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**Stony Brook University**

Statistics for the Behavioral and Social Sciences: A Brief Course Fifth Edition Arthur Aron, Elaine N. Aron, Elliot Coups Prepared by: Genna Hymowitz Stony Brook University This multimedia product and its contents are protected under copyright law. The following are prohibited by law: -any public performance or display, including transmission of any image over a network; -preparation of any derivative work, including the extraction, in whole or in part, of any images; -any rental, lease, or lending of the program. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Introduction to the t Test**

Chapter 8 Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Chapter Outline The t Test for a Single Sample**

The t Test for Dependent Means Assumptions of the t Test for a Single Sample and t Test for Dependent Means Effect Size and Power for the t Test for Dependent Means Single-Sample t Tests and Dependent Means t Tests in Research Articles Copyright © 2011 by Pearson Education, Inc. All rights reserved

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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

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**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 S2 = ∑(X – M)2 N – 1 Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Degrees of Freedom (df)**

The number by which you divide to get the estimated population variance Number of scores free to vary when estimating a population parameter If you know the mean of the population and all but one of the scores in the sample, you can figure out the score you don’t know. Once you know the mean, one of the scores in the sample is not free to have any possible value and the degrees of freedom then would = N – 1 Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**The Standard Deviation of the Distribution of Means**

After finding the estimated population variance, you can calculate the standard deviation of the comparison distribution. The variance of a distribution of means is the variance of the population of individuals divided by the sample size. The standard deviation of the distribution of means based on an estimated population variance is the square root of the variance of the distribution of means based on an estimated population variance. S is used instead of Population SD when the population variance is estimated. S2M = S2 / N SM = √S2M Copyright © 2011 by Pearson Education, Inc. All rights reserved

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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. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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Using the t Table There is a different t distribution for any particular degrees of freedom. The t table is a table of cutoff scores on the t distribution for various degrees of freedom, significance levels, and one- and two-tailed tests. The t table only shows positive scores. A portion of a t table might look like this: One-Tailed Tests Two-Tailed Tests df .10 .05 .01 1 3.078 6.314 31.821 12.706 63.657 2 1.886 2.920 6.965 4.303 9.925 3 1.638 2.353 4.541 3.182 5.841 Copyright © 2011 by Pearson Education, Inc. All rights reserved

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When Using a t Table… Determine whether you have a one- or a two-tailed test. If you are using a one-tailed test, decide whether your cutoff score is a positive or a negative t score. If your one-tailed test is testing whether the mean of Population 1 is greater than the mean of Population 2, the cutoff t score is positive. If the one-tailed test is testing whether the mean of Population 1 is less than the mean of Population 2, the cutoff t score is negative. Decide which significance level you will use. Find the column labeled with the significance level you are using. Go down to the row for the appropriate degrees of freedom. If your study has degrees of freedom between two of the higher values on the table, you should use the degree of freedom that is nearest to yours and less than yours. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**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. t = M – Population M SM 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

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**Deciding Whether to Reject the Null Hypothesis**

This is exactly the same as for the other hypothesis-testing procedures discussed in earlier chapters. You will compare the t score for your sample to the cutoff score found using the t table to decide whether to reject the null hypothesis. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**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. population mean This is the same as the known population mean. population variance Figure the estimated population variance. S2 = [∑(X – M)2] / df Figure the variance of the distribution of means. S2M = S2 / N standard deviation of the distribution of means Figure the standard deviation of the distribution of means. S2M = √S2M shape of the comparison distribution t distribution with N – 1 degrees of freedom Determine the significance cutoff. Decide the significance level and whether to use a one- or a two-tailed test. Look up the appropriate cutoff in a t table. Determine your sample’s score on the comparison distribution. t = (M – Population M) / SM Decide whether to reject the null hypothesis. Compare the t score of your sample and the cutoff score from the t table. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Example of Hypothesis Testing When the Population Variance Is Unknown: Step 1**

A survey at your university showed that students at your school study an average of 17 hours a week. You surveyed 16 students in your dorm and found that they each study 21 hours per week. Restate the question about the research hypothesis and a null hypothesis about the populations. Population 1: the kind of students who live in your dormitory Population 2: the kind of students in general at your university The research hypothesis is that Population 1 students study more than Population 2 students. The null hypothesis is that Population 1 students do not study more than Population 2 students. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Example of Hypothesis Testing When the Population Variance Is Unknown: Step 2**

Determine the characteristics of the comparison distribution. population mean = 17 This is the same as the known population mean. population variance Figure the estimated population variance. S2 = [∑(X – M)2] / df = 694 / (16 – 1) = 694 / 15 = 46.27 Figure the variance of the distribution of means. S2M = S2 / N = / 16 = 2.89 standard deviation of the distribution of means Figure the standard deviation of the distribution of means. S2M = √2.89 = 1.70 shape of the comparison distribution t distribution with N – 1 = 15 degrees of freedom Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Example of Hypothesis Testing When the Population Variance Is Unknown: Step 3**

Determine the significance cutoff. Decide the significance level and whether to use a one- or a two-tailed test. Look up the appropriate cutoff in a t table. In this example, with 15 degrees of freedom, a significance level of .05, and a one-tailed test, using a t table you will find that the crucial cutoff is Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Example of Hypothesis Testing When the Population Variance Is Unknown: Step 4**

Determine your sample’s score on the comparison distribution. t = (M – Population M) / SM t = (21 – 17) / 1.70 = 4 / 1.70 = 2.35 Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Example of Hypothesis Testing When the Population Variance Is Unknown: Step 5**

Decide whether to reject the null hypothesis. The cutoff score is Your sample’s score is 2.35. You can reject the null hypothesis. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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How Are You Doing? How does a sample’s variance differ from the population’s? How do we adjust for bias when estimating the population variance? What does N – 1 represent? What is a t distribution? What is a t score? How is a t score calculated? Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**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 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. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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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

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**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

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**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. 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. S2 = [∑(X – M)2] / df Figure the variance of the distribution of means of difference scores. S2M = S2 / N Figure the standard deviation of the distribution of means of difference scores. S2M = √S2M The shape is a t distribution with N – 1 degrees of freedom. 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. Look up the appropriate cutoff in a t table. Determine the sample’s score on the comparison distribution. t = (M – Population M) / SM Decide whether to reject the null hypothesis. Compare the t score for your sample to the cutoff score found using the t table. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Example of a t Test for Dependent Means: Step 1**

Use the brain activation example from the text (Aron, Fisher, Mashek, Strong, & Brown, 2005). Restate the question as a research hypothesis and a null hypothesis about the populations. Population 1: individuals like those tested in this study Population 2: individuals whose brain activation in the caudate area of interest is the same whether looking at a picture of their beloved or a picture of a familiar, neutral person Research hypothesis: Population 1’s mean difference score (brain activation when viewing the beloved’s picture minus brain activation when viewing a neutral person’s picture Null hypothesis: Population 1’s mean difference score is not different from Population 2’s mean difference score. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Example of a t Test for Dependent Means: Step 2**

Determine the characteristics of the comparison distribution. Make each person’s two scores into a difference score. difference = brain activation for beloved – brain activation for control Do all of the remaining steps using these difference scores. Figure the mean of the difference scores. Sum of the difference scores = 12 Number of difference scores = 10 M = 12 / 10 = 1.200 Assume the mean of the distribution of means of difference scores = 0. Population M = 0 Find the standard deviation of the distribution of means of difference scores. Figure the estimated population variance of difference scores. S2 = [∑(X – M)2] / df = / (10 – 1) = .438 Figure the variance of the distribution of means of difference scores. S2M = S2 / N = .438 / 10 = .044 Figure the standard deviation of the distribution of means of difference scores. S2M = √S2M = √.044 = .210 The shape is a t distribution with 9 degrees of freedom. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Example of a t Test for Dependent Means: Step 3**

Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected. You chose to use a .05 significance level. You will use a one-tailed test because your hypothesis is directional. You have 9 degrees of freedom. Look up the appropriate cutoff in a t table. The cutoff t score is Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Example of a t Test for Dependent Means: Step 4**

Determine the sample’s score on the comparison distribution. t = (M – Population M) / SM t = (1.200 – 0) / .210 = 5.71 Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Example of a t Test for Dependent Means: Step 5**

Decide whether to reject the null hypothesis. Compare the t score for your sample to the cutoff score found using the t table. The sample’s score of 5.71 is more extreme than the cutoff score of You can reject the null hypothesis. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Review of the Z test, t Test for a Single Sample, and t Test for Dependent Means**

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 SDM The best estimate of the population mean is the sample mean. t Test for a Single Sample Population variance is not known. The comparison distribution is a t distribution. df = N – 1 Formula t = (M – Population M) / Population SM t Test for Dependent Means Population mean is not known. There are 2 scores for each participant. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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How Are You Doing? What is an example of a research study for which you would need a t test for dependent means? What is a difference score? Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**Assumptions of the t Test for a Single Sample and t Test for Dependent Means**

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

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**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/S 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

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Power Power for a t test of dependent means can be calculated using a power software program, a power calculator, or a power table. Table 8-9 in your textbook shows an example of a power table for a .05 significance level. To use a power table: Decide whether you need a one- or a two-tailed test. Determine from previous research what effect size (small, medium, or large) you might expect from your study. Determine what sample size you plan to have. Look up what level of power you can expect given the planned sample size, the expected effect size, and whether you will use a one- or a two-tailed test. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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Planning a Sample Size A power table can be used to see how many participants you would need to have enough power. Many studies use 80% as the power needed to make the study worth conducting. To use a power table to determine the number of participants needed in a sample: Decide whether you need a one- or a two-tailed test. Determine the expected effect size. Determine the level of power you want to achieve (usually .80). Use this information to guide you to the appropriate columns and rows on the power table. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**The Power of Studies Using a t Test for Dependent Means**

Studies using a repeated-measures design (using difference scores) often have much larger effect sizes than studies using other research designs. There is more power with this type of study than if the participants were divided into groups and each group was tested under each condition of the study. The higher power is due to a smaller standard deviation that occurs in these type of studies. The smaller variation is because you are comparing participants to themselves. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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**t Tests in Research Articles**

Results from t tests are generally reported in the following format: t (df) = x.xx, p < .05 x.xx represents the t score. Commonly, the significance level will be set at p < .05, but it is also often set at p < .01. Research more commonly uses the t test for dependent means. It is rare to see a study that uses a t test for a single sample. Often a t test for dependent means will be given in the text, but sometimes results are reported in a table format. Copyright © 2011 by Pearson Education, Inc. All rights reserved

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Key Points When you have to estimate the population variance from scores in a sample, you will use a formula that divides the sum of square deviation scores by the degrees of freedom. With an estimated population variance, the comparison distribution is a t distribution; it is close to normal, but varies depending on the associated degrees of freedom. A t score is a sample’s number of deviations from the mean of the comparison distribution this is used in situation when the population variance is estimated. A t test for a single sample is used when the population mean is known but the population variance is unknown. A researcher would use a t test for dependent means when there is more than one score for each participant. In this case you would use difference scores. An assumption of the t test is that the population distribution is normal, but even if the distribution is not normal, the results are fairly accurate. When testing hypotheses with t tests for dependent means, the mean of Population 2 is assumed to be 0. effect size for t tests = mean of the difference scores/standard deviation of the difference scores Power or sample size can be looked up using a power table. The power with a repeated-measures design is usually much higher than that of most other designs with the same number of participants. t tests for dependent means are often found in the text or in a table of a research article in this format: t (df) = x.xx, p < .05 Copyright © 2011 by Pearson Education, Inc. All rights reserved

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