Aron chpt 9 ed t test independent samples
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Aron chpt 9 ed t test independent samples Presentation Transcript

  • 1.  
  • 2.
    • Hypothesis-testing procedure used for studies with two sets of scores
      • Each set of scores is from an entirely different group of people and the population variance is not known.
        • e.g., a study that compares a treatment group to a control group
  • 3.
    • When you have one score for each person with two different groups of people, you can compare the mean of one group to the mean of the other group.
      • The t test for independent means focuses on the difference between the means of the two groups.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 4.
    • The null hypothesis is that Population M 1 = Population M 2
      • If the null hypothesis is true, the two population means from which the samples are drawn are the same.
    • The population variances are estimated from the sample scores.
    • The variance of the distribution of differences between means is based on estimated population variances.
      • The goal of a t test for independent means is to decide whether the difference between means of your two actual samples is a more extreme difference than the cutoff difference on this distribution of differences between means.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 5.
    • With a t test for independent means, two populations are considered.
      • An experimental group is taken from one of these populations and a control group is taken from the other population.
    • If the null hypothesis is true:
      • The populations have equal means.
      • The distribution of differences between means has a mean of 0.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 6.
    • In a t test for independent means, you calculate two estimates of the population variance.
      • Each estimate is weighted by a proportion consisting of its sample’s degrees of freedom divided by the total degrees of freedom for both samples.
        • The estimates are weighted to account for differences in sample size.
      • The weighted estimates are averaged.
        • This is known as the pooled estimate of the population variance.
          • S 2 Pooled = df 1 (S 2 1 ) + df 2 (S 2 2 )
            • df Total df Total
          • df Total = df 1 + df 2
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 7.
    • The pooled estimate of the population variance is the best estimate for both populations.
    • Even though the two populations have the same variance, if the samples are not the same size, the distributions of means taken from them do not have the same variance.
      • This is because the variance of a distribution of means is the population variance divided by the sample size.
        • S 2 M 1 = S 2 Pooled / N 1
        • S 2 M 2 = S 2 Pooled / N 2
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 8.
    • The Variance of the distribution of differences between means (S 2 Difference ) is the variance of Population 1’s distribution of means plus the variance of Population 2’s distribution of means.
      • S 2 Difference = S 2 M 1 + S 2 M 2
    • The standard deviation of the distribution of difference between means (S Difference ) is the square root of the variance.
      • S Difference = √S 2 Diifference
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 9.
    • Figure the estimated population variances based on each sample.
      • S 2 = [∑(X – M) 2 ] / (N – 1)
    • Figure the pooled estimate of the population variance.
        • S 2 Pooled = df 1 (S 2 1 ) + df 2 (S 2 2 )
            • df Total df Total
            • df 1 = N 1 – 1 and df 2 = N 2 – 1; df Total = df 1 + df 2
    • Figure the variance of each distribution of means.
        • S 2 M 1 = S 2 Pooled / N 1
        • S 2 M 2 = S 2 Pooled / N 2
    • Figure the variance of the distribution of differences between means.
      • S 2 Difference = S 2 M 1 + S 2 M 2
    • Figure the standard deviation of the distribution of differences between means.
      • S Difference =√ S 2 Difference
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 10.
    • Since the distribution of differences between means is based on estimated population variances:
      • The distribution of differences between means is a t distribution.
      • The variance of this distribution is figured based on population variance estimates from two samples.
        • The degrees of freedom of this t distribution are the sum of the degrees of freedom of the two samples.
          • df Total = df 1 + df 2
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 11.
    • Figure the difference between your two samples’ means.
    • Figure out where this difference is on the distribution of differences between means.
      • t = M 1 – M 2 / S Difference
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 12.
    • The comparison distribution is a distribution of differences between means.
    • The degrees of freedom for finding the cutoff on the t table is based on two samples.
    • Your samples’ score on the comparison distribution is based on the difference between your two means.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 13.
    • Use the expressive writing study example from the text. Page 287
      • You have a sample of 20 students who were recruited to take part in the study.
      • 10 students were randomly assigned to the expressive writing group and wrote about their thoughts and feelings associated with their most traumatic life events.
      • 10 students were randomly assigned to the control group and wrote about their plans for the day.
      • One month later, all of the students rated their overall level of physical health on a scale from 0 (very poor health) to 100 (perfect health).
    Copyright © 2011 by Pearson Education, Inc. All rights reserved Example of The t Test for Independent Means: Background
  • 14.
      • Population 1: students who do expressive writing
      • Population 2: students who write about a neutral topic (their plans for the day)
      • Research hypothesis: Population 1 students would rate their health differently from Population 2 students (two-tailed tests).
      • Null hypothesis: Population 1 students would rate their health the same as Population 2 students.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 15.
      • The comparison distribution is a distributions of differences between means.
      • Its mean = 0.
      • Figure the estimate population variances based on each sample.
      • S 2 1 = 94.44 and S 2 2 = 111.33
      • Figure the pooled estimate of the population variance.
      • S 2 Pooled = 102.89
      • Figure the variance of each distribution of means.
      • S 2 Pooled / N = S 2 M
      • S 2 M1 = 10.29
      • S 2 M2 = 10.29
      • Figure the variance of the distribution of differences between means.
      • Adding up the variances of the two distributions of means would come out to S 2 Difference = 20.58
      • Figure the standard deviation of the distribution of difference between means.
      • S Difference = √S 2 Difference = √20.58 = 4.54
      • The shape of the comparison distribution will be a t distribution with a total of 18 degrees of freedom.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 16.
    • Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
      • You will use a two-tailed test.
      • If you also chose a significance level of .05, the cutoff scores from the t table would be 2.101 and -2.101.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 17.
    • Determine your sample’s score on the comparison distribution .
      • t = (M 1 – M 2 ) / S Difference
      • (79.00 – 68.00) / 4.54
      • 11.00 / 4.54 = 2.42
    Copyright © 2011 by Pearson Education, Inc. All rights reserved Step Four: Determine your sample’s score (calculate t)
  • 18.
      • Compare your samples’ score on the comparison distribution to the cutoff t score.
      • Your samples’ score is 2.42, which is larger than the cutoff score of 2.10.
      • You can reject the null hypothesis.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 19.
    • 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.
      • t test is carried out on a difference score.
      • df = N – 1
      • Formula t = (M – Population M) / Population S M
    • t Test for Independent Means
      • Population variance is not known.
      • Population mean is not known.
      • There is 1 score for each participant.
      • The comparison distribution is a t distribution.
      • df total = df1 + df2 (df1 = N1 – 1; df2 = N 2 – 1)
      • Formula t = (M 1 – M 2 ) / S Difference
    Copyright © 2011 by Pearson Education, Inc. All rights reserved