September 11, 2023

What would happen to the rate of prize winning if the bar was much narrower? Why?

Confidence intervals can be difficult to understand. Beyond understanding the concept of what a confidence interval is, students also wrestle with understanding the effects of variability (e.g., due to sample size) and desired level of confidence (typically 95 or 99%, corresponding to alphas of .05 and .01, respectively) on the width of a confidence interval.

The following activity will help convey the basic principle of what a confidence interval is. In brief, of course, a confidence interval is an estimate of a parameter based on a sample. The estimate is given in the form of a range that reflects the variability in the sample and the level of confidence desired. Each range (interval) gives a specific degree of certainty that the true population mean (or other desired parameter) is captured within its boundaries. A range is better than a single point because any one sample is unlikely to be a perfect representation of its population (unless every score in the population is identical). It also is important to stress the point that the population value is fixed (unless the population changes) and the goal is to be reasonably certain it is captured by the proposed interval.

I will use The Price is Right’s “Range Game” to illustrate the basic features of confidence intervals. This game involves a translucent red bar (covering a $150 range every time) that slowly moves up a number line. The real price of the prize the contestant is trying to win is somewhere along the line (the contestant does not know where; that is only revealed after the bar is stopped). If the contestant stops the red bar and the price of the prize is within the bar’s area, the contestant wins the prize. Thus, the actual price of the prize represents the population value you are trying to “capture” (estimate) and the top and bottom prices indicated by the moving red bar represent the upper and lower limits, respectively, of a confidence interval.