I271B The t distribution and the independent sample t-test

Some Data Considerations for Final Exam: Two metric variables Correlation One binary categorical variable, one metric variable T-test Two categorical variables Chi-Square (crosstabulations) Polytomous dependent variable, metric IV ANOVA Metric dependent variable, multiple IV’s Linear Regression

How do we test to see if the means between two sample populations are, in fact, different?

Null and Alternative Hypotheses Null Hypothesis:  H 0 : μ 1 = μ c Alternative Hypotheses:  H 1 : μ 1 < μ c  H 2 : μ 1 > μ c  H 3 : μ 1 ≠ μ c

One and Two-Tailed Tests: Defining Critical Regions

Why use the t distribution? Useful for testing mean differences when the N is not very large In very large samples, a t-test is the same as a z- score test. Why?

The t-test  Where: M = mean S DM = Standard error of the difference between means N = number of subjects in group s = Standard Deviation of group df = degrees of freedom Essentially, the t-value is the difference between group means (mean difference), divided by the variability of the two groups (standard error of difference).

Standard Deviations vs. Standard Errors Standard deviation is a measure of spread for a given measure. Standard error of a given statistic is the standard deviation of the sampling distribution of that statistic. Standard Error of the mean

Degrees of freedom d.f. = the number of independent pieces of information from the data collected in a study.  Example: Choosing 10 numbers that add up to 100.  This kind of restriction is the same idea: we had 10 choices but the restriction reduced our independent selections to N-1.  In statistics, further restrictions reduce the degrees of freedom.  In the t-test, since we deal with two means, our degrees of freedom are reduced by two (df = n 1 + n 2 – 2)

Z-distribution versus t-distribution

t distribution As the degrees of freedom increase (towards infinity), the t distribution approaches the z distribution (i.e., a normal distribution) Because N plays such a prominent role in the calculation of the t- statistic, note that for very large N’s, the sample standard deviation (s) begins to closely approximate the population standard deviation (σ)

Assumptions Underlying the Independent Sample t-test Assumption of Normality  Variables are normally distributed within each group Assumption of Homogeneity of Variance  Variation of scores in the two groups should not be very different.

What if the Variances are not equal? We can test to see if the variances are significantly different between groups (equality of variance test) If they *are* significantly different, we can use a modified t-test that corrects for this problem (Welch’s t-test)  Essentially, the Welch’s t-test uses a different estimation of the degrees of freedom and also avoids using the pooled variance.

Calculating Effect Size Note that the SD in the effect size calculation is the pooled (combined) standard deviation for both groups Effect size allows you to compare relative effect of one test to another (using same dependent variable) Conceptually, think of effect size as increasing the probability that you could guess which group a given person came from. Values of.8 or greater are usually large effects,.5 medium,.2 small. Effect size is a measure of practical significance, not statistical significance.

t-tests using STATA t-test.do auto.dta dataset grades.dta dataset