Question

The age of a students in a class is a normal random variable. There are 80 students in our class. I select 9 students randomly and calculate the mean of their ages (sample mean). I repeat this experiment 1,000,000 times. Then I calculate the mean and standard deviation of the 1,000,000 sample means that I measured; the calculated values are 22 and 4, respectively. What is the probability that the age of a randomly selected student in the class is below 20?

Answer 1

Since every student follows a normal distribution then the experiment of selecting 9 students from 80 students also follows normal distribution and 1000000experiments also follows normal distribution with mean 22 and standard deviation 4 as given in the question

So the procedure goes as follows , firstly we need to convert the normal distribution in standard normal distribution and then check the probability from the tables

Z= standard normal variate , Z~ standard normal distribution

X = normal variate , X~ Normal distribution

E(X)= mean of X

SD(X)= Standard deviation of X

P(randomly selected student in the class is below 20) = standard normal probability {{X - E(X) } / S.D(X) }

P(randomly selected student in the class is below 20) = standard normal probability{ (20 - 22)/ 4 }

P(randomly selected student in the class is below 20) = standard normal probability( - 0.5)

Now , checking from tables

standard normal probability( - 0.5) = 1 - standard normal probability( 0.5) = 1 - 0.69146 = 0.30854

So ,  there is a probability of 0.30854 that the age of a randomly selected student in the class is below 20