Background on Confidence Intervals

A confidence interval is used to estimate some numerical characteristic in a population by observing or measuring that characteristic on a sample from the population of interest. Such an estimate is needed because it is not generally possible to observe or measure every member of a population. If your question of interest begins "What is the value ... " or "How much ... " then you can use a confidence interval to answer your question. (If your question could be answered with a "yes" or "no" then you probably want to use another important technique in statistical inference - a hypothesis test. That will be the subject of the next lesson.

Confidence intervals are often used as alternatives to hypothesis testing (something we'll cover in the next lesson).

Population parameters, like the true population mean, may be inferred by setting up a range of values within which the true parameter may fall. We simply establish a degree of confidence that the population mean will fall within this interval, hence the term confidence interval. Whether you realized it or not, you have been doing this for most of your life. Now we are just going to do it with a little more intention.

After all, if you took another sample of 16 10-year-old girls it would not be surprising that the mean weight was a little bit different from the first sample - perhaps 93 pounds. Since each sample of 16 girls will yield a slightly different mean weight, our point estimate is highly dependent on the particular sample we obtain. Therefore we need a way to estimate how much variation there might be among the many possible samples we could obtain.

 

Since it would have been impractical to weigh all the 10-year old girls in the United States, you took a sample of 16 and found that the mean weight was 90 pounds. This sample mean of 90 is a point estimate of the population mean. Taken by itself, this is of limited usefulness because it does not reveal the uncertainty associated with the estimate; you do not have a good sense of how far this sample mean may be from the population mean. For example, can you be confident that the population mean is within 5 pounds of 90? You simply do not know.

A confidence interval gives us an estimate of how much sample-to-sample variation we might expect as we attempt to determine the actual mean weight of 10-year-old girls. Remember the elderly couple sitting on the park bench? We could guess they have been married 52 years, but there isn't a very high likelihood that particular guess is correct. However, if we estimate they have been married 47 to 53 years, we have a much better chance of having identified the true answer as being in this interval. Of course there is a cost to this increase in our chances of being correct - the estimate isn't quite as precise as stating the single estimate of 52 years. This is where the concept of "confidence level" comes in.

A confidence level is a probability statement about how likely the confidence interval procedure will produce an interval that captures the truth (common confidence levels include 95% and 99%). For example, a 95% confidence level means the interval will be computed based on a procedure that in repeated sampling would produce intervals capturing the true population mean about 95 times out of every 100 samples. In other words, if you could obtain 100 different samples of sixteen 10-year-old girls, and then computed 100 confidence intervals (one for each sample), 95% confidence means you can expect about 95 of those intervals to have actually captured the true (but unknown) mean weight of the entire population. If you want to be more confident (i.e. 99% rather than 95%) you have to pay a cost by accepting a slightly wider interval which will be a little less precise.

Whether you choose to compute an interval based on 95% or 99% confidence, you don't ever know if your one sample will lead to one of the lucky 95 or 99 "correct" intervals, or one of the "unlucky" intervals that missed the target. That's the nature of any probability-based calculation and the nature of statistical inference.

 

  

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