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Chapter 9 HYPOTHESIS TESTING USING THE TWO-SAMPLE t-TEST Going Forward Your goals in this chapter are to learn: • The logic of a two-sample experiment • The difference between independent samples and related samples • When and how to perform the independentsamples t-tests • When and how to perform the relatedsamples t-test • What effect size is and how it is measured 2 using Cohen’s d or rpb Understanding the Two-Sample Experiment Two-Sample Experiment • Participants’ scores are measured under two conditions of the independent variable • Condition 1 produces sample mean X 1 representing 1 • Condition 2 produces sample mean X 2 representing 2 Two-Sample t-Test • The parametric statistical procedure for determining whether the results of a two-sample experiment are significant is the two-sample t-test • The two versions of the two-sample t-test are – The independent-samples t-test – The related-samples t-test Relationship in the Population in a Two-sample Experiment The Independent Samples t-Test Independent Samples t-Test • The parametric procedure used for testing two sample means from independent samples • Independent samples result when we randomly select participants for a condition without regard to who else has been selected for either condition Assumptions of the Independent Samples t-Test • The dependent scores are normally distributed interval or ratio scores. • The populations have homogeneous variance. Homogeneity of variance means the variance of the populations being 2 represented ( X ) are equal. Statistical Hypotheses • For a two-tailed test, the statistical hypotheses are H 0 : 1 2 0 H a : 1 2 0 • H0 implies both samples represent the same population of scores • Ha implies the means from our conditions each represent a different population of scores Sampling Distribution The sampling distribution of differences between the means is the distribution of all possible differences between two means when both samples are drawn from the one raw score population that H0 says we are representing. Performing the Independent Samples t-Test 1. Compute the mean and estimated population variance for each condition Remember: The formula for the estimated variance in each condition is ( X ) X 2 n sX n 1 2 2 Performing the Independent Samples t-Test 2. Compute the pooled variance using the formula s 2 pool (n1 1) s (n2 1) s (n1 1) (n2 1) 2 1 2 2 Performing the Independent Samples t-Test 3. Compute the standard error of the difference. This is the standard deviation of the sampling distribution of differences between means. The formula is s X1 X 2 ( s 2 pool 1 1 ) n1 n2 Performing the Independent Samples t-Test 4. Compute tobt for two independent samples using the formula tobt ( X 1 X 2 ) ( 1 2 ) s X1 X 2 One-Tailed Tests The statistical hypotheses for a one-tailed test of independent samples are H 0 : 1 2 0 H a : 1 2 0 If 1 is expected to be larger than 2 OR H 0 : 1 2 0 H a : 1 2 0 If 2 is expected to be larger than 1 One-Tailed Tests Conduct one-tailed tests only when you can confidently predict the direction the dependent scores will change. One-Tailed Tests 1. Decide which X and corresponding is expected to be larger 2. Arbitrarily decide which condition to subtract from the other 3. Decide whether the difference will be positive or negative 4. Create Ha and H0 to match this prediction 5. Locate the region of rejection 6. Complete the t-test as described previously Critical Values Critical values for the independent samples ttest (tcrit) are determined based on • degrees of freedom df = (n1 – 1) + (n2 – 1), • the selected a, and • whether a one-tailed or two-tailed test is used Interpreting the Independent-Samples t-Test • In a two-tailed t-test of independent samples, reject H0 if tobt is greater than (beyond) +tcrit or if tobt is less than (beyond) –tcrit • Otherwise, fail to reject H0 The Related Samples t-Test Related-Samples t-Test The related-samples t-test is used when we have two sample means from two related samples • Related samples occur when we pair each score in one sample with a particular score in the other sample • Two types of research designs producing related samples are the matched-samples design and the repeated-measures design Matched-Samples Design • The researcher matches each participant in one condition with a particular participant in the other condition • We do this so we have more comparable samples Repeated-Measures Design • Each participant is tested under both conditions of the independent variable • That is, each participant is measured under condition 1 and again under condition 2 Transforming the Raw Scores • In a related samples t-test, the raw scores are transformed by finding each difference score • The difference score is the difference between the two raw scores in a pair • The symbol for a difference score is D Statistical Hypotheses The statistical hypotheses for a two-tailed related-samples t-test are H0 : D 0 Ha : D 0 Sampling Distribution The sampling distribution of mean differences shows all possible values of the population mean of the difference scores ( D ) that occur when samples are drawn from the population of difference scores that H0 says we are representing. Performing the Related-Samples t-Test 1. Compute the estimated variance of the 2 difference scores ( s D ) using the formula ( D ) D N sD2 N 1 2 2 where N equals the number of difference scores Performing the Related-Samples t-Test 2. Compute the standard error of the mean difference ( s D ) using the formula sD sD2 N Performing the Related-Samples t-Test 3. Find tobt using the formula tobt D D sD One-Tailed Tests The statistical hypotheses for a one-tailed t-test of related samples are H0 : D 0 H0 : D 0 Ha : D 0 Ha : D 0 If we expect the difference to be larger than 0 If we expect the difference to be less than 0 Critical Values The critical value (tcrit) is determined based on • degrees of freedom df = N – 1 where N is the number of difference scores • the selected a, and • whether a one-tailed or two-tailed test is used Interpreting the Related-Samples t-Test • In a two-tailed test of related samples, reject H0 if tobt is greater than (beyond) +tcrit or if tobt is less than (beyond) –tcrit • Otherwise, fail to reject H0 Describing Effect Size Effect Size • Effect size indicates the amount of influence changing the conditions of the independent variable had on dependent scores • The larger the effect size, the more scientifically important the independent variable is Computing Effect Size Cohen’s d is used to compute effect size Independent Samples t-Test d X1 X2 2 s pool Related Samples t-test d D 2 sD Interpreting Effect Size We interpret the Cohen’s d using a small, medium, or large effect size classification • d = 0.2 is a small effect • d = 0.5 is a medium effect • d = 0.8 is a large effect Proportion of Variance Accounted For • The proportion of variance accounted for is the proportion of the differences in scores that can be attributed to changing the conditions in the independent variable • We use the formula for the squared pointbiserial correlation coefficient 2 ( t ) 2 obt rpb (t obt ) 2 df Example 1 Using the following data set, conduct an independent-samples t-test. Use a = 0.05 and a two-tailed test. Sample 1 Sample 2 14 14 13 15 11 15 13 10 12 13 14 13 14 15 17 14 14 15 Example 1 X 1 13.556 X 2 13.778 s 2pool s1 1.944 s2 1.302 n1 9 n2 9 (n1 1) s12 (n2 1) s22 (n1 1) (n2 1) (9 1)3.779 (9 1)1.695 (9 1) (9 1) 2.737 Example 1 The standard error of the difference is s X1 X 2 ( s 2 pool 1 1 ) n1 n2 1 1 (2.737) 9 9 (2.737)(0.222) 0.780 Example 1 tobt ( X 1 X 2 ) ( 1 2 ) s X1 X 2 (13.556 13.778) 0 0.780 0.222 0.285 0.780 Example 1 • tcrit for df = (9 – 1) + (9 – 1) = 16 with a = .05 and a two-tailed test is 2.120. • Reject H0 if tobt is greater than +2.120 or if tobt is less than –2.120. • Because tobt of – 0.285 is not beyond the –tcrit of –2.120, it does not lie within the rejection region. We fail to reject H0. Example 2 Using the following data set, conduct a related-samples t-test. Use a = 0.05 and a two-tailed test. Sample 1 Sample 2 14 14 13 15 16 15 13 10 12 16 14 13 14 15 17 18 17 19 Example 2 First, we find the differences between the matched scores Sample 1 Sample 2 Differences 14 14 13 15 16 15 -1 -2 -2 13 10 12 16 14 13 -3 -4 -1 14 15 17 18 17 19 -4 -2 -2 Example 2 tobt D D 2 D s N 2.333 0 2.333 6.260 0.373 1.25 9 Example 2 • Using a = 0.05 and df = 8, tcrit = 2.306. • Reject H0 if tobt is greater than +2.306 or if tobt is less than –2.306. • Because tobt of –6.260 is beyond the –tcrit value of –2.306, it lies within the rejection region. We reject H0. Example 2 Effect size d D 2 sD 2.333 1.25 2.087 Example 2 Proportion of variance accounted for 2 ( t ) 2 obt rpb 2 (tobt ) df 6.26 6.26 2 8 39.188 47.188 0.830 2