American Airlines claims that the average number of people who pay for in-flight movies when the plane is fully loaded is $42$ with a standard deviation of $8$. A sample of $36$ fully loaded planes is taken. What is the probability that fewer than 38 people paid for in-flight movies?
Ans is $0.0013$. We get this answer if we find $p(\mu<38)$, but I don't understand why we have taken the distribution of mean and not summation as the question has asked for the number of people who have paid not average no of people who paid. But if I find the total number of people who paid for in-flight movies less than $38$ by CLT, I get $P(z<-50.)$ something which is $0$.