Does a random sample have mean 520?
In an industrial process, some measurement, X, is normally distributed with standard deviation σ = 10. Its mean should be µ = 520 but can drift from this, so samples of n = 10 measurements are regularly collected as part of quality control.
If one such sample had mean 529, does the process need to be adjusted? The question can be reexpressed as:
If the underlying population mean was really µ = 520, what is the chance a sample of 10 values having a mean as far from 520 as 529?
Simulation
We can base our answer on the distribution of the sample mean, assuming that X has a normal distribution with µ = 520 and σ = 10. Simulations of 10 values from this distribution can be used to get an approximate distribution.
From the 200 simulated samples above, it seems very unlikely that a sample mean of 529 would have been recorded if the process meanhad been µ = 520. We therefore conclude that:
There is strong evidence that the process no longer has a mean of µ = 520 and needs to be adjusted.