The t Test for Two Independent Samples

What Does a t Test for Independent Samples Mean? We will look at difference scores between two samples. A research design that uses a separate sample for each treatment condition (or for each population) is called an independent-measures research design or a between-subjects design This is in contrast to repeated measures or within-subjects designs

What Do Our Hypotheses Look Like For These Tests? Null: H0: μ1 = μ2 (No difference between the population means) Same as μ1 - μ2 = 0 Alternative H1: μ1 ≠ μ2 (There is a mean difference) Same as μ1 - μ2 ≠ 0

What is the Formula for Two Sample t – tests? It is actually very similar to the one sample test… t = [(M1 – M2) – (μ1 - μ2)] / s(M1 – M2) This says that t is equal to the mean observed difference minus the mean expected difference all divided by the standard error This begs the question… What is the standard error for two samples?

What Is the Standard Error for Two Samples? We know that M1 approximates μ1 with some error Also, M2 approximates μ2 with some error Therefore we have two sources of error We pool this error with the following formula s(M1 – M2) = √[(s12/n1) + (s22/n2)]

But There Is a Problem… Does anyone know the problem with this standard error? It only works for n1 = n2. When this isn’t the case we need to use pooled estimates of variance, otherwise we will have a biased statistic. So what we have to do is pool the variance. What does this mean?

What Is the Pooled Variance of Two Samples? To correct for the bias in the sample variances, the independent-measures t statistic will combine the two sample variances into a single value called the pooled variance. The formula for pooled variance is: sp2 = (SS1 + SS2) / (df1 + df2) This allows us to calculate an estimate of the standard error

What Is Our New Estimate of Standard Error? For this we use the pooled variance in place of the sample variance s(M1 – M2) = √[(sp2/n1) + (sp2/n2)] What does the pooled standard error tell us? It is a measure of the standard discrepancy between a sample statistics (M1 – M2) and the corresponding population parameter (μ1 - μ2) Now all we need are the df.

How Do We Calculate the df? We need to take into account both samples df1 = n1 – 1 df2 = n2 – 1 Finally, the dftot = df1 + df2

An Example Group 1 Group 2 {19, 20, 24, 30, 31, 32, 30, 27, 22, 25} SS1 = 200 Group 2 {23, 22, 15, 16, 18, 12, 16, 19, 14, 25} n2 = 10 M2 = 18 SS2 = 160

Step 1: State Your Hypotheses Null H0: μ1 = μ2 Alternative H1: μ1 ≠ μ2 State your alpha α = .05

Step 2: Find tcrit First find the df dftot = df1 + df2 = 9 + 9 = 18 Find the two tailed critical t value for df = 18 and α = .05 tcrit = 2.101

Step 3: Sample Data and Test Statistics SS1 = 200 n2 = 10 M2 = 18 SS2 = 160 sp2 = (SS1 + SS2) / (df1 + df2) = 20 s(M1 – M2) = √[(sp2/n1) + (sp2/n2)] = 2 tobs = [(M1 – M2) – (μ1 - μ2)] / s(M1 – M2) = 4

Step 4: Make a Decision Is our observed t (tobs) greater than, or less than the critical value for t (tcrit) 4 > 2.101 Therefore we make the decision t(18) = 4.00, p<.05

Effect Size d = (M1 – M2) / √sp2 r2 = t2/(t2 + df) r2 = PRE = variability explained by treatment / total variability

Confidence Intervals Point Estimate Interval Estimate

Assumptions! There are always assumptions underlying statistical tests. We need to make sure to know these assumptions to make sure we don’t violate them and get misleading results. So what are the t-test assumptions? The observations within each sample must be independent. The two populations from which the samples are selected must be normal. The two populations from which the samples are selected must have equal variances.