Confidence Intervals for Single Population Parameters

When real estate boards post statements about the average rent in a particular city or area, how do they come up with that number?  Are they going around to every single renter in that area, asking them what they pay for rent each month, and then calculating the average?  In a very, very small town, this might be possible.  But in large cities with lots of renters, such an undertaking would be very time consuming and very expensive.  Instead, a sample of rents is taken, and the average of the sample is used to estimate the average rent in that city or town.  The average from the sample is called a point estimate of the actual average rent for the entire population.  We would not expect the point estimate to equal that actual average rent, and there is a possibility that the point estimate is not particularly close to the actual average rent.  Is there a way to improve the result in order to get a better estimate of the average rent?  The answer is yes, by creating an interval estimate.

In the run-up to any election, polls are taken to estimate what percentage of the population will vote for the different people or parties involved.  Pollsters call a random sample of the population, ask the people in the sample how they plan to vote, and, from that data, create an estimate of what the voting public will do on election day.  The percentages obtained from the sample are called point estimates of the actual population percent.  Because polls are based on sample data and not the entire population, there are gaps between the sample statistic (the point estimate) and the corresponding population parameter.  Sometimes, these gaps are quite large, which means the sample statistic is not a good estimate of the population parameter.  Instead of relying on just the point estimate, pollsters use the point estimate to create an interval estimate called a confidence interval.  We can see this confidence interval at work when polls are posted in the media—in the fine print, and there are usually statements saying something like “plus or minus [latex]2.5\%[/latex], [latex]19[/latex] times out of [latex]20[/latex]“.  In other words, the pollsters are saying that there is a [latex]95\%[/latex] probability that the actual percentage of the population that will vote for a particular party falls inside the interval created by adding and subtracting [latex]2.5\%[/latex] from the point estimate.

This type of statistics, using sample data to make generalizations about an unknown population parameter, is called inferential statistics.  In this case, the type of inferential statistics we are interested in are called confidence intervals.  A random sample is taken from the population under study, and the sample statistic is calculated as a point estimate of a population parameter.  Because the point estimate is most likely not the exact value of the population parameter, we construct an interval estimate, a confidence interval.  With a certain probability, we will be able to say that the population parameter is inside the confidence interval.

“7.1 Introduction to Confidence Intervals” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.