Z

Fig. 1.2. Normal and Poisson approximations: p = 0.01, n = 100

same value of p, we can estimate p by p = (X\ + X2)/(n1 + n2). In large samples, when the null hypothesis is true, the estimator of p will be very close to the true value, so one can show that Z = (p1 — p2)/[p(1 — i>)(1/n1 + 1/n2)]1/2 is also approximately N(0,1). We can use Z to test the hypothesis p1 = p2 by rejecting that hypothesis if \Z| > z, where for a two-sided test the threshold for a 0.05 level of significance is z = 1.96.

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