Chapter 12

• μ1


  Mean of the first sample
• μ2


  Mean of the second sample
• σ1


  Estimated population standard deviation of the first sample
• σ2


  Estimated population standard deviation of the second sample
• n1


  Number of scores in the first sample
• n2


  Number of scores in the second sample
• Two-Sample Experiment


  • Participants' scores are measured under two conditions of the independent variable
  • Condition 1 produces sample mean μ1 that represents μ1
  • Condition 2 produces sample mean μ2 that represents μ2
• Two-Sample t-Test


  Parametric statistical procedure for determining whether the results of a two-sample experiment are significant
• Versions of the two-sample t-test

  • Independent samples t-test
  • Related samples t-test
• Independent Samples t-Test


  • Parametric procedure used for testing two sample means from independent samples
  • Two samples are independent when we randomly select participants for a sample, without regard to who else has been selected for either sample
• Assumptions of the Independent Samples t-Test

  • The dependent scores measure an interval or ratio variable
  • The populations of raw scores form at least roughly normal distributions
  • The populations have homogeneous variance
  • While ns may be different, they should not be massively unequal
• Statistical Hypotheses


  • Two-tailed test
  • One-tailed test
• Sampling Distribution


  The distribution of all possible differences between two means when they are drawn from the raw score population described by H0
• Computing the Independent Samples t-Test

  1. Calculate the estimated population variance for each condition

  2. Compute the pooled variance

  3. Compute the standard error of the difference between means

  4. Compute tobt for two independent samples
• Critical Values


  Determined based on degrees of freedom df = (n1 - 1) + (n2 - 1), the selected a, and whether a one-tailed or two-tailed test is used
• Describing the Relationship


  1. Compute a confidence interval

  2. Compute the effect size

  3. Graph the relationship
• Confidence Interval


  Computed when the t-test for independent samples is significant
• Effect Size


  Indicates the amount of influence changing the conditions of the independent variable has on dependent scores
• Ways to measure effect size

  • Cohen's d
  • Proportion of Variance Accounted For
• Cohen's d


  Measures effect size as the magnitude of the difference between the conditions, relative to the population standard deviation
• Proportion of Variance Accounted For

  The squared point-biserial correlation coefficient indicates the proportion of variance accounted for in a two-sample experiment
• Power


  • Maximize the difference produced by the two conditions
  • Minimize the variability of the raw scores
  • Maximize the sample ns
• Steps for Independent Samples t test

  1. Decide if one- or two tailed

  2. State null hypothesis

  3. State alternative hypothesis

  4. Determine α

  5. Determine degrees of freedom

  6. Find tcrit

  7. State rejection rule

  8. Calculate t obt

  9. Reject or fail to reject the null

  10. Draw conclusion

  11. If you reject, compute confidence interval, compute Cohen's d and rpb, graph means, interpret
• Related Samples


  • The related samples t-test is the parametric inferential procedure used with two related samples
  • Related samples occur when we pair each score in one sample with a particular score in the other sample
  • Two types of research designs that produce related samples are matched samples design and repeated measures design
• Matched Samples Design


  • The researcher matches each participant in one condition with a participant in the other condition
• Repeated-Measures Design


  • Each participant is tested under all conditions of the independent variable
• Assumptions of the Related Samples t-Test

  • The dependent variable involves an interval or ratio scale
  • The raw score populations are at least approximately normally distributed
  • The populations being represented have homogeneous variance
  • Because related samples form pairs of scores, the n in the two samples must be equal
• Difference score (D)


  The difference between the two raw scores in a pair
• Statistical Hypotheses


  • Two-tailed test
  • One-tailed test
• Estimated Population Variance of the Difference Scores

  The formula for the estimated population variance of the difference scores
• Standard Error of the Mean Difference

  The formula for the standard error of the mean difference
• Computing the Related Samples t-Test

  The computational formula for the related samples t-test
• Critical Values


  Determined based on degrees of freedom df = N - 1 where N equals the number of difference scores, the selected a, and whether a one-tailed or two-tailed test is used
• Confidence Interval


  Computed when the t-test for related samples is significant
• Power


  • The related samples t-test is intrinsically more powerful than an independent samples t-test
  • Maximize the differences in scores between the conditions
  • Minimize the variability of the scores within each condition
  • Maximize the size of N