Chapter 9 Test for Independent Means Between-Subjects Design

Chapter Outline The Distribution of Differences Between Means Estimating the Population Variance Hypothesis Testing with a t Test for Independent Means Assumptions of the t Test for Independent Means Effect Size and Power for the t Test for Independent Means Review and Comparison of the Three Kinds of Tests The t Test for Independent Means in Research Articles

t Tests for Independent Means Hypothesis-testing procedure used for studies with two sets of scores or two groups Each set of scores is from an entirely different group of people and the population variance is unknown Study: Compares Experimental to Control group

t -Test for Independent Means Like the t test for dependent means… Used to compare two groups of scores when you do not know the population variance Unlike the t test for dependent means… Two groups of scores come from two entirely different groups of people The scores are “independent” of one another Can’t take difference scores: Not same subject Not even the same N of subjects in the two conditions

Between-subjects Designs Principle: Study with two different groups of subjects Does Caffeine affect Sleeping patterns?

Distribution of Differences Between Means Compare scores obtained from two different groups of people: Compare the two means (against each other) t test for independent means focuses on difference between the means of two groups. Comparison distribution: Distribution of differences between means. Randomly select one mean from the distribution of means from the first group’s population & randomly select one mean from the distribution of means for the second group’s population Subtract the mean from the second distribution of means from the mean from the first distribution of means to create a difference score between the two selected means Repeat process a large number of times & you will have a distribution of differences between means

Logic of a Distribution of Differences Between Means H0: μ1 = μ2 If H0 true: population means from which the samples are drawn are the same S2 estimated from the sample scores Variance of the distribution of differences between means is based on estimated population variances

Mean of the Distribution of Differences Between Means 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 H0 true: The populations have equal means. The distribution of differences between means has a mean of 0.

The hypothesis testing question Are means of these two groups different enough to permit us to conclude that the two populations they represent have different means? H0 : µ1 = µ2 HA :µ1 ≠ µ2 or HA : µ1 < µ2 The t Test for Independent means answers the following Question(s): Are these two means from two different populations? Is there a significant difference between the two groups? Are the means from the two groups different enough to conclude that the two populations they represent have different means?

Estimating Population Variance Step 2 of hypothesis testing process. Assume both populations have same variance Compute S2 from each sample Average the two estimates. If samples sizes differ, larger sample provides better estimate.

S2pooled (Pooled estimate of the pop. Variance) S2 based on the weighted average of the two estimates, giving emphasis in proportion to the degrees of freedom each sample contributes Estimates are weighted to account for differences in sample size (More weight given to larger sample) S2pooled (Pooled estimate of the pop. Variance)

Figuring the Variance of Each of the Two Distributions of Means The S2Pooled 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. S2M1 = S2Pooled / N1 S2M2 = S2Pooled / N2 Copyright © 2011 by Pearson Education, Inc. All rights reserved

Copyright © 2011 by Pearson Education, Inc. All rights reserved The Variance and Standard Deviation of the Distribution of Differences Between Means The Variance of the distribution of differences between means (S2Difference) is the variance of Population 1’s distribution of means plus the variance of Population 2’s distribution of means. S2Difference = S2M1 + S2M2 The standard deviation of the distribution of difference between means (SDifference ) is the square root of the variance. SDifference = √S2Diifference Copyright © 2011 by Pearson Education, Inc. All rights reserved

Estimated Population Variance of a Distribution of Differences Between Means S of the distribution of differences between means (Estimated standard error of the differences)

Assumptions of the t Test for Independent Means The population distributions are normal. The two populations have the same variance. Even if the distributions are not exactly normal or the variances are not exactly equal, the t test is still pretty accurate. If the populations are very far from normal, the variances are very different, or the variances are very different and the populations are far from normal, then the t test does not give accurate results and alternative tests should be used.

Shape of the Distribution of Differences Between Means Distribution of differences between means based on estimated population variances: The distribution of differences between means = t distribution Variance of this distribution is figured based on population variance estimates from two samples df for this t distribution: dfTotal = df1 + df2

t Score for the Difference Between the Two Actual Means Compute the difference between your two samples’ means Figure out where this difference is on the distribution of differences between means.

T-Test Calculator for 2 Independent Means http://www.socscistatistics.com/tests/studentttest/Default2.aspx

Expressive Writing study from the text N = 20 10 students randomly assigned to expressive writing group: wrote about thoughts and feelings associated with most traumatic life events. 10 were randomly assigned to control group: wrote about plans for the day. One month later, all students rated their overall level of physical health on a scale from 0 (very poor health) to 100 (perfect health). STEP 1 Restate the question as a Ha and H0 about the populations. Population 1: students who do expressive writing Population 2: students who write about a neutral topic (their plans for the day) Ha: Pop. 1 would rate their health differently from Pop. 2 H0: Pop. 1 would rate their health the same as Pop. 2

comparison distribution (distribution of differences between means) STEP 2 Determine the characteristics of the comparison distribution. The comparison distribution is a distributions of differences between means. Mean = 0. comparison distribution (distribution of differences between means) Estimate pop. Variance = Weighted Average S2pooled = 102.89 Mean = 0 S Difference = 4.54 Shape = t (DF = 18)

STEP 3 Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected A two-tailed test. Chose a p-value = .05 Cutoff scores from t table (DF = 18) = 2.101 & -2.101.

Step 4 Determine your sample’s score on the comparison distribution. t = (M1 – M2) / SDifference t = (79.00 – 68.00) / 4.54 t = 11.00 / 4.54 = 2.42

Reject the H0. Step 5 Decide whether to reject the null hypothesis. 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. Reject the H0.

Effect Size and Power for the t Test for Independent Means Estimated Effect Size = (M1 – M2) / Spooled The estimated effect size is the difference between the sample means divided by the pooled estimate of the population's standard deviation. Power determined using a power table, computer software, or a power calculator (found online) Copyright © 2011 by Pearson Education, Inc. All rights reserved

Power When Sample Sizes Are Not Equal Harmonic Mean special average influenced more by smaller numbers It is used in a t test for independent means when the number of scores in the two groups differ. In such cases, the harmonic mean is used as the equivalent of each group’s sample size when determining power. It gives the equivalent sample size for what you would have if you had two equal samples. It is two times the first sample size multiplied by the second sample size—all divided by the sum of the two sample sizes. Harmonic Mean = [2(N1)(N2)] / (N1 + N2) Copyright © 2011 by Pearson Education, Inc. All rights reserved

Copyright © 2011 by Pearson Education, Inc. All rights reserved Review of the t Test for a Single Sample, t Test for Dependent Means, and the t Test for Independent Means 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 SM t Test for Dependent Means Population variance is not known. Population mean is not known. There are 2 scores for each participant. t test is carried out on a difference score. t Test for Independent Means df total = df1 + df2 (df1 = N1 – 1; df2 = N2 – 1) Formula t = (M1 – M2) / SDifference Copyright © 2011 by Pearson Education, Inc. All rights reserved

The t Test for Independent Means in Research Articles When found in research articles, the results of these tests are accompanied by reporting of the means and sometimes the standard deviations. t = (dftotal) = x.xx, p < .01 Copyright © 2011 by Pearson Education, Inc. All rights reserved

Copyright © 2011 by Pearson Education, Inc. All rights reserved Key Points A t test for independent means is used for hypothesis testing with scores from two entirely separate groups of people. The comparison distribution is a distribution of differences between means of samples. The distribution can be thought of as being built up in two steps. Each population of individuals produces a distribution of means, and then a new distribution is created of differences between pairs of means selected from these two distributions of means. The distribution of differences between means has a mean of 0 and a t distribution with the total of the degrees of freedom from the two samples. Its standard deviation is figured in several steps. Figure the estimated population variance based on each sample. Figure the pooled estimate of the population variance. Figure the variance of each distribution of means. Figure the variance of the distribution of differences between means. Figure the standard deviation of the distribution of differences between means. The assumptions of the t test for independent means are that the two populations have a normal distribution and have the same variance. However, the t test gives fairly accurate results when the distribution is not exactly normal or the variances are not exactly equal. Estimated effect size for a t test for independent means is the difference between the samples’ means divided by the pooled estimate of the population standard deviation. Power is greatest when the sample sizes of the two groups are equal. When they are not equal, you use the harmonic mean of the two sample sizes when looking up power on a table. Power for a t test for independent means can be determined using a t table, a power software package, or an internet power calculator. t tests for independent means are usually reported in research articles with the means of the two groups and the degrees of freedom, t score, and significance level. Results may also be reported in a table where significant differences are noted by asterisks. Copyright © 2011 by Pearson Education, Inc. All rights reserved