Probability that the average is within range I'm currently preparing for an exam, and I just can't seem to grasp how to go about the question given the following information:
Information:
A brick factory produce bricks whose length $X_i$ can be assumed normally distributed with mean length of $228 mm$ and standard deviation $4 mm$ .. 
Question:
At the brick factory a random sample of $50$ bricks is taken and the average of the lengths calculated by $$\bar X=\frac{1}{50}\sum^n_{i=1}X_i$$
What is the probability that the average of the observed lengths $\bar X$, is within the range of $[227;229]$.
My thoughts:
In the question before this, I was asked to calculate the 0.95 CI and found it to be in the range: $220 mm < X < 236 mm$, so I would assume when we averaged the random sample of 50 bricks, the probability would be over $90$ pct. 
I hope someone can point me in the right direction.
 A: You need to use some basic results about means and variances


*

*The expectation of a sum is the sum of the expectations

*The expectation of a constant times a random variable is the constant times the expectation of the variable

*The variance of a sum of independent random variables is the sum of the variances 

*The variance of a constant times a random variable is the square of the constant times the variance of the variable
The two results about expectation mean that the distribution of a sample mean of identically distributed quantities has the same mean as the original variable. The two results about variance mean that the variance of a sample mean (of independent identically distributed variates) has variance that's the original variance divided by the sample size.
Hence the sample mean of a sample of 50 bricks has mean $228$ mm and standard deviation $\frac{4}{\sqrt{50}}$ mm.


*

*You also need that linear combinations of independent normal random variates will also be normally distributed.


As a result, the sample mean will be $\sim N(228,0.5657)$
From there you should be able to find the probability called for in the question.
