# Distribution of the mean of normally distributed data

I have an exercise which requires the following:

In water production, 1000 ml bottles are filled. The actual fill content is a random variable X. n = 20 bottles have been sampled (independent identically distributed), and their content has been measured. It is known from past experience that a normal distribution N(µ, σ^2) is suitable for the data. What is the distribution of the sample mean X from the n = 20 bottles? Be as specific as possible?

How should I begin solving this problem and what could be the specific information which is require in it?

• Since this is an exercise, the question qualifies as self-study: you should indicate which resolution you attempted and where you got stuck. Jan 22 '19 at 20:25
• thank you @Xi'an for the comment, I'm new here so I'll have that in mind, as I mentioned in the problem, I don't now how to begin with, so that's all the information I could provide Jan 22 '19 at 20:29
• Well, you can at least add the self-study tag to the question by clicking on edit Jan 22 '19 at 20:32

## 1 Answer

Your samples are i.i.d. normal random variables, so their sum will also be normally distributed. It doesn't matter if $$n=20$$ or not. Even the sum of two independent normal random variables is normal. Mean is just a scaling applied to the sum, and scaling a normal RV produces another normal random variable. Your job is to find the mean and variance of $$\bar{X}$$, i.e. sample mean, based on the mean and variance of $$X$$.

• thank you for your response, how would I go about solving them? Jan 22 '19 at 21:08
• As a first step, calculate the mean and variance of $X_1+X_2$, where $X_1,X_2$ are iid normal RVs. Then try for $X_1,X_2,X_3$ and increase $n$ to $20$. As you go, you'll see that it's fairly easy. Jan 22 '19 at 21:11