Marks obtained by certain number of students are assumed to be normally distributed with mean 65 and variance 25. If three students are taken at random, what is the probability that exactly two of them will have marks over 70?
The textbook way to solve it is: Finding the probability (p) that a student gets more than 70 marks. Then find $3(C)2 * p^2 * q$
To find the probability(p) the solution first calculates z=$(70-65)/5$
My confusion is that why did it use the standard deviation of the population(5) instead of using mean of the sampling distribution of sample mean which would have been $5/sqrt(3)$?
In general how do I know when to use what because a lot of questions related to normal distribution first calculate the standard error of mean to calculate the z score.