HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 613

Testing the difference between two proportions Suppose that we want to test the null hypothesis, H0: p1-p2 = p0, where the p's represents the probability of obtaining a successful outcome in any given repetition of a Bernoulli trial for two populations 1 and 2. To test the hypothesis, we perform n1 repetitions of the experiment from population 1, and find that k1 successful outcomes are recorded. Also, we find k2 successful outcomes out of n2 trials in sample 2. Thus, estimates of p1 and p2 are given, respectively, by p1' = k1/n1, and p2' = k2/n2. The variances for the samples will be estimated, respectively, as s12 = p1'(1-p1')/n1 = k1⋅(n1-k1)/n13, and s22 = p2'(1-p2')/n2 = k2⋅(n2-k2)/n23. And the variance of the difference of proportions is estimated from: sp2 = s12 + s22 . Assume that the Z score, Z = (p1-p2-p0)/sp, follows the standard normal distribution, i.e., Z ~ N(0,1). The particular value of the statistic to test is z0 = (p1'-p2'-p0)/sp. Two-tailed test If using a two-tailed test we will find the value of z α/2, from Pr[Z> zα/2] = 1-Φ(zα/2) = α/2, or Φ(z α/2) = 1- α/2, where Φ(z) is the cumulative distribution function (CDF) of the standard normal distribution. Reject the null hypothesis, H0, if z0 >zα/2, or if z0 < - zα/2. In other words, the rejection region is R = { |z0| > zα/2 }, while the acceptance region is A = {|z0| < zα/2 }. One-tailed test If using a one-tailed test we will find the value of za, from Page 18-42