How to explain a statistical interval confidently

My youngest daughter, being a glutton for punishment like her father, is struggling on the stretch run of her senior year in high school with three AP (advanced placement) college-level classes, one of which is Statistics. That means more fun for me trying to answer questions she has about her homework, many of which are very tricky.

Possibly the most perplexing thing to learn is how one should express a confidence interval. For example, this month’s issue (#49) of Stats, “the magazine for students of statistics,” states this common misconception: “After you compute a 95% confidence interval for the mean, you can say the probability is 95% that it contains the population mean.” *

I confess that after learning statistics on the job as chemical engineer in the 1970s, I would have agreed with this statement. It wasn’t until the advent of applets allowing one to simulate any number of random samples taken from a normal population and generating confidence intervals that I literally saw how they really worked.

For example, see the screen shot taken the tenth time I took one-hundred random samples of 5 using an applet programmed by my oldest son.** I got lucky on this run by missing the true mean of 50 only the expected 5 times out of one hundred (the red intervals). However, the total of 954 successes on the total of one thousand trials reflects the nature of statistics nearly always being a bit off true kilter. The varying results, albeit somewhat unnerving like any natural variations (deterministic outcomes being much more comforting!), are very instructive for statistical concepts like confidence intervals.

For a great discussion on how to properly describe a confidence interval see this thread posted at the Math Forum of Drexel University by Doctor Wilko (aka Dr. Math). It may help you from falling into this particular trap, one of many as noted in the Stats article, that riddle the field of statistics. Be careful out there!

*(Jessica Utts interview with Jackie Miller titled “Busting Statistical Myths,” page 10.)

**(Stat-Ease provides this applet and others to students of its Statistics for Technical Professionals workshop.)