2What 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 designThis is in contrast to repeated measures or within-subjects designs
3What Do Our Hypotheses Look Like For These Tests? Null:H0: μ1 = μ2 (No difference between the population means)Same as μ1 - μ2 = 0AlternativeH1: μ1 ≠ μ2 (There is a mean difference)Same as μ1 - μ2 ≠ 0
4What 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 errorThis begs the question…What is the standard error for two samples?
5What Is the Standard Error for Two Samples? We know that M1 approximates μ1 with some errorAlso, M2 approximates μ2 with some errorTherefore we have two sources of errorWe pool this error with the following formulas(M1 – M2) = √[(s12/n1) + (s22/n2)]
6But 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?
7What 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
8What Is Our New Estimate of Standard Error? For this we use the pooled variance in place of the sample variances(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.
9How Do We Calculate the df? We need to take into account both samplesdf1 = n1 – 1df2 = n2 – 1Finally, the dftot = df1 + df2
19Assumptions!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.