t-Test with Two Independent Samples

Part 2 Overview: t Test for Independent Samples

Used with between groups designs Now instead of comparing a sample mean to a population mean (as in the single sample t-test), we compare one sample mean to another sample mean (X1 - X2 that is, we take a difference) We can do this because the central limit theorem holds: if 2 populations are normally distributed, then the sampling distribution of X1 - X2 will be normally distributed

Variance (σ2) and Standard Deviation (σ)

Another standard measure of variability is: Variance: the average squared deviations of the mean, or average of (X─μ)2, notated with σ2 (for a population) So what is the σ that it is the square of? σ is the square root of variance, which is called the standard deviation.

What if my two groups are not exactly equal?

For an independent samples t-test, n1 (sample 1) and n2 (sample 2) need not be exactly equal but should be "close." -In designing an experiment you should try to get the sample sizes as close as possible. -This t-test can be run even if you lose participants. (Ex: There is attrition so that one sample has 10 people and the other has 9 people.) -Heads up for future lectures: With a dependent samples t-test, the size of both samples must be equal.

Estimating the Population Variance cont.

If one sample is larger than the other, the estimate it provides is likely to be more accurate (because it is based on more information). Some adjustment must be made in the averaging of our population variances to give more weight to the larger sample (weighted average). Remember, the amount of information each sample provides is not its number of scores, but its degrees of freedom (number of scores minus 1). So, your weighted average needs to be based on the degrees of freedom each sample provides (just like for the single samples t-test).

Effect Size - Cohen's d

If your findings are significant, you can compute effect size: The proportion of variance in the dependent variable that is accounted for by the manipulation of the independent variable. Small = .20 Medium = .50 Large = .80

What order should the groups be in: who is group 1 and who is group 2?

It doesn't matter! (If you can tell which sample has larger mean, choosing that as sample 1 so you get a positive t; it will be easier.) If you reverse the order of the groups, all you do is change the signs of the observed t AND the comparison value at the same time. And the test is a matter of comparing MAGNITUDES: if the magnitude of the observed t is greater than the critical t you reject the null.

Assume the Null Hypothesis

The null hypothesis assumes that BOTH samples come from the same population. X1¯ is the mean of a sample of size N1 from that population. X2¯ is the mean of a sample of size N2 from that population. The population is assumed to have a variance σ2 How do you estimate that variance? Use a pooled estimate from the variances of both samples.

A Quick Review of Variability

Variability: How much the scores in a set differ from one another Example: Two classes might have the same average exam score, but one set of scores might be much more spread out Standard deviation (SD): Average distance of scores from the mean (how much individual scores vary from the average/middle of the distribution)

Final Formula that Emerges:

We now need to take the square root of the variance to get the standard deviation. So if s1 and s2 are given. Now how do you get it from raw scores?

Estimating the Population Variance

With a single sample t-test we estimated the population variance from the scores in our sample. For an independent samples t-test, we have to assume that the populations of the two samples come from the same variance (homogeneity of variance). Thus, with two samples, we get two separate estimates of what should be the same number. FYI: In practice, the two estimates will almost never be exactly identical. Since they are both supposed to be estimating the same thing, the best solution is to average the two estimates to get the best single overall estimate (pooled estimate of the population variance). This is an option that we will not be using in this class.

Use Differences of Means

With this t-test we are comparing the mean of group 1 to the mean of group 2 to see if the two means are significantly different enough to reject the null hypothesis. The test statistic is X1 - X2 the difference of sample means, and is divided by its standard deviation. The null and alternative hypotheses are about μ1 ─ μ2, the difference of population means.

Now for the Pooled Estimates

σ2 is estimated both by the sample variance s12 of the first sample and the sample variance s22 of the second sample. Those two estimates are weighted by sample size and averaged, so the first is multiplied by N1 ─ 1 (= df for that sample) and the second by N2 ─ 1 (df for that sample). Then the sum is divided by (N1 ─ 1) + (N2 ─ 1) = N1 + N2 ─ 2 (total df)