How to find a probility that the sample mean of a population lies in a particular range?

Question

Assume that X is a random variable with mean x_mean and standard deviation x_sd. If we take a sample of n items from a population at random, what is the probability that this sample mean (say sample_x_mean) lies between left and right values?

Answer 1

The distribution of the sample mean $X_n$ is approximately normal with parameters $μ=x_{mean}$ and $σ/\sqrt{n}=x_{sd}/\sqrt{n}$, in symbols $$X_n \sim N\left(μ, \dfrac{σ}{\sqrt{n}}\right)$$ approximately (where the approximation is better as $n$ grows to $\infty$). This follows from the Central Limit Theorem and you do not need any further particular assumptions on the distribution of the population from which you are sampling.