**The t Test for Two Independent Samples**• Compare means of two groups • Experimental—treatment versus control • Existing groups—males versus females • Notation—subscripts indicate group • M1, s1, n1 M2, s2, n2 • Null and alternative hypotheses • translates into • translates into**Criteria for use**• Dependent variable is quantitative, interval/ratio • Independent variable between-subjects • Independent variable has two levels • t-test • Basic form • One sample**Two sample**• Difference between sample means M1 - M2 • Population parameter • Sampling distribution of the difference • Difference between M1 and M2 drawn from population**Standard error of the difference**• Population variance known • Sum of • Estimate from samples • Differences more variable than scores**Variability of mean differences**• Randomly generated set of 1000 means • Μ= 50, σM = 10 • Take difference between pairs**S2pooled Pooled Variance**• Homogeneity of variance • Assume two samples come from populations with equal σ2’s • Two estimates of σ2 — and • Weighted average**Hypothesis testing**• Two-tailed • H0: µ1 = µ2, µ1 - µ2 = 0 • H1: µ1 ≠ µ2, µ1 - µ2 ≠ 0 • One-tailed • H0: µ1 ≥ µ2, µ1 - µ2 ≥ 0 • H1: µ1 < µ2, µ1 - µ2 < 0 • Determine α • Critical value of t • df = n1 + n2 - 2**Assumptions**• Random and independent samples • Normality • Homogeneity of variance • SPSS—test for equality of variances, unequal variances t test • t-test is robust**H0: µ1 = µ2, µ1 - µ2 = 0**H1: µ1 ≠ µ2, µ1 - µ2 ≠ 0 df = n1 + n2 - 2 =10 + 7 – 2 = 15 =.05 t(15) = 2.131 Example 1**df = n1 + n2 - 2 = 15 + 15 – 2 = 28**=.05, t(28) = 2.049 Example 2**Confidence Interval for the Difference**• Example 1 • -3.257 - (2.131*1.401) < µ1 - µ2 < -3.257 + (2.131*1.401) = -6.243 < µ1 - µ2 < -0.272 • Example 2 • -0.867 - (1.701*5.221) < µ1 - µ2 < -0.867 + (1.701*5.221) = -9.748 < µ1 - µ2 < 8.014 • Includes 0 retain H0**SPSS**• Analyze • Compare Means • Independent-Samples T Test • Dependent variable(s)—Test Variable(s) • Independent variable—Grouping Variable • Define Groups • Cut point value • Output • Levene’s Test for Equality of Variances • t Tests • Equal variances assumed • Equal variances not assumed**Effect size**• Cohen’s d = • Example 1 Cohen’s d • Example 2 Cohen’s d • r2 or η2 • G = grand mean**Factors Influencing t–test and Effect Size**• Mean difference M1 – M2 • Larger difference, larger t • Larger difference, larger r2 and Cohen’s d**Example 1, subtract 1 from first group, add 2 to second**group • M1 – M2 increases from –3.257 to –6.257 • unaffected t increases from –2.325 to –4.466 • r2increases from**Magnitude of sample variances**• As sample variances increase: • t decreases • Cohen’s d and r2 decreases • SSExplainedunchanged, SSErrorand SSTotal increases, S2pooled increases**Sample size**• Larger sample smaller t affects • No effect on Cohen’s d, minimal effect on r2 • First example increase n1from 10 to 30 and n2 from 7 to 21