Confidence Intervals

 Teaching Tips

• It is useful to think of confidence intervals as a range of "plausible" values for the parameter.

• Interpreting the confidence interval is different from interpreting the confidence level. For example, suppose we've taken a random sample of 10 ice-cream cones, and determined that a 95% confidence interval for the mean caloric contents of a single scoop of ice-cream is (260,310). Interpret the confidence level: If we repeatedly took samples of size 10 and then formed confidence intervals, we would expect 95% of them to contain the true (but unknown) mean. Interpret this particular confidence interval: we are 95% confident that the true mean caloric content lies between 260 and 310.

• Some books will talk about point estimates (the average caloric content of an ice cream cone is 285) and interval estimates (the 95% confidence interval is 260 to 310).

• Pay close attention to the assumptions and/or conditions required to achieve the correct confidence levels. If the assumptions are violated, the confidence level (i.e. "95%") will be wrong, and your interpretations might not be meaningful.

• There are several infamous misinterpretations of a confidence interval. The most infamous is this: there is a 95% probability that the true mean caloric content lies between 260 and 310. The runner-up is: if we take repeated samples, 95% of them will have the sample mean between 260 and 310. Be on guard.

• Three concepts that should not be confused: margin of error, standard error and standard deviation. Margin of error is half the width of a confidence interval; standard error measures the variation of a statistic; standard deviation measures the variation within a population or sample.

• Confidence intervals are a relatively new concept, invented in the mid-twentieth century. They are obtained from the sampling distribution of the statistic. Every statistic has a different formula for the confidence interval -- but to keep things from getting too complicated, we will focus on confidence intervals based on statistics for which the central limit theorem applies.

• One way to explain confidence intervals that might stick in students' heads is this. A dog is tied to a tree, and this dog's leash is three standard errors long. The dog likes the shade of the tree, and 68% of the time you'll find the dog within one standard error of the tree. 95% of the time the dog will be two standard errors from the tree and on rare occaisons, maybe when a cat comes by, the dog is 3 standard errors away. Now for some reason, the tree has become invisible and all you see is the dog. Where would you say the tree is? You'd be 95% confident it was within 2 standard errors of the dog, wouldn't you?

• It is tempting to rely on the formula for the CI alone. But what we're trying to teach here is the method of construction, and the behavior of this method from sample to sample. For this reason it's important that students construct confidence intervals for many different random samples from the same population to understand how confidence intervals vary. You might do this by having each student in your class take their own random sample, or you might make use of technology to do simulations. But, just like in the last unit, in practice we do not get to take many different random samples; we get just the one.

• Make sure your students have some practice calculating the sample size needed to achieve a pre-determined confidence level.

• It is possible that an approximate 95% confidence interval will include nonsensical values in its range. For example, your estimate for a proportion might include negative values. This is a sign that the normal model wasn't a good fit to the data. The moral is that you need to pay attention to the assumptions underlying the confidence intervals.

• Students need to be shown that the particular confidence interval you get depends on the sample you take. Too often everyone does the same homework problem and gets the same confidence interval. It is very worthwhile to let students collect their own data from the same population (and again, a simulation makes this easy) and notice that they all get different confidence intervals, but most of them (about 95% of them but, of course, not exactly 95% of them) contain the true mean.