Unit 10: Confidence Intervals

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 Teaching Tips

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

• 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.

• 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.

• Try an intuitive explanation of a confidence interval. One way to explain them 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 confidence interval 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.

• Do a lot of practice looking at the effect of changing various components of the confidence interval and make sure students understand intuition behind why things change the way they do; for example you might explain the following:
  • Effect of sample size: The bigger the sample, the more accurate the sample estimate is because we have more information.
  • Effect of confidence level: We’re 100% confidence that the true parameter can be anything, we’re 95% confident that the true parameter can be in a given interval, but we’re only 1% confidence that the true parameter can be in some tiny interval.
  • Effect of sample standard deviation: If the sample we take has values that are very spread out (high s.d.), then it’s hard to tell where the true population parameter is; if the sample is pretty concentrated (low s.d.), then we might have a slightly better idea of where the true population parameter lies.
• Do a lot of practice identifying good interpretations of confidence intervals from bad interpretations; practice will help students understand (see practice problems).

• 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.

Student Misconceptions and Confusions

• Constructing the confidence interval is not too difficult, but interpreting it is.

• 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.

• Below are examples of incorrect interpretations. Students should understand why these are incorrect.
  • We are 95% confident that our sample estimate is in this interval
    • We are 100% confident of this; confidence intervals are built around our sample estimate.
  • If we sample repeatedly, 95% of all sample estimates will be in this interval
    • Confidence intervals do not make inference about other samples; they make inference about populations.
  • The true population parameter will be in this interval 95% of the time
    • The population parameter is fixed; so it is either 100% in the interval or 100% not.
• 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.

• 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.

• Students forget that population parameters are fixed. Don’t let them lose sight of the fact that we are trying to estimate what the true value is using a single sample, and that the sample gives us the best possible estimate (assuming it is random and all the assumptions are met).

Resources

Berkeley confidence interval applet (see activity)

Rice confidence interval applet