Section 9.3 Interpreting Confidence Intervals
When we interpret a confidence interval, we say that “we are \(90\%\) confident that the interval we calculated captures the population mean”. We DO NOT say “there is a \(90\%\) chance that the population mean falls between A and B”. It might sound the same, but it is not. The latter implies that the population mean is a variable that we can say something about. But it is not -- it is fixed, and we do not know what it is.
Remember that we develop our confidence interval based on the sampling distribution of the sample mean. We draw a sample and calculate an interval around the sample mean and say that we think we have “captured” the population mean within the interval -- we do NOT know anything about the population mean. We can only say something about our sample mean and interval.
If we were to repeatedly draw samples, each has the same margin of error (since the sample size and \(\sigma\) are constant for all samples), but each sample mean likely varies. After calculating all intervals, \(90\%\) of them would result in intervals that include the population mean, but \(10\%\) of them would have sample means so extreme (in the tail of the sampling distribution) that they would not include the population mean.
The figure below is about proportions instead of means, but it illustrates this idea well.
