8.15 Test for difference between proportions
If two samples are drawn from different populations, we may be interested in finding out whether the difference between the proportion of successes is significant or not. Let x_{1} and x_{2} be the number of items possessing the attribute A, in the random sampling of sizes n_{1} and n_{2} from two populations respectively.
Then the sample proportions of successes are if P_{1} and P_{2} are proportion of successes in the two populations and Q_{1} = 1  P_{1}, Q_{2} = 1  P_{2} then
Under the hypothesis that the proportions in two populations are equal.
i.e. P_{1} = P_{2} = P Þ Q_{1} = Q_{2} = Q (say) then
In general, however, we do not know the population’s proportion of success. In such a case we can replace P by its best estimate P = the pooled estimate of the actual proportion in the population, where
Pooled estimate (P) =
Example A machine produced 20 defective
articles in a batch of 500. After overhauling it produced 3 defective
in a batch of 100. Has the machine improved ?
Solution: H_{o} : P_{1}
= P_{2} i.e. The machine has not improved after overhauling
H_{a} : P_{1} ¹ P2
Now P_{1} = = 0.032 and P_{2} = = 0.030
Pooled estimate of actual proportion in the population is given by
Q = 1  P = 0.968
H_{o} is true i.e. the machine has not improved significantly.
