Chapter Twelve The Two-Sample t-Test

Copyright © Houghton Mifflin Company. All rights reserved.Chapter is the mean of the first sample is the mean of the second sample is the estimated population standard deviation of the first sample is the estimated population standard deviation of the second sample is the number of scores in the first sample is the number of scores in the second sample New Statistical Notation

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Understanding the Two-Sample Experiment

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Participants’ scores are measured under two conditions of the independent variable Condition 1 produces sample mean that represents Condition 2 produces sample mean that represents Two-Sample Experiment

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Two-Sample t-Test The parametric statistical procedure for determining whether the results of a two-sample experiment are significant is the two-sample t-test There are two versions of the two- sample t-test –The independent samples t-test –The related samples t-test.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Relationship in the Population in a Two-sample Experiment

Copyright © Houghton Mifflin Company. All rights reserved.Chapter The Independent Samples t-Test

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Independent Samples t-Test The independent samples t-test is the parametric procedure used for significance testing of two sample means from independent samples Two samples are independent when we randomly select and assign a participant to a sample

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Assumptions of the Independent Samples t-Test 1.The dependent scores measure an interval or ratio variable 2.The populations of raw scores form normal distributions 3.The populations have homogeneous variance. Homogeneity of variance means that the variance of all populations being represented are equal. 4.While N s may be different, they should not be massively unequal.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Two-tailed test One-tailed test –If  1 is expected to If  2 is expected to be larger than  2 be larger than  1 Statistical Hypotheses

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Critical Values Critical values for the independent samples t-test (t crit ) are determined based on degrees of freedom df = ( n 1 - 1) + ( n 2 - 1), the selected , and whether a one-tailed or two-tailed test is used

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Sampling Distribution The sampling distribution of differences between means is the distribution of all possible differences between two means when they are drawn from the raw score population described by H 0

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Computing the Independent Samples t-Test 1.Calculate the estimated population variance for each condition

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Computing the Independent Samples t-Test 2.Compute the pooled variance

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Computing the Independent Samples t-Test 3.Compute the standard error of the difference

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Computing the Independent Samples t-Test 4.Compute t obt for two independent samples

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Computing the Independent Samples t-Test These steps can be combined into the following computational formula for the independent samples t-test

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Confidence Interval When the t-test for independent samples is significant, a confidence interval for the difference between two ms should be computed

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Power To maximize power in the independent samples t-test, you should maximize the difference produced by the two conditions Minimize the variability of the raw scores Maximize the sample n s

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Related Samples t-Test

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Related Samples The related samples t-test is the parametric inferential procedure used when we have two sample means from 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Matched Samples Design In a matched samples design, the researcher matches each participant in one condition with a participant in the other condition We do this so that we have more comparable participants in the conditions

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Repeated Measures Design In a repeated measures design, each participant is tested under all conditions of the independent variable.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Assumptions of the Related Samples t-Test The assumptions of the related samples t-test are when 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

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Transforming the Raw Scores In a related samples t-test, the raw scores are transformed by finding each difference score The difference score is the difference between the two raw scores in a pair The symbol for a difference score is D

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Statistical Hypotheses Two-tailed test One-tailed testIf we expect the difference to be larger than 0less than 0

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Estimated Population Variance of the Difference Scores The formula for the estimated population variance of the difference scores is

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Standard Error of the Mean Difference The formula for the standard error of the mean difference is

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Computing the Related Samples t-Test The computational formula for the related samples t-test is

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Critical Values The critical value (t crit ) is determined based on degrees of freedom df = N - 1 The selected , and whether a one-tailed or two-tailed test is used

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Confidence Interval When the t-test for related samples is significant, a confidence interval for  D should be computed

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Power The related samples t-test is intrinsically more powerful than an independent samples t-test To maximize the power you should –Maximize the differences in scores between the conditions. –Minimize the variability of the scores within each condition. –Maximize the size of N.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Describing the Relationship in a Two-Sample Experiment

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Describing the Relationship Once a t-test has been shown to be significant, the next step is to describe the relationship In order to describe the relationship, you should –Compute a confidence interval –Graph the relationship –Compute the effect size –Compute the appropriate correlation coefficient to determine the strength of the relationship

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Sample Line Graphs Describing a Significant Relationship

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Using r pb Because a two-sample t-test involves one dichotomous X variable (the two conditions of the independent variable) and one continuous interval or ratio Y variable, the point-biserial correlation coefficient is the appropriate coefficient to use

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Degrees of Freedom for the r pb For independent samples, df = ( n 1 - 1) + ( n 2 - 1), where n is the number of scores in a sample For related samples, df = N - 1, where N is the number of difference scores

Copyright © Houghton Mifflin Company. All rights reserved.Chapter In a two-sample experiment, equals the proportion of the variance accounted for This proportion of variance accounted for also is called the effect size in an experiment Effect size indicates how big a role changing the conditions of the independent variable plays in determining differences in dependent scores Effect Size

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Sample 1 Sample Example 1 Using the following data set, conduct an independent samples t-test. Use  = 0.05 and a two-tailed test.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Example 1

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Example 1

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Example 1 Because t obt does not lie within the rejection region, we fail to reject H 0

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Sample 1 Sample Example 2 Using the following data set, conduct a related samples t-test. Use  = 0.05 and a two-tailed test.

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Sample 1 Sample 2 Differences Example 2 First, we find the differences between the matched scores

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Example 2

Copyright © Houghton Mifflin Company. All rights reserved.Chapter Example 2 Using  = 0.05 and df = 8, t crit = Because t obt does not lie within the rejection region, we fail to reject H 0