Sara Gordon is heading a fund raising drive for a college. She wishes to concentrate on the current 10th reunion class and hopes to get contributions from 36% of the 250 members of that class. Past data indicate that those who contribute to the 10th year reunion will donate 4% of their annual salaries. Sara believes that the reunion class members' average salaries have an avg of 32000 dollars and a std. deviation of 9600 dollars. If her expectations are met (36% of class contributing 4% of their salaries), what is the probability that the reunion gift will be between 110000 dollars and 120000 dollars?
Should I take the std error of sampling distribution and calculate the z score? If I do that, the z score comes to ~ 6.1
If X is the distribution of the donation,
$$\mathbb{E}[X]=\frac{32,000}{25}=1,280$$
$$\sigma_X=\sqrt{\frac{9,600^2}{25^2}}=384$$
Now, if $36\%$ of the 250 guys donate, you have that 90 guys donate.
Lt's Y indicate the distribution of the sum donated, we have
$$\mathbb{E}[Y]=1,280\times 90=115,200$$
$$\sigma_Y=\sqrt{384^2\times 90}\approx3,643$$
As 90 is large enough you can use the gaussian approx and thus your expected probability is
$$\mathbb{P}[110,000<Y<120,000]=\Phi[\frac{120,000-115,200}{3,643}]-\Phi[\frac{110,000-115,200}{3,643}]=$$
$$=\Phi[1.32]-\Phi[-1.43]\approx 83\%$$