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The T-Test for Two Related Samples (Dependent Samples) Introduction to Statistics Chapter 11 March 31, 2009 Class #20 Beyond the one-sample t test Are people happier before or after therapy? In married couples, is the husband or wife happier? Among people with the same IQ, is performance higher on multiple choice or essay exams? How to do this? For each person or pair of people, create difference score Ignore the original scores Calculate a t, comparing the mean of difference scores to 0 df = number of pairs -1 The t Test for Dependent Samples Repeated-Measures Design • When you have two sets of scores from the same person in your sample, you have a repeated-measures, or within-subjects design The t Test for Dependent Samples Related-Measures Design • • When each score in one sample is paired, on a oneto-one basis, with a single score in the other sample, you have a related-measures or matched samples design. You use a related-measures design by matching pairs of different subjects in terms of some uncontrolled variable that appears to have a considerable impact on the dependent variable. The t Test for Dependent Samples You do a t test for dependent samples the same way you do a t test for a single sample, except that: • You use difference scores • You assume the population mean is 0 • See formulas on page 292 Difference Scores The way to handle two scores per person, or a matched pair, is to make difference scores. • • For each person, or each pair, you subtract one score from the other. Once you have a difference score for each person, or pair, in the study, you treat the study as if there were a single sample of scores (scores that in this situation happen to be difference scores). A Population of Difference Scores with a Mean of 0 The null hypothesis in a repeated-measures design is that on the average there is no difference between the two groups of scores. This is the same as saying that the mean of the population of the difference scores is 0. HO : D 0 H A : D 0 The t Test for Dependent Samples: Example 1 Step 1 State the statistical hypotheses: HO : D 0 H A : D 0 Step 2 Set and locate the critical region. .05 df number of difference scores 1 8 1 7 t crit 2.365 Step 3: Shade in critical region Step 4: Calculate the t statistic Individual 1 2 3 4 5 6 7 8 Before After D D2 Step 4: Calculate the t statistic Calculate the sample mean of the difference scores D MD n -16 8 = - 2.00 Step 4: Calculate the t statistic Calculate the sample variance (see page 292 for formula) • s2 = SS/df = SS/(n-1) First you need to calculate the SS (use computational formula from page 93) • SS = 42 – 162 8 = 42- 256 = 42-32=10 8 Step 4: Calculate the t statistic Then you plug numbers into variance formula: s2 = SS/df = SS/(n-1) = 10/7 = 1.42 Step 4: Calculate the t statistic Now compute the estimated standard error (see formula on page 292) estimated standard error = s MD = s2 n Step 4: Calculate the t statistic estimated standard error = 1.42 = .421 8 Step 4: Calculate the t statistic See new t formula (page 292) t t = 2 - 0 = 4.750 .421 M D D s MD Step 5: Your decision Make a decision – compare t computed in Step 3 (tOBTAINED) with tCRITICAL found in the t table. • • If tOBT > tCRIT (ignoring signs) Reject HO If tOBT < tCRIT (ignoring signs) Fail to reject HO t obtained = 4.750 t critical = + 2.365 • Answer: Reject HO Interpret your results. • After the pro-socialized medicine lecture, individuals’ attitudes toward socialized were significantly different (more positive) than before the lecture. The t Test for Dependent Samples: Example 2 At the Olympic level of competition, even the smallest factors can make the difference between winning and losing. For example, Pelton (1983) has shown that Olympic marksmen shoot much better if they fire between heartbeats, rather than squeezing the trigger during a heartbeat. The small vibration caused by a heartbeat seems to be sufficient to affect the marksman’s aim. The following hypothetical data demonstrate this phenomenon. A sample of 6 Olympic marksmen fires a series of rounds while a researcher records heartbeats. For each marksman, an accuracy score (out of 100) is recorded for shots fired during heartbeats and for shots fired between heartbeats. Do the data indicate a significant difference? Test with an alpha of .05. During Heartbeats Between Heartbeats 93 90 95 92 95 91 98 94 96 91 97 97 Go through the same five steps as in Example 1 Step 3: Shade in critical region Issues with Repeated Measures Designs Order effects. • • Use counterbalancing in order to eliminate any potential bias in favor of one condition because most subjects happen to experience it first (order effects). Randomly assign half of the subjects to experience the two conditions in a particular order. Practice effects. • Do not repeat measurement if effects linger. Credits http://myweb.liu.edu/~nfrye/psy53/ch11.ppt http://psy.ucsd.edu/~sky/Psyc%2060%20t%20Tests.ppt#2