Question

Consider a hypothesis test of difference of proportions for two independent populations. Suppose random samples produce r1 successes out of n1 trials for the first population and r2 successes out of n2 trials for the second population. What is the sample test statistic for the test

Answer 1

Explanation:

`p_1 = (r_1)/(n_1) \\p_2= (r_2)/(n_2) \\`

Pooled proportion

`p = (r_1+r_2)/(n_1+n_2)`

The hypothesis null would be

`H_0: p_1-p_2 =0`

Alternate can be either not equal to or < or > according to need

Std error for difference of proportions

`=\sqrt{p(1-p)((1)/(n_1) +(1)/(n_2)}`

Hence test statistic = p difference/std error

=
`\frac{(r_1)/(n_1)- (r_2)/(n_2)}{\sqrt{p(1-p)((1)/(n_1) +(1)/(n_2)}}`

where p is given as above as combined proportion.