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Aron chpt 9 ed t test independent samples

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Aron chpt 9 ed t test independent samples

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

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