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## Confidence Intervals

Why are these students stacking candy hearts? What possible educational purpose can that serve?

Probably none, but they are having fun. This student will never get all of those hearts in one stack.

This will work a lot better. Stacking candy seemed to amused them.

The candy itself was left over from Valentine's Day and was used in a previous class to illustrate confidence intervals. We were trying to estimate the average number of candy hearts in the individual bags. The sample itself was not a random sample--the individual bags came from a couple of big bags of little bags from a local store that had discounted them to get rid of them. So the results may not be valid, but the point was to show in a sweet, non-abstract way how a procedure worked.

Each student got a couple of bags to count, and then we entered the results into a spreadsheet. Using the sample mean, the standard error, and a t-value obtained from a t-value calculator, we got an interval estimate, an estimate that involved hearing, sight, touch, taste, and maybe even smell. With a sample of 30 bags, we had a mean of 18.7 and a standard deviation of 2.053592311. This implies a standard error of 0.374932944. We know that if we take another sample of 30 bags, we will probably not get the same as we got the first time. Each time we take a sample, we almost always be a bit too big or a bit too small.

The standard error is an estimate of how much we will be off on average. Using that, we can construct an interval in a way so that 95% of the time when we construct this interval, we will get the true mean. For this example, we would be 95% confident in the interval 17.9 to 19.5. We cannot tell if the average per bag is 18 or 19, but we are pretty sure it is not 17 or 20.

Except that one student spilled his candy on the floor. He picked it up and counted it, but another student may have added some bits to his pile. He got 26, which was the highest of any student. If we include his bag, we change our interval to 18.1 to 19.8. But should we count it?

I hope the story of this 100-cents, five-sense experience makes sense to you.

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