There is a process whose output X is a random variable that

There is a process whose output X is a random variable that you know (never mind how you know this) to be normally distributed with = 50 and = 30. You are going to collect a sample of size 20 from this process and calculate a sample mean from those data. What is the probability that the value you calculate will fall somewhere between 45 and 55?

Solution

Mean ( u ) =50
Standard Deviation ( sd )=30
Number ( n ) = 20
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
To find P(a <= Z <=b) = F(b) - F(a)
P(X < 45) = (45-50)/30/ Sqrt ( 20 )
= -5/6.7082
= -0.7454
= P ( Z <-0.7454) From Standard Normal Table
= 0.22803
P(X < 55) = (55-50)/30/ Sqrt ( 20 )
= 5/6.7082 = 0.7454
= P ( Z <0.7454) From Standard Normal Table
= 0.77197
P(45 < X < 55) = 0.77197-0.22803 = 0.5439