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Aron chpt 8 ed

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  • 1. Introduction to the t Test
    • Chapter 8
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 2. t Tests
    • Hypothesis-testing procedure in which the population variance is unknown
      • compares t scores from a sample to a comparison distribution called a t distribution
    • t Test for a single sample
      • hypothesis-testing procedure in which a sample mean is being compared to a known population mean but the population variance is unknown
      • Works basically the same way as a Z test, but:
        • because the population variance is unknown, with a t test you have to estimate the population variance
        • With an estimated variance, the shape of the distribution is not normal, so a special table is used to find cutoff scores.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 3. Basic Principle of the t Test: Estimating the Population Variance from the Sample Scores
    • You can estimate the variance of the population of individuals from the scores of people in your sample.
      • The variance of the scores from your sample will be slightly smaller than the variance of scores from the population.
        • Using the variance of the sample to estimate the variance of the population produces a biased estimate.
    • Unbiased Estimate
      • estimate of the population variance based on sample scores, which has been corrected so that it is equally likely to overestimate or underestimate the true population variance
        • The bias is corrected by dividing the sum of squared deviation by the sample size minus 1
          • S 2 = ∑(X – M) 2
            • N – 1
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 4. The t Distribution
    • When the population variance is estimated, you have less true information and more room for error.
      • The shape of the comparison distribution will not be a normal curve; it will be a t distribution.
        • t distributions look like the normal curve—they are bell shaped, unimodal, and symmetrical—but there are more extreme scores in t distributions.
          • Their tails are higher.
        • There are many t distributions, the shapes of which vary according to the degrees of freedom used to calculate the distribution.
          • There is only one t distribution for any particular degrees of freedom.
  • 5. The t Score
    • The sample’s mean score on the comparison distribution
    • It is calculated in the same way as a Z score, but it is used when the variance of the comparison distribution is estimated.
    • It is the sample’s mean minus the population mean divided by the standard deviation of the distribution of means.
    • If your sample’s mean was 35, the population mean was 46, and the estimated standard deviation was 5, then the t score for this example would be -2.2.
      • This sample’s mean is 2.2 standard deviations below the mean.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 6. Hypothesis Testing When the Population Variance Is Unknown
    • Restate the question about the research hypothesis and a null hypothesis about the populations.
    • Determine the characteristics of the comparison distribution.
    • Determine the significance cutoff.
    • Determine your sample’s score on the comparison distribution.
    • Decide whether to reject the null hypothesis.
      • Compare the t score of your sample and the cutoff score from the t table.
      • Population mean
      • Population variance
      • Standard deviation of the distribution of means
      • Shape of the comparison distribution
  • 7. EXAMPLE– Aron, Coups, Aron (pg 264)
      • Eight participants are tested after being given an experimental procedure. Their scores are 14, 8, 6, 5, 13, 10, 10, 6. The population (of people not given this procedure) is normally distributed with a mean of 6. Using the .05 level, does the experimental procedure make a difference? (a) Use the five steps of hypothesis testing and (b) sketch the distinctions involved.
  • 8. Step 1: Restate the question about the research hypothesis and a null hypothesis about the populations.
      • Research Question – “Does the experimental procedure make a difference for the participants?”
  • 9. Step 1: Restate the question about the research hypothesis and a null hypothesis about the populations.
      • Research Question – “Does the experimental procedure make a difference for the participants?”
      • Population 1: Participants given the experimental procedure.
      • Population 2: The general population
    • H a : “Participants given the experimental procedure will score differently than the general population”
    • H o : “Participants given the experimental procedure will not score differently than the general population”
  • 10. Step 2: Determine the characteristics of the comparison distribution
      • population mean
        • This is the same as the known population mean.
      • population variance
        • Figure the estimated population variance.
            • (df = N-1)
            • (N = # scores in sample)
        • Figure the variance of the distribution of means.
          • (N = # scores in sample)
      • standard deviation of the distribution of means
        • Figure the standard deviation of the distribution of means.
      • shape of the comparison distribution
        • t distribution with N – 1 degrees of freedom
  • 11. Step 2: Determine the characteristics of the comparison distribution (cont.)
      • population mean = 6
      • population variance
    X M   (X-M) (X-M) 2 14 - 9 " = 5 25 8 - 9 " = " -1 1 6 - 9 " = " -3 9 5 - 9 " = " -4 16 13 - 9 " = 4 16 10 - 9 " = 1 1 10 - 9 " = 1 1 6 - 9 " = " -3 9 72 78 8 Sample M = 9   df = 8-1 (7)
  • 12. Step 2: Determine the characteristics of the comparison distribution (cont.)
      • Figure the variance of the distribution of means.
      • standard deviation of the distribution of means
        • Figure the standard deviation of the distribution of means.
    S 2 = 11.14 N = 8 (# scores in sample)
  • 13. Step 2: Determine the characteristics of the comparison distribution (cont.)
      • shape of the comparison distribution
        • t distribution with N – 1 df
        • t distribution with 7 df
  • 14. Step 3: Determine the significance cutoff
      • Decide the significance level and whether to use a one- or a two-tailed test.
      • p< .05 (two-tailed)
      • Look up the appropriate cutoff in a t table.
      • Cutoffs = -2.365 & 2.365
    H a = “Participants given the experimental procedure will score differently than the general population”
  • 15. Step 4: Determine your sample’s score on the comparison distribution. M = 9 Pop. M = 6 S M = 1.18
  • 16. Step 5: Decide whether to reject the null hypothesis.
      • Compare the t score of your sample and the cutoff score from the t table.
      • t of 2.54 is more extreme than the t cutoff of 2.365.
      • Reject H o
      • Found support for H a
      • The experimental procedure makes a difference.
  • 17. The t Test for Dependent Means
    • It is common when conducting research to have two sets of scores and not to know the mean of the population.
    • Repeated Measures Design (Within Subjects Design)
      • research design in which each person is tested more than once
      • For this type of design, a t test for dependent means is used.
        • The means for each group of scores are from the same people and are dependent on each other.
        • A t test for dependent means is calculated the same way as a t test for a single sample; however:
          • Difference scores are used .
          • You assume that the population mean is 0.
  • 18. Difference Scores
    • For each person, you subtract one score from the other.
    • If the difference compares before versus after, difference scores are also called change scores.
    • Once you have the difference score for each person in the study, you do the rest of the hypothesis testing with difference scores.
      • You treat the study as if there were a single sample of scores.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 19. Population of Difference Scores with a Mean of 0
    • Null hypothesis in a repeated measured design
      • On average, there is no difference between the two groups of scores.
        • When working with difference scores, you compare the population of difference scores from which your sample of difference scores comes (Population 1) to a population of difference scores (Population 2) with a mean of 0.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 20. Steps for a t Test for Dependent Means
    • Restate the question as a research hypothesis and a null hypothesis about the populations.
    • Determine the characteristics of the comparison distribution.
    • Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
    • Determine the sample’s score on the comparison distribution.
    • Decide whether to reject the null hypothesis.
  • 21. EXAMPLE– Aron, Coups, Aron (pg 265)
      • A researcher tests 10 individuals before and after an experimental procedure.
      • Test the hypothesis that there is an increase in scores, using the .05 significance level. (a) Use the five steps of hypothesis testing and (b) sketch the distribution involved.
  • 22. Step 1: Restate the question about the research hypothesis and a null hypothesis about the populations.
      • Population 1: Participants given the experimental procedure.
      • Population 2: Those who show no change from before or after the procedure
    • H a : “Participants given the experimental procedure will have an increase in scores following the procedure”
    • H o : “Participants given the experimental procedure will not have an increase in scores following the procedure”
  • 23. Step 2: Determine the characteristics of the comparison distribution
      • Make each person’s two scores into a difference score.
        • Do all of the remaining steps using these difference scores.
      • Figure the mean of the difference scores.
      • Assume the mean of the distribution of means of difference scores = 0.
      • Find the standard deviation of the distribution of means of difference scores.
        • Figure the estimated population variance of difference scores.
          • df = N-1
        • Figure the variance of the distribution of means of difference scores.
        • Figure the standard deviation of the distribution of means of difference scores.
      • The shape is a t distribution with N – 1 degrees of freedom.
  • 24. Step 2: Determine the characteristics of the comparison distribution (cont.)
      • Make each person’s two scores into a difference score.
      • Figure the mean of the difference scores.
    Participant Before After Difference (After-Before) 1 10.4 10.8 0.4 2 12.6 12.1 -0.5 3 11.2 12.1 0.9 4 10.9 11.4 0.5 5 14.3 13.9 -0.4 6 13.2 13.5 0.3 7 9.7 10.9 1.2 8 11.5 11.5 0 9 10.8 10.4 -0.4 10 13.1 12.5 -0.6 117.7 119.1 1.4
  • 25. Step 2: (cont.)
      • Find the standard deviation of the distribution of means of difference scores.
        • Figure the estimated population variance of difference scores.
    (df = N-1) Participant Before After Difference (After-Before) Deviation (Difference - M) Squared Deviation 1 10.4 10.8 0.4 0.26 0.068 2 12.6 12.1 -0.5 -0.64 0.410 3 11.2 12.1 0.9 0.76 0.578 4 10.9 11.4 0.5 0.36 0.130 5 14.3 13.9 -0.4 -0.54 0.292 6 13.2 13.5 0.3 0.16 0.026 7 9.7 10.9 1.2 1.06 1.124 8 11.5 11.5 0 -0.14 0.020 9 10.8 10.4 -0.4 -0.54 0.292 10 13.1 12.5 -0.6 -0.74 0.548 117.7 119.1 1.4 3.484 M = 0.14
  • 26. Step 2: (cont.)
        • Figure the variance of the distribution of means of difference scores.
        • Figure the standard deviation of the distribution of means of difference scores.
      • The shape is a t distribution with N – 1 degrees of freedom.
        • t distribution with 9 df
  • 27. Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
      • Decide the significance level and whether to use a one- or a two-tailed test.
        • p<.05 (one-tailed)
      • Look up the appropriate cutoff in a t table.
    “ Participants given the experimental procedure will have an increase in scores following the procedure”
  • 28. Step 4: Determine the sample’s score on the comparison distribution.
      • Assume the mean of the distribution of means of difference scores = 0.
  • 29. Step 5: Decide whether to reject the null hypothesis.
      • Compare the t score for your sample to the cutoff score found using the t tables
      • t of .71 is less extreme than the t cutoff of 1.833.
      • Fail to reject H o
      • Findings inconclusive for H a
      • Cannot say the experimental procedure made a difference.
  • 30. Review of the Z test, t Test for a Single Sample, and t Test for Dependent Means
    • Z Test
      • Population variance is known.
      • Population mean is known.
      • There is 1 score for each participant.
      • The comparison distribution is a Z distribution.
      • Formula Z = (M – Population M) / Population SD M
      • The best estimate of the population mean is the sample mean.
    • t Test for a Single Sample
      • Population variance is not known.
      • Population mean is known.
      • There is 1 score for each participant.
      • The comparison distribution is a t distribution.
      • df = N – 1
      • Formula t = (M – Population M) / Population S M
    • t Test for Dependent Means
      • Population variance is not known.
      • Population mean is not known.
      • There are 2 scores for each participant.
      • The comparison distribution is a t distribution.
      • df = N – 1
      • Formula t = (M – Population M) / Population S M
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 31. Assumptions of the t Test for a Single Sample and t Test for Dependent Means
    • Assumption
      • a condition required for carrying out a particular hypothesis-testing procedure
      • It is part of the mathematical foundation for the accuracy of the tables used in determining cutoff values.
    • A normal population distribution is an assumption of the t test.
      • It is a requirement within the logic and mathematics for a t test.
      • It is a requirement that must be met for the t test to be accurate.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 32. Effect Size for the t Test for Dependent Means
    • Mean of the difference scores divided by the estimated standard deviation of the population of difference scores
        • estimated effect size =
        • M = mean of the difference scores
        • S = estimated standard deviation of the population of individual difference scores
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 33. Step 2: (cont.)
      • Find the standard deviation of the distribution of means of difference scores.
        • Figure the estimated population variance of difference scores.
    (df = N-1) Participant Before After Difference (After-Before) Deviation (Difference - M) Squared Deviation 1 10.4 10.8 0.4 0.26 0.068 2 12.6 12.1 -0.5 -0.64 0.410 3 11.2 12.1 0.9 0.76 0.578 4 10.9 11.4 0.5 0.36 0.130 5 14.3 13.9 -0.4 -0.54 0.292 6 13.2 13.5 0.3 0.16 0.026 7 9.7 10.9 1.2 1.06 1.124 8 11.5 11.5 0 -0.14 0.020 9 10.8 10.4 -0.4 -0.54 0.292 10 13.1 12.5 -0.6 -0.74 0.548 117.7 119.1 1.4 3.484 M = 0.14
  • 34. Effect Size for the t Test for Dependent Means
    • Mean of the difference scores divided by the estimated standard deviation of the population of difference scores
        • estimated effect size =
    Copyright © 2011 by Pearson Education, Inc. All rights reserved