Essentials

50 ESSENTIALS OF STATISTICSsults could be due entirely to chance (e.g., it may be that the blood pressure increasesyou observed in the participants of both groups are due to fear reactions,and that by accident the drug control group wound up with more fearful participantsthan the placebo group). In order to dismiss the possibility that you gotyour good-looking results by accident (i.e., the null hypothesis), you first have tofigure out the distribution of null experiments (i.e., experiments just like yours exceptneither group gets any treatment).In the two-group case, you could find this distribution (i.e., the NHD) by takingtwo samples from the same distribution, subtracting their means, and doingthis over and over. Fortunately, you don’t have to. We know that these differencesof means usually will pile up into an approximate normal distribution, and if thetwo samples are coming from the same population (or two populations with thesame mean—we will point out the distinction later), this distribution will be centeredat zero. So all we need to know for our hypothesis test is the standard deviationof this NHD, which is called the standard error of the difference ( x1 –x 2).Then we can create a z score for our experimental results as follows:(X 1 X 2) ( 1 2)z (3.1)X1X 2Compared to the one-group test (Formula 2.2, Chapter 2), it’s like seeing double.In the one-group case, the numerator of the z score is the difference betweenyour specially treated (or selected) group and the mean you would expect to getif the null hypothesis were true. The denominator (i.e., the standard error of themean) is a typical difference you would get for the numerator when the null hypothesisis true (according to the normal distribution, about two thirds of the numeratordifferences will be smaller than the typical difference in the denominator,and one third will be larger). In the two-group case, the numerator is thedifference between the difference of groups you observed (X 1– X 2) and the differenceyou expected ( 1– 2). Because the expected difference (i.e., the null hypothesis)is almost always zero for the two-group case (H 0: 1 2, so 1– 2 0), we will leave that term out of the numerator in the future. The denominatoris a typical difference of two sample means when H 0is true.The Standard Error of the DifferenceTo find X(the standard error of the mean) in Chapter 2 we used a simple law, X/N. There is a similar law that applies to the two-group case. But tomake sure you see the similarity, let us change the formula for the standard errorof the mean to this equivalent form: