Basic Statistics

TESTS OF HYPOTHESIS FOR POPULATION PROPORTIONS 135

• .10 0 .lo P*-P2
  -2.0 a 2.0 2

Figure 10.6 Two-sided test of HO : TI = 7r2.

The null hypothesis is Ho : 7il = TZ and we are merely asking whether or not the
two recovery rates are the same, so a two-sided test is appropriate. We choose cy = .05.
Using the fact that pl - p2 has a normal distribution with mean O(r1 - 7rz = 0) if Ho
is true, we wish to find P, the shaded area in Figure 10.6, which represents the chance
of obtaining a difference as far away from 0 as .10 is if the true mean difference is
actually 0.
To change to the z-scale in order to use the normal tables, we need to know the
standard deviation of pl - p2. This is

which is unknown since the population proportions are unknown. It must be estimated.
Under the null hypothesis, 7r1 and ~2 are equal, and the best estimate for each
of them obtainable from the two samples is 170/200 = .85, since in both samples
combined, 90 + 80 = 170 of the 200 patients recovered. This combined estimate can
be called p. It is a weighted average of the pl and pz. That is,

p= nlpl + nzpz

n1+ 12’2

When nl = 71.2, p is simply the average of pl and pz.

The estimate of the standard deviation is

In our example, apl -p2 is estimated by

apl-pz = J.85(1 - .85)(1/100 + 1/100) = J2(.1275)/100 = .050

Now, since we have an estimate of the apl -pz, we can calculate z: