Essentials

X2 NTHE TWO-GROUP t TEST 51(the square root of a ratio equals the ratio of the two square roots).The formula for the standard error of the difference is2 X1 X 2 1 22n 1n (3.2)2The notation n 1and n 2represents the sample sizes for the two groups and allowsfor the fact that your drug and placebo groups could have different numbers ofparticipants. The subscripts on 2 allow for the case in which you are samplingfrom two different populations with the same mean but different variances. Thisis a situation that will be explained later under the topic of homogeneity of variance.Calculating a Test Comparing Two GroupsIn the case of blood pressure, we can use the variance in the general populationfor both 2 and 1 2 2. Let’s say we are measuring systolic blood pressure (the peakpressure in each heart beat cycle) in mm Hg (millimeters of mercury), and weknow the standard deviation in the population to be 20, so 2 400. If after takingthe drug, 15 participants average 143 mm Hg, and after taking placebos, a separateset of 10 participants averages 125, the z score would be143 125 18 18z 2.2.26.7 408 .17 4 00 154 001 0With an 18-point difference between the two means we can reject the null hypothesis(2.20 > 1.96) and conclude that the drug led to significantly higher bloodpressure.The problem with Formula 3.2 is that it requires that you know the populationvariance. There are very few variables that meet this criterion, other than wellstudiedaspects of human physiology. If two groups of participants memorize alist of words created for your study—one group while listening to happy music,and the other while listening to sad music—you cannot use Formula 3.2 to findthe standard error for comparing their means for recall of those words: No oneknows what 2 is in this case! However, if your groups are very large (e.g., if bothare in the hundreds), the variances of the recall scores within the groups are good