Essentials

124 ESSENTIALS OF STATISTICStribution in this case, but because it is not centered on zero it is called a noncentralt distribution.The Alternative Hypothesis DistributionThe value at the center of the AHD is called the noncentrality parameter and it is usuallysymbolized by the Greek letter delta (). Delta depends, in part, on just whichH 1is true (there are an infinite number of possibilities). The larger delta is, themore likely it is that your study will lead to a statistically significant result, so it isimportant to look at all of the elements that contribute to delta, as we do next.Delta can be understood in a very concrete way. If you repeat the same twogroupexperiment over and over again, using new samples (from the same population)each time, and calculating t each time, the average of these t values shouldbe about equal to delta, which can be defined as the expected t value (the averageof the t values if you repeat the experiment an infinite number of times). Let’s lookat the t formula and see what can be expected in the long run. Our purpose willbe best suited by beginning with a t formula that requires equal sample sizes andhomogeneity of variance, and is based on Formula 3.6:t n 2 X 1 X 2(6.1)s pwhere s pis the square root of the pooled-variance. If the t test is repeated infinitely,X 1will average out to 1, X 2will average out to 2, and s pwill average out to . Sothe average of all the possible t’s, the expected t (or delta), is given by the followingformula: n 2( 1 2) (6.2)Population Effect SizeThe sample size is not changing, of course, so what Formula 6.2 is really showingus is that g, the sample effect size, (X 1– X 2)/s p, averages out to ( 1– 2)/,which is the population effect size and is symbolized by d:d ( 1 2) (6.3)We would prefer to use a Greek letter to symbolize the effect size in a population,but the use of d was popularized by Jacob Cohen (1988), who did a great deal to