1 of 65 Inferential Statistics I: The t-test Experimental Methods and Statistics Department of Cognitive Science Michael J. Kalsher

2 of 65 Outline Definitions Descriptive vs. Inferential Statistics The t-test - One-group t-test - Dependent-groups t-test - Independent-groups t-test

3 of 65 The t-test: Basic Concepts Types of t-tests - Independent Groups vs. Dependent Groups Rationale for the tests - Assumptions Interpretation Reporting results Calculating an Effect Size t-tests as GLM

4 of 65 William Sealy Gosset (1876–1937) Famous as a statistician, best known by his pen name Student and for his work on Student's t-distribution. Beer and Statistics: A Winning Combination!

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7 of 65 The One Group t test The One-group t test is used to compare a sample mean to a specific value (e.g., a population parameter; a neutral point on a Likert-type scale). Examples: 1.A study investigating whether stock brokers differ from the general population on some rating scale where the mean for the general population is known. 2. An observational study to investigate whether scores differ from some neutral point on a Likert-type scale. Calculation of t y : t y = Mean Difference Standard Error (of the mean difference) Note: The symbol t y indicates this is a t test for a single group mean.

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10 of 65 Assumptions The one-group t test requires the following statistical assumptions: 1.Random and Independent sampling. 2.Data are from normally distributed populations. Note: The one-group t test is generally considered robust against violation of this assumption once N > 30.

11 of 65 Computing the one-group t test by hand

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14 of 65 Critical Values: One-Group t test Note: Degrees of Freedom = N - 1

15 of 65 Computing the one-group t test using SPSS

16 of 65 Move DV to box labeled “Test variable(s): Type in “3” as a proxy for the population mean.

17 of 65 SPSS Output

18 of 65 Reporting the Results: One Group t test The results showed that the students’ rated level of agreement with the statement “I feel good about myself” (M=3.4) was not significantly different from the scale’s neutral point (M=3.0), t(4)=.784. However, it is important to note several important limitations with this result, including the use of self-report measures and the small sample size (five participants). Additional research is needed to confirm, or refute, this initial finding.

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37 of Select both Time 1 and Time 2, then move to the box labeled “Paired Variables.” 12. Next, “click”, “Paste”.

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39 of 65 The Independent Groups t test: Between-subjects designs Assumption: Participants contributing to the two means come from different groups; therefore, each person contributes only one score to the data. Calculation of t: t = Mean Difference Standard Error (of the mean difference)

40 of 65 Standard Error: How well does my sample represent the population? When someone takes a sample from a population, they are taking one of many possible samples-- each of which has its own mean (and s.d.). We can plot the sample means as a frequency distribution or sampling distribution. Sample Mean Frequency Sampling Distribution

41 of 65 Standard Error: How well does my sample represent the population? The Standard Error, or Standard Error of the Mean, is an estimate of the standard deviation of the sampling distribution of means, based on the data from one or more random samples. Large values tell us that sample means can be quite different, and therefore, a given sample may not be representative of the population. Small values tell us that the sample is likely to be a reasonably accurate reflection of the population. An approximation of the standard error can be calculated by dividing the sample standard deviation by the square root of the sample size SE =  N

42 of 65 Standard Error: Applied to Differences We can extend the concept of standard error to situations in which we’re examining differences between means. The standard error of the differences estimates the extent to which we’d expect sample means to differ by chance alone-- it is a measure of the unsystematic variance, or variance not caused by the experiment. An estimate of the standard error can be calculated by dividing the sample standard deviation by the square root of the sample size. SE =  N

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44 of 65 Computing the independent- groups t test by hand

45 of 65 Sample Problem A college administrator reads an article in USA Today suggesting that liberal arts professors tend to be more anxious than faculty members from other disciplines within the humanities and social sciences. To test whether this is true at her university, she carries out a study to determine whether professors teaching liberal arts courses are more anxious than professors teaching behavioral science courses. Sample data are gathered on two variables: type of professor and level of anxiety. Anxiety Scores Liberal Arts Behavioral Science

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49 of 65 Critical Values: Independent Groups t test Note: Degrees of Freedom = N 1 + N 2 - 2

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51 of 65 On average, the mean level of anxiety among a sample of liberal arts professors (M = 55.7) was significantly lower than the mean level of anxiety among a sample of behavioral science professors (M = 63.4), t(18) = -2.54, p <.05, r 2 =.26. The effect size estimate indicates that the difference in anxiety level between the two groups of professors represents a large effect. Reporting the Results: Independent Groups t test

52 of 65 Computing the independent- groups t test using SPSS

53 of 65 Sample Problem A researcher is interested in comparing the appetite suppression effects of two drugs, fenfluramine and amphetamine, in rat pups. Five- day-old rat pups are randomly assigned to be injected with one of the two drugs. After injection, pups are allowed to eat for two hours. Percent weight gain is then measured. Compute the independent groups t- test using the data at right. Is this a true experiment, quasi- experiment, or observational study? Percent Weight Gain FenfluramineAmphetamine

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57 of 65 SPSS Output: Independent-Groups t test

58 of 65 Calculating Effect Size: Independent Samples t test r = t 2 t 2 + df (-2.819) 2 (-2.819) r =.5534 r 2 =.306 Note: Degrees of freedom calculated by adding the two sample sizes and then subtracting the number of samples: df = – 2 = 18

59 of 65 On average, the percent weight gain of five-day- old rat pups receiving amphetamine (M = 7.4, SE =.85) was significantly higher than the percent weight gain of rat pups receiving fenfluramine (M = 4.6, SE =.52), t(18) = -2.82, p <.05, r 2 =.31. The effect size estimate indicates that the difference in weight gain caused by the type of drug given represents a large, and therefore substantive, effect. Reporting the Results: Independent Groups t test

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