Basic Statistics

134 CATEGORICAL DATA: PROPORTIONS

70 80 90 P

-1.77 0 1.77 L

Figure 10.5 Two-sided test of Ho : ri = .80.

We wish to calculate P, the probability that a sample proportion p will be .70 or
less or will be .90 or more if actually 7r = 230. We need the shaded area under the
normal curve in Figure 10.5. First, we find the z corresponding to p = .70 using the
following binomial formula:
P - To
QP

z=-

or

.70- .80 -.lo

z= - - - = -1.77

.0566 .0566

From Table A.2, we find that the area below z = +1.77 is .9616, so that the area of

the two shaded portions is 2(l- ,9616) = ,0768. That is, P = .0768, and we accept

the null hypothesis using a significance level of cy = .05. In other words, we decide

that the recovery rate for this disease may not have changed.

We may prefer a one-sided test. Possibly, we may really believe that the recovery

rate is lower now than in the past; then we are interested in finding an answer to the

question: Is the population proportion < .80? We will test the null hypothesis that

HO : TO 1 30. We then compute the probability in the lower side of the distribution

only, so that we have P = .0384, and we reject the null hypothesis that T 2 .80 at

the .05 level. Our conclusion is that the new recovery rate is lower than the recovery

rate for the past.

It should be noted that from the same set of data, two different tests have led to

different conclusions. This happens occasionally. Clearly, the test to be used should

be planned in advance; one should not subject the data to a variety of tests and then

pick the most agreeable conclusion. In the example, we picked the test according to

the question we wished to answer. For more accurate results the correction factor

given in Section 10.3.2 should be used, especially if n is small.

10.6.2

When two sample proportions have been calculated, one from each of two populations,
then often, instead of giving a confidence interval for the difference in population
proportions, a test is made to determine if the difference is 0. The example of the
doctor with 100 patients under each of two treatments will be used to illustrate the test
procedure. In this example, the two sample proportions were pl = .90 and p2 = .80,
and the sample sizes were nl = nz = 100.

Testing the Equality of Two Population Proportions