This is what I think confidence interval for population mean is.
As stated above no one point estimator will be exactly the same each time, there will be some error. We use confidence intervals to better predict the population parameter by stating it will fall with a certain interval based on the point estimate. It is also dependent upon our sample size and how confident we would like to be in knowing our true population parameter falls within the interval.
We typically use 90%, 95% and 99% confidence intervals but in fact can use any percentage we choose. Remember the empirical rule (68-95-99.7)? Let’s take the 95% confidence interval for example. We know that based on the empirical rule 95% of the data will fall within 2 standard deviations of the mean. This leaves 5% outside of the interval. Since the normal distribution is symmetrical half of the 5% or 2.5% will fall above the interval and 2.5% will fall below the interval. We call 5% α (alpha). It is the % of data that will be falling outside our confidence interval.
Now if we look up the z-score for .025 (2.5%) we would get 1.96 which called the critical value corresponding to a 95% confidence interval. The other critical value would be -1.96 would be critical value on the lower end. See figure below. Notice the E in the figure. When data from a sample is used to estimate a mean there is a margin of error as stated above. This error is found by using the following formula. It is based on the z-score of the confidence, the standard deviation of the population and the sample size. Looking at these variables you can note that the only one you truly have control over is the sample size. The larger the sample size the smaller the margin of error. This formula assumes you know the standard deviation of the population. We will use a different distribution and formula when we do not know the population standard deviation.
Margin of Error E = (1.96 * sigma)/sqrt(n)
We need to know the margin of error to calculate the 95% confidence intervals. Notice the 1.96 σ in the numerator of the fraction. Remember what % of data falls within 2 standard deviations of the mean…….95%!
We take the margin of error and add it and subtract it from the sample mean to get the interval we believe will contain the true mean. As stated before if we were to conduct a study 100 times we would find that 95 out of 100 of the intervals we calculate would contain the true mean.
Confidence interval for mean Example
We conduct a study and collect a sample of 100 and find the mean of this sample to be 75 with a population standard deviation of 8.
The Margin of Error is E = (1.96 * 8)/sqrt(100) = 1.568
The margin of Error is then added and subtracted from the sample mean to get the confidence interval
(75 – 1.568, 75 + 1.568); (73.432, 76.568)
which is the 95% confidence interval. When we change the percentage for the confidence interval the only thing that changes is the z value (1.96).