Confidence Interval

A probability sample can provide a point estimate of an unknown population parameter and the standard error of that point estimate. This information can be used to [Page 127]construct a confidence interval to give an estimated range of values around the point estimate that is likely to include the unknown population parameter.

For example, assume that a soda can-filling plant fills soda cans at an average rate of 1,000 to 1,500 cans per minute. Several filling nozzles are simultaneously used to fill the cans. Electronic sensors are used to ensure that the filled amount is within specified limits. Due to inherent variability in the filling process, it is impossible to fill an exact amount (355 milliliters [ml]) of soda in each can. As a final quality control measure, a quality assurance inspector wants to estimate the mean amount of soda filled in one particular batch of 120,000 cans. To do so, one extreme option would be to open all the cans and measure the contents. Clearly, this approach is not cost-effective because doing so will destroy all the cans and contaminate the soda. A reasonable approach would be to take a random sample of, say, 20 cans, measure their contents, and calculate the average amount of soda in each can. In survey sampling terminology, this average is known as the “sample mean.” The average amount of soda in each of the 120,000 cans is called the “population mean.”

It is a common practice to use the sample mean as a point estimate of the population mean. Suppose the sample mean is calculated to be 352 ml. Does it make sense to infer that the population mean also is 352 ml? If “Yes,” then what is the margin of error in drawing such an inference? If another random sample of 100 cans yields a sample mean of 355.8 ml, then the inspector will have more confidence in making an inference about the population mean as compared with an inference based on a random sample of 20 cans because she or he will be using more information in the inference. If the inspector had additional information that the filled amount of soda does not vary much from can to can (i.e. information that the population standard deviation of the filled amount of soda is quite small), then a random sample of 20 cans may be sufficient to draw a conclusion about the population mean with reasonable confidence. On the other hand, if the filled amount of soda varies a lot from can to can (i.e. the population standard deviation of the filled amount is very large), then even a random sample of 100 cans may not be sufficient to draw any conclusion about the population mean with desired confidence.

This example shows that point estimates alone are not sufficient for drawing conclusions about a population characteristic unless accompanied by some additional information regarding the level of confidence and margin of error involved in the estimation process. It would be more informative if the inspector could make a statement, such as “I am 95% confident that, on average, between 354.5 ml to 355.3 ml of soda is present in the 120,000 cans.” Such statements are facilitated by adopting the method of confidence intervals for estimation or statistical inference purposes.

...