1. Testing for differences between 2 means Does the mean weight of cats in Toledo differ from the mean weight of cats in Cleveland? Do the mean quiz scores of female and male Biostatistics students differ? Does the mean growth rate of plants given P fertilizer differ from those not given P fertilizer More from your own research ……..

2. Two tailed test Null; H 0 : xbar 1 = xbar 2 Alternative; H A : xbar1 ≠ xbar2

3. Similar to one-sample, but…. t = X 1 – X 2 t-statistic SE of diff between sample means Sample1 Mean Sample2 Mean s x 1 – x 2

4. The variance of the difference between two (independent) variables equals the sum of the variances of the two variables Calculate from your sample data Calculate pooled variance of the sample as best estimate of the true population variance pooled variance = SS 1 + SS 2 df 1 + df 2

5. s x 1 – x 2 SE of diff between sample means = n1n1 pooled var n2n2 + n1n1 n2n2 + t = X 1 – X 2 Excel demo

6. Ex. Test a new fertilizer (independent/predictor variable) and measure plant height (dependent/response variable) Diff between means

7. 40 42 44 46 48 50 52 54 56 58 present fertnew fert plant height (cm) Series1 Now imagine that the variance was higher Ratio of  means and SE of difference Ratio gets smaller -dividing by larger number

8. Diff is significant (because diff large relative to variance), but is this difference biologically important?

9. Compare to very large diff between means, but large SE 47 cm 4.5 cm (last example)

10. Violations of the 2 sample t-test assumptions 1) Both samples come from normal populations with equal variance 2) Samples collected randomly T-test robust, especially at large sample size How to from SAS HELP: The underlying assumption of the t test in all three cases is that the observations are random samples drawn from normally distributed populations. This assumption can be checked using the UNIVARIATE procedure; if the normality assumptions for the t test are not satisfied, you should analyze your data using the NPAR1WAY procedure.

11. Violations of the 2 sample t-test assumptions 3) Populations have equal varince If population (sample) variances unequal then higher Type I error than stated  (heteroscedastic) called the Behrens-Fisher problem Corrected t available when equal variance cannot be assumed How to from SAS HELP: PROC TTEST computes the group comparison t statistic based on the assumption that the variances of the two groups are equal. It also computes an approximate t based on the assumption that the variances are unequal (the Behrens-Fisher problem). The degrees of freedom and probability level are given for each; Satterthwaite's (1946) approximation is used to compute the degrees of freedom associated with the approximate t. In addition, you can request the Cochran and Cox (1950) approximation of the probability level for the approximate t. The folded form of the F statistic is computed to test for equality of the two variances (Steel and Torrie 1980).

12. 4) The two populations must also be independent Dealing with dependent samples on Friday