It oftentimes happens in statistical studies that one is required to study a sample taken from a large population. Since the sample is chosen randomly, it is clear that the sample mean is a random variable.
The central limit theorem tells us that the sample mean follows the normal distribution. Therefore, we can calculate the probabilities for the values taken by the sample mean on the basis of the population mean and population standard deviation.
The calculator below finds the probability that the sample mean is greater than or less than a given value. Simply input the value of the population mean, population standard deviation, sample size, and value of X in order to find the associated probability.
P(x ≤ 6): 0.63602
P(x ≥ 6): 0.36398