# The continuous probability dilemma

If X is a random variable with a mean of 85.43 and a standard deviation of 17.23, what is the mean of the sample means calculated from samples of size 12 drawn from this population? Give your answer to two decimal places in the form xx.xx

I understand the mean is given and the standard dev is given for the random variable X. The part of the question I don't understand is the part where it says "mean of the sample means", how would you go about calculating the mean of the sample mean, if only a sample space is given, what kind of distribution method do I use?

• The sample mean with a sample size $n$ is defined as $\displaystyle \frac {1} {n} \sum_{i=1}^n X_i$, so you are finding the expected value of this sample mean. As you may have noticed, you only need the value of $E[X]$ to answer this question. – BGM May 10 at 9:24
• @BGM so you mean i have to divide the mean 85.43 by 12? – Ashwin Sarith May 10 at 9:33
• Also how would you , rationalise this entirely – Ashwin Sarith May 10 at 10:14
• You are taking the mean of sample means from samples of size $12$. These sample means are made from $12$ iid observations of your random variable with mean $85.43$. You can use linearity of expectation to take $1/n$ outside the expectation, and take the sum outside as well. Then it's easy to work out the mean of sample means – George Dewhirst May 10 at 10:39
• The "sample mean" is itself a random variable. (Obviously: since it varies if you try the experiment many times, i.e. take different samples.) So ignore the word "mean" in its name, think of $Y = \frac1n \sum^n_{i=1} X_i$ as a r.v. and calculate its mean. – antkam May 10 at 13:43