DDBA 8438: The t Test for Independent Samples Video Podcast Transcript JENNIFER ANN MORROW: Welcome to The t Test for Independent Samples. My name is Dr. Jennifer Ann Morrow. In today's demonstration, I will go over with you the definition for t-test for independent samples. I will give you the alternative names for this test. I will go over some sample research questions that can be addressed using a t-test for independent samples. I will give you the formulas. I will go over the assumptions, discuss effect size, and then give you examples using both the formula and SPSS. Okay, let's get started. Definition JENNIFER ANN MORROW: A t-test for independent samples is a statistic that is used when you have two separate groups-- levels of your independent variable-- and you want to compare them on your dependent variable. These separate groups have different participants within them. Alternative Names JENNIFER ANN MORROW: There are many different names for this particular statistic: independent t-test, independent samples t-test, between subjects t-test, and between groups t-test. All of these are different names for the t-test for independent samples. Sample Research Questions JENNIFER ANN MORROW: Here are a couple of sample research questions that can be addressed using a t-test for independent samples. The first research question is, is there a difference in level of self-esteem between participants in the control group and those in the experimental group? In this case, my independent variable would be group, and my levels or groups would be control and experimental, and my dependent variable is self-esteem. My second question that can be addressed using a t-test for independent samples is, is there a gender difference in spirituality? And in this
case, my independent variable would be gender, and my levels or groups would be female and male, and my dependent variable would be spirituality. Formulas Basic Formula JENNIFER ANN MORROW: The basic formula for a t-test for independent samples is as follows: X sub 1 minus X sub 2 divided by the standard error where X sub 1 is the mean for the first group and X sub 2 is the mean for the second group and your standard error is equal to the pooled variance divided by the sample size for the first group plus the pooled variance divided by the sample size for the second group, and you take the square root of that. To find the pooled variance, all you have to do is take the sum of squares for the first group, add it to the sum of squares for the second group, and divide that by the degrees of freedom for the first group plus the degrees of freedom for the second group. And for this statistic, the t- test for independent samples, the degrees of freedom equals the sample size for the first group minus 1 plus the sample size for the second group minus 1. There are also additional formulas that you can use to calculate a t-test for independent samples. Raw Score Formula JENNIFER ANN MORROW: You can use the raw score formula where you have the mean of the first group minus the mean of the second group divided by the square root of the multiplication of sum of squares for the first group plus the sum of squares for the second group divided by the sample size for the first group plus the sample size for the second group minus 2. You multiply that by the addition of 1 over the sample size for the first group plus 1 over the sample size for the second group. And the formula for a sum of squares is below, where you take the sum of all of the X squares for the first group minus the sum of all the values, and you square that, and you divide that by the sample size for the first group. And that will give you the sum of squares for the first group. Using the same formula, plug in the information for your second group, and you'll get the sum of squares for the second group. There's also another formula that you can use.
Deviation Formula JENNIFER ANN MORROW: The next formula is called the deviation formula. And here you take the mean of the first group, subtract the mean of the second group, and you divide it by the square root of the sum of the deviations for the first group, each score minus the mean of that group and the deviations for the second group, each score minus the mean of that group. Those are both squared. You divide that by the sample size for the first group plus the sample size for the second group, minus 2. Multiply that by the sum of 1 over the sample size for the first group, 1 over the sample size for the second group, and then you take the square root of that. All of these formulas will give you the same result. Choose the one that's easiest for you to use. Recap JENNIFER ANN MORROW: Let's recap. So far we've learned the definition for a t-test for independent samples. We've learned some different names for this particular statistic. We've gone over some sample research questions that can be addressed using this analysis. And we've learned the different formulas that we need to use to analyze using a t-test for independent samples. Now let's go over a t-test for independent samples in more detail. Assumptions JENNIFER ANN MORROW: There are three basic assumptions that must be met in order to use a t-test for independent samples. The first assumption is, the observations within each sample must be independent. The scores cannot be related to other scores in your sample. The second assumption is the two populations from which the samples are selected must be normally distributed. And lastly, the two populations from which the samples are selected must have equal variances, and this is known as homogeneity of variance. If you violate any of these assumptions, you should not use the t-test for independent samples.
Effect Size JENNIFER ANN MORROW: For a t-test for independent samples, you can report two measures of effect size, Cohen's D and the percentage of variance explained or R squared. To calculate Cohen's D, you take the mean of the first group minus the mean of the second group and divide that by the square root of your pooled variance. To get the percentage of variance explained, you take your t value, you square it and divide that by your t value squared plus your degrees of freedom. Examples Using Formula JENNIFER ANN MORROW: Let's do an example using the formula. My research question is, Do girls perform better on a math test compared to boys? My null hypothesis is mu 1 minus mu 2 equals 0: there is no difference in math performance between girls and boys. My alternative hypothesis or my research hypothesis is mu 1 minus mu 2 does not equal 0. There is a difference in math performance between girls and boys. I have ten scores... for the boys: 24, 23, 16, 17, 19, 13, 17, 20, 15, and 26. And I also have ten scores for the girls: 18, 19, 23, 29, 30, 31, 29, 26, 21, and 24. My mean for the boys equals 19, and my sum of squares for the boys equals 160. My mean for the girls equals 25, and my sum of squares for the girls equals 200. My degrees of freedom for this analysis, again, is n1 plus n2 minus 2, which is 20 minus 2, 18. And when I look in my t distribution table for the critical value for a t-test with 18 degrees of freedom, an alpha level of 0.05, and a two-tailed test, I find that the critical value that I need to surpass is equal to plus or minus 2.101. So my t value needs to surpass that in order for it to be significant. Now let's calculate the t statistic. So, again, my t-test for independent samples equals the mean of the first group minus the mean of the second group over the standard error. And to calculate my standard error, I need to find my pooled variance. And my pooled variance is equal to my sum of squares of the first group plus my sum of squares for the second group divided by my degrees of freedom for the first group and my degrees of freedom for the second group. My pooled variance is equal to 20, and then to get your standard error... is my pooled variance divided by the sample size for the first group and my pooled variance divided by the sample size for the second group.
You take the square root of that. And so my standard error is 2.00. So let's plug in this information in my t statistic. So t equals the mean of the first group for boys is 19, the mean of my second group, girls, is 25. Divided by my standard error, which is 2, I get a t value of negative 3.00. So here in this case, my t value has surpassed a critical value. And, remember, my critical value is 2.101, plus or minus. So to write that up, I would be t, my degrees of freedom, which is 18, equals my t value, negative 3.00, P less than 0.05. Again, the alpha level that I chose a priori, two-tailed. So what this would say is that girls perform significantly better on a math test compared to boys. So I could put girls' mean equals 25... performed significantly better... than the boys, and their mean was 19. I can also calculate effect size for this example. My Cohen's D, again, is the mean difference, which would be minus 6 divided by the square root of your pooled variance, which in this case is 20, so my Cohen's D is negative 1.34, which is a large effect size. I can also calculate my percent of variance, which, again, is t squared divided by t squared plus your degrees of freedom. And in this case, my percent of variance is equal to 0.33. Using SPSS JENNIFER ANN MORROW: Now let's use an example using SPSS. First open up your SPSS program and find the file that you want to use to conduct your analysis. Click on File. Click on Open. Click on Data and find the data set that you want to use. Once you have found the data set, click on it, and click on Open. And make sure your data view window appears on the screen. Now, for this example, I want to conduct an independent samples t-test looking at gender differences in stress, so in this case, my independent variable would be gender, and my dependent variable would be stress, so let me show you how to do that. I'm going to click on Analyze, Compare Means. Click on Independent Samples t-test, and your independent samples t-test dialog box will appear on the screen. Now, the first thing you should do is input your independent variable, and, in this case, it's gender. So in the box on the left, find Gender and click on it, and then click on the right arrow key where it says Grouping Variable. That is another name for your independent variable. Once you click on that, you'll see that these two question marks appear in the box. What you have to do is, you have to tell SPSS what the values you have for each of your levels in your independent variable. So click on Define Groups. Now, I know that for my values of female and male, females are 0
and males are 1. And if you don't remember that, you can go back to your data set and check that. So here you put 0 for group one and 1 for group two, and click on Continue. The next thing you need to do is find your dependent variable here in the box on the left. Scroll down until you can see the variable Stress. Click on Stress. Click on the right arrow key to move that to the variable box that says Test Variables. And now the variable Stress has appeared in the dialog box here on the right. Now click Okay. As you can see, SPSS is now giving you an independent samples t-test where you have the dependent variable of stress. You have the independent variable of gender of participant, where 0 are females and 1 are males. You have 97 females and 51 males. For your females, the mean for stress is 2.43, and the mean for males is 2.45. Your standard deviation for females is 0.73, and your standard deviation for males is 0.70. And you have your standard error of the mean for females is 0.07. And the standard error of the mean for males is, if you round this up, 0.10. In your next table, SPSS will give you the results of the independent samples t-test. The first thing that you should look at is something called a Levene's test for equality of variances. This is testing out your assumption of homogeneity of variance. You want this here where it says Sig, the significance for this particular assumption test, to be nonsignificant. You do not want significance here. If you have nonsignificance-- if it's greater than P less than 0.05-- in this case it is. It does not go below P less than 0.05. Saying that, yes, we do have equal variance is assumed for this analysis, so then you focus on the first row in this table, and that is your T statistic result. So here in this case, we have a t value of negative 0.104. We'll just round that to negative 0.10. We have 146 degrees of freedom, and my significance, again, for two-tailed test-- alpha level of 0.05-- the value is 0.917. Here this is nonsignificant because it's not P less than 0.05. So how would I write this up? You would put t, 146 degrees of freedom equals negative 0.10, which is your t value, comma, NS--which stands for Nonsignificant-- comma, two-tailed. And what this would be saying is that there are no significant differences between the means for females for stress and the mean for males for stress, so I would write that up as... females--and here the mean equals 2.43-- and my standard deviation is 0.73. Report... similar levels... of stress... compared to males. And my mean is equal to 2.45, and a standard deviation is 0.71. So here I did not achieve significance for this T statistic. I can also calculate the percent of variance accounted for. Again, that's just t squared divided by t squared plus your degrees of freedom. And here in this case, it would be negative 0.10 squared divided by negative 0.10 squared plus your
degrees of freedom. And my percent of variance accounted for actually would equal to 0.00. There is no effect size, which makes sense, because we have a very low t value, and we have a nonsignificant result. Recap JENNIFER ANN MORROW: Okay, let's recap. We learned about the assumptions of the t-test for independent samples. We went over how to calculate two measures of effect size, and we did an example using the formula and an example using SPSS. We have now come to the end of our demonstration. Practice calculating conducting a t- test for independent samples on your own. Thank you, and have a great day.