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Introduction to Confidence Intervals

Introduction to Confidence Intervals

Author: Dan Laub

Define confidence intervals.

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[MUSIC PLAYING] Hi. Dan Laub, here. In this lesson, we're going to talk about confidence intervals. And before we get started, let's discuss the objective for this lesson. By the end of the lesson, you should be able to a correctly identify the definition of a confidence interval. So let's get started.

When attempting to estimate a quantity of a population variable, we can use a single value as an estimate for that quantity. Such a value is referred to as a point estimate. Additionally, we can also provide a range to estimate such a quantity. And this range is referred to as an interval estimate. In the event that we can be 100% certain that the quantity lies in this range, we can say the range is a 100% confidence interval.

However, achieving a 100% confidence interval is impossible without sampling the entire population. So we can really only provide confidence intervals with levels of confidence lower than 100%. In fact, a 95% confidence interval is most commonly used. Meaning that we, as researchers, are 95% certain that the quantity that we are estimating falls within this range. Or that if the experiment was repeated 20 times, the population variable would fall within our 95% confidence interval in 19 of these times.

Recall from previous lessons that a population is a large group of observations, while a sample is a small group of observations drawn from a population. It is desirable to randomly select that sample so is to get a relatively representative portion of the population, which allows one to make more accurate predictions about the population as a whole.

One uses the mean of a data set to get an idea of the center of the data, regardless of whether the data set is a population or a sample. Typically, the mean of a randomly drawn sample will not be the same as the population mean. So it is helpful to use a 95% confidence interval to estimate this population mean instead. Such a 95% confidence interval is equal to a sample mean plus or minus a margin of error.

By seeking the interval for a value, we are actually trying to estimate the range of values that fit within the interval we are looking at. In estimating an interval for a range of values, one must first consider a single value, and then think about how much above and how much below that specific value they are willing to look.

For example, what if we were interested in estimating the mean hours of television that children watch per day? How would we go about estimating such a value by using sample data? Well, we can estimate this by using a point estimate, which let's assume in this case is 4.6 hours per day. We could also use a range to estimate this value, or an interval estimate, which maybe something along the lines of 3.8 hours per day to 5.4 hours per day.

The difference between the point estimate and the end values in the interval estimate helps us determine the margin of error we are working with. So we could express the interval estimate as 4.6 hours per day, plus or minus 0.8 hours per day.

To consider another example, let's look at estimating how much Americans spend on dining out per week. In this case, while we are interested in the overall population, we realize that asking everyone to detail their dining expenses is impossible. So we opt instead to select a random sample.

So suppose that we were to take such a sample of 500 people and determine a sample mean of $77.48 per week was spent on dining out. This would be considered a point estimate for the mean of the overall population. Now suppose that there is a margin of error of $10.21. In this case, the 95% confidence interval would be calculated by subtracting $10.21 from $77.48, and also adding $10.21 to $77.48.

This would give us a 95% confidence interval of $67.27 to $87.69. This 95% confidence interval provides us with a likely range for the population mean, which we would be 95% certain that the actual mean of the population, in this case, the average amount that Americans spend dining out per week, falls within this range. So if we were to continue repeating this experiment, the 95% confidence interval states that 95% of our intervals would contain the actual population mean.

So let's go back to our objective just to make sure we covered what we said we would. We wanted to be able to, by the end of this lesson, correctly identify the definition of a confidence interval, which we did. And we went over several different examples to illustrate that point.

So again, my name is Dan Laub. And hopefully, you got some value from this lesson.

Formulas to Know
Confidence Interval

s a m p l e space m e a n space plus-or-minus space m a r g i n space o f space e r r o r