11-41. Suppose PX, PY are sample proportions determined by drawing nX, nY items, respectively, from two separate...

11-41. Suppose ˆPX, ˆPY are sample proportions determined by drawing nX, nY items, respectively, from two separate binomial populations with parameters pX, pY. In addition, let nX, nY be sufficiently large so that ˆPX, ˆPY are approximately normally distributed. Verify that ˆPX − ˆPY is approximately N(μˆPX−ˆPY , σˆPX−ˆPY ), where μˆPX−ˆPY = μˆPX − μˆPY and σˆPX−ˆPY = ˆPX(1 − ˆPX)>nX + ˆPY(1 − ˆPY)>nY.