Confidence Interval Overview
The Confidence Interval might not be what you think it is.
It is NOT the probability that the true population parameter is within the calculated sample Confidence Interval. Let’s start at the beginning.
The area of statistics that this applies in is called Statistical Inference. This is the case where it is too onerous to determine population parameters such as the mean (μ) by measuring the whole population. So instead, you sample the population in order to estimate the population mean ( x̅ ). This is where the Confidence Interval comes into play. Remember, we do not know and will never know what the true value of μ is.
First of all, we are making an assumption that the population distribution is normal. There is a statistical test for normality called the Anderson-Darling Normality Test. If using this test or plotting a histogram convinces you that the distribution is normal then you are in good shape. If it is not normal, maybe the mean is not the right parameter to give you the information you need.
Now use the sampled data to calculate the Confidence Interval. This is not the right venue to go into mathematical detail but I will give you a high-level sketch of the equation:
x̅ = +/- z (σ/√n)
x̅ is the sample mean
z is the critical z score that correlates to the Confidence Interval value selected
σ is the estimated standard deviation using the sample calculated standard deviation value “s”.
n is the sample size
This is the calculation for one Confidence Interval. The Confidence Interval actually requires multiple blocks of samples to be collected and tested the same way as I show you below. Now you have enough information to use the Confidence Interval.
Here is an example conclusion you can make from the above analysis. For instance, if you specify a 90% Confidence Interval then do analysis on 10 samples, you expect the range of nine of those samples to include the population mean, and one of those sample ranges to not include the population mean.
If you specify a 95% Confidence Interval it means that if you perform 20 Confidence Interval tests on 20 samples you would expect the range of 19 of those Confidence Intervals to include the population mean, and one of them to exclude the population mean.
This is a short overview of the complex selection of a Confidence Interval. For help with the Confidence Interval you should have onboard a strong Lean Six Sigma Black Belt.