Students who calculate statistics by hand are better able to select the appropriate statistical test to use than students who do not calculate statistics by hand. • Null hypothesis: • Students who calculate statistics by hand are not better (i.e., are no different from or are less able) to select the appropriate statistical test than students who do not calculate stats by hand. C. Statistical hypotheses: • HA: hand - SPSS > 0 • H0: hand - SPSS< 0

  37. Step 2. Set the significance level  = .05. Determine tcrit. 1. Is it a one-tailed or a two-tailed test? • one-tailed 2. What is the alpha level? • .05 3. What are the degrees of freedom? n1 = 5; n2 = 5 • df = (n1 – 1) + (n2 – 1) = (5 -1) + (5-1) = 4 + 4 = 8 tcrit = 1.860

  38. Step 3. Select and compute the appropriate statistic. Independent-Samples t-Test

  39. 1) calculate the estimated variance of each population

  40. 1a) calculate the pooled variance

  41. 2) calculate the estimated standard error of the difference between the means

  42. 3) calculate tobt

  43. Step 4. Make a decision. • If +tobt > +tcrit  Reject Ho • If tobt < +tcrit Retain Ho tcrit = + 1.860 tobt = 2.77

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