The sampling distribution for the difference in sample proportions will be approximately normal since each sample is sufficiently large. (There are at least 5 children of each type in each sample.)
We state our hypotheses symbolically as
H0: p1 = p2 HA: p1 does not equal p2Under the null hypothesis, each population has a common proportion. Instead of estimating the SE for the difference in sample proportions by plugging the estimates in individually, we can do (a little) better by pooling the information from both samples to estimate the common p.
p = (x1 + x2) / (n1 + n2)or
p = (38 + 48) / (100 + 120) = 0.39We may estimate the standard error by plugging in this estimate for p for both populations, finding = 0.0660
The z statistic is
z = (.38 - .40) / 0.0660 = -0.30The two-sided p-value is twice the area to the left of -0.30, or 2(.3821) = .7642.
There is no evidence against the null hypothesis, so we do not reject it.
Bret Larget, firstname.lastname@example.org