The distribution for scores of a particular IQ test for adults is normally distributed with mean 100 and standard deviation 15.
Let Y be the number of people in a random sample of size 12 that gave an IQ of less than 106. What is the distribution of Y? In a random sample of size 12, what is your he probability that at least 5 will have IQs less than 106?
Please Help!!!!!
First note that $X\sim N(100,15)$ then $Y$ is obviously having a binomial distribution with parameter p where p is $Pr(X<106)=\Phi(\frac{106-100}{15})=\Phi(0.4)$ where $\Phi$ denotes on standard normal CDF. Therefore $$p(Y=y)=\binom{12}{y}p^y(1-p)^{12-y}$$ and the requested probability is $$Pr(Y>4)=1-Pr(Y<5)=1-Pr(Y=0)-Pr(Y=1)-Pr(Y=2)-Pr(Y=3)-Pr(Y=4)$$