Understanding T-Tests: Analyzing Independent & Dependent Samples

T(EA) for Two: Tests between the means of different groups Determining Correct Statistic:  Most statistical tests have underlying assumptions. o T-test assumes equal variability in each group (homogeneity of variance). o Smaller sample sizes may pose challenges if this assumption is violated. One-Sample Tests:  Two common inferential tests: o One-sample Z test o One-sample t test. o Both compare the mean score of a sample with another score (often the population mean).  Despite using different calculations and tables, they yield similar conclusions. Independent samples:  If the values in one sample reveal no information about or are unrelated to those of the other sample, then the samples are independent Dependent samples:  If the values in one sample affect or are related to the values in the other sample, then the samples are dependent Equal VS Unequal Variance  For independent samples, you have to specify variance of the two groups o Can run specialized test (Welch t-test) to examine variances o As a general rule, you can assume equal variance if the ratio of the sample variances (s 1 2 /s 2 2 ) is between 0.5 and 2 o i.e. the variances of the two samples are no more than double each other

Paired two sample for means two-sample assuming equal/unequal variance

Calculating two-sample T-Test: Independent Samples – by hand

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Example for two-sample t-test: independent samples Researchers wish to know which treatment protocol is more effective at helping Alzheimer’s patients remember words. They divide a group of Alzheimer’s patients into two groups. Group 1 was taught using visuals, and Group 2 was taught using visuals and intense verbal rehearsal.

1. State the null and alternative hypothesis: a. The null hypothesis: the mean of group one will equal the mean of group two. i. H 0 : 𝑥 = 𝑥 b. Alternative/research hypothesis: the mean of group one will be different than the mean of group two-nondirectional hypothesis. i. H 1 : 𝑥 ≠ 𝑥 2. State the test statistic formula: Unequal t-test because we know the standard deviation and our sample is 30 or less. The standard deviations are not equal from the data provided a. 3. State the level of significance: .05 a. for this class our default level of significance is .05, unless stated otherwise. 4. Compute the test statistic 5. Determine the critical value or p-value 6. Determine the statistical conclusion 7. State the experimental conclusion Limitations:  Only evaluate means o Can not make conclusions about individual scores  Affected by sample size o Results of analyses from different studies can not be compared Computing the T Test Statistic The formula for computing the t value for the t test for independent means. All these symbols really just provide two important values to computing the T-Test Statistic. First is the difference between the two means makes up the numerator, and the denominator (top of the equation) makes up the amount of variance within and between each of the two groups. 1. State the null and research hypotheses:

2. Set the level of risk (or the level of significance or chance of Type I error) associated with the null hypothesis. 3. Select the appropriat test statistic 4. Compute the test statistic value (called the obtained value) 5. Determine the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic. 6. Compare the obtained value and the critical value 7. Decide if the obtained value is more extreme than the critical value, the null hypothesis shoud NOT be accepted. If the obtained value does not exceed the critical value, the null hypothesis is the most attractive explanation. Interpreting the t test statistic value: t (58) = -0.14, p > .05  T represents the test statistic that was used  58 is the number of degrees of freedom  -0.14 is the obtained value, calculated using the formula  P > .05 indicates that the probability is greater than 5% that on any one test of the null hypothesis, the two groups do not differ because of the way they were taught. Note that p > .05 can also appear as p= ns for nonsignificant The effect size and T(EA) for two Effect size is a measure of how strongly variables relate to one another-with group comparisons, it’s a measure of the magnitude of the difference.  Sample size in not considered

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Computing and Understanding the Effect Size  Divide the difference between the means by any one of the standard deviations.  This assumes that the standard deviations (and amount of variance) between groups are equal to one another. Two Effect Size Calculators:  Lee Beckers ES Calculator  Drs. Wolfgang & Alexandra Lenhards Calculator Interpreting Effect sizes: Using Excel’s T.TEST function  Excel does not have a function that computes the t value for the difference between two independent groups.

 T.TEST returns the probability of that value occuring, which is very useful 1. Enter the indivual scores into columns in a worksheet a. Label one column as Group 1 and one as Group 2 2. Select the cell into which you want to enter the T.TEST function. 3. Now click Formulas -> More Functions -> Statistical menu option and scroll down to select T.TEST. The function looks like this: Data Analysis Computes the T Value: 1. Use the t test: two-sample equal variances then you should see the descriptive statistics dialog box 2. In the variable 1 range, enter the cell addresses for the first group of data 3. In the variable 2 range, enter the cell addresses for the second group of data 4. Click the labels box so that labels are included in the output that Excel generates 5. Click the output range button and enter an address on the same worksheet as the data where you want the output located 6. Click OK, you will get a summary of important data relating to the analysis. Guide to Output of t Test: Two-Sample Assuming Equal Variances:

Two-sample tests for dependent samples Dependent Samples  If the values in one sample are related to the values in the other sample, then the samples are dependent  Example: To compare the average scores of Exam 2 between Exam 1 & Exam 2 for my Downtown Class Computing two-sample t-tests: dependent samples  T-test for dependent samples will compare the difference in means between 2 groups  Also called matched sample or paired t-test  Groups are matched, so the sample size will be equal  Natural extension of the one-sample t-test

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 t.test(array1, array2, 2 (2-tailed), 1 (paired t-test)  T.TEST result is not the t-test statistic, it is the p-value*

T(EA) for two (AGAIN): Tests between the means of related groups Select the appropriat test statistic. For the t test for dependent means follow along highlighted sequence of steps. Computing the t test statistic The t test for dependent means involves a comparison of means from each group of scores and focuses on the differences between the scores. The sum of the differences between the two tests forms the numerator and reflects the difference between groups of scores: 1. State the null and research hypotheses 2. Set the level of ris (or the level of significance or Type I error) associated with the null hypothesis 3. Select the appropriat test statistic 4. Compute the test statistic value (called the obtained value) 5. Determine the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic. - either Appendix B or Table B a. Determine the degrees of freedom (df) which approximates the sample size (n-1) 6. Compare the obtained value (test statistic value) with the critical value (found in appendix or table B)

7. Decide if the obtained value is more extreme than the critical value, the null hypothesis cannot be accepted. If the obtained vlaue does not exceed the critical value, the null hypothesis is the most attractive explanation. 8. Compute the effect size to interrept effect size:

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Ch. 12 and 13 Notes

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