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Applying Confidence Intervals

Applying Confidence Intervals

Author: Dan Laub

Explain the 95% confidence interval in a given scenario.

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Source: Image of tall man, PD,; Image of measuring tape, PD,; Image of coffee, PD,

Video Transcription

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[MUSIC PLAYING] Hi, Dan Laub here. And in this lesson, we're going to discuss applying confidence intervals. But before we do so, let's cover the objective for the lesson. By the end of this lesson, you should be able to interpret a 95% confidence interval. So let's get started.

Now 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 as 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 likely range for a population mean is the 95% confidence interval. Meaning that if enough random samples are drawn from the population, the actual population mean will occur in this range in approximately 95% of the samples.

To put it another way, if we were to conduct 20 experiments, 19 of the confidence intervals we would obtain would contain the actual population mean. It is important to note that the population mean is a fixed value that does not change and that we are simply trying to estimate it by using random samples. For example, suppose we were to take a sample of 100 adults and measured how tall they were in the attempt to estimate the mean height of all adults, if the sample mean of the height of 100 of adults is 67.5 inches with a 95% confidence interval being approximately 59.1 inches to 75.9 inches.

If we were to draw another sample, we would quite likely find a different sample mean. Suppose that a second example of 100 adults was drawn. And the sample mean in this case is 68.3 inches with a 95% confidence interval of 59.9 inches to 76.7 inches. If we were to continue drawing random samples of 100 adults, we would most likely arrive at different sample means and 95% confidence intervals for each instance.

For example, maybe a third sample is drawn with a sample mean of 67.9 inches and a 95% confidence interval of 59.5 inches to 76.3 inches, or a fourth sample is drawn with a sample mean of 66.1 inches and a 95% confidence interval of 57.7 inches to 74.5 inches. In fact, if we were to continue drawing samples to the point where we had 100 of them, we would expect the actual population mean to fall in the range of 95% of those samples.

As another example, suppose that we were interested in determining the mean number of cups of coffee that Americans drink per day, if we were to draw multiple random samples of, let's say, 250 people in an effort to estimate this mean. Let's suppose that the sample mean is 2.1 cups per day with a 95% confidence interval of 1.8 cups to 2.4 cups per day.

A second sample of 250 people yields a sample mean of 1.8 cups per day and a 95% confidence interval of 1.5 cups to 2.1 cups per day. If we were to repeat this process to the point of drawing 60 samples, we would expect that the actual population mean would fall in the range of 95% of those samples, or 57 of them. So let's go back to the objective just to make sure we covered what we said would.

By the end of this lesson, we wanted to be able to interpret a 95% confidence interval. And we did. We went through and used two different examples to illustrate what a 95% confidence interval means. So again, my name is Dan Laub. And hopefully, you've got some value from this lesson.