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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?

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    $\begingroup$ Since this is an exercise, the question qualifies as self-study: you should indicate which resolution you attempted and where you got stuck. $\endgroup$
    – Xi'an
    Jan 22 '19 at 20:25
  • $\begingroup$ 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 $\endgroup$
    – donots
    Jan 22 '19 at 20:29
  • $\begingroup$ Well, you can at least add the self-study tag to the question by clicking on edit $\endgroup$
    – Xi'an
    Jan 22 '19 at 20:32
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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 $X$.

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  • $\begingroup$ thank you for your response, how would I go about solving them? $\endgroup$
    – donots
    Jan 22 '19 at 21:08
  • $\begingroup$ 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. $\endgroup$
    – gunes
    Jan 22 '19 at 21:11

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