Using the t Distribution to Find Confidence Intervals

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  • 0:01 Sample Standard Deviation
  • 0:31 Case Scenarios
  • 1:36 The T Distribution
  • 2:51 Calculating Confidence…
  • 3:47 Example
  • 6:01 Lesson Summary
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Lesson Transcript
Instructor: Artem Cheprasov

Artem has a doctor of veterinary medicine degree.

In this lesson, you're going to learn how we find confidence intervals for normally distributed populations where the population standard deviation is not known. Work through the sample, then test your understanding with a brief quiz.

Sample Standard Deviation

More than likely, we will not have the privilege of knowing the entire population's standard deviation, which is denoted by the symbol sigma. The standard deviation is the variability of individual observations around their mean. Therefore, we pretty much always use the sample standard deviation, denoted by the symbol s, to calculate the confidence interval for a population mean (mu).

Let's see how we do this with an example.

Case Scenarios

Before we work out the example, we need to consider the fact that there are three possible case scenarios we might encounter where we need to construct a confidence interval. They are as follows:

Case I:

  • The population standard deviation is not known
  • The sample size is small, meaning n (the sample size) is less than 30
  • The population is normally distributed

Case II:

  • The population standard deviation is not known
  • The sample size is large; in other words, n is greater than or equal to 30

Case III:

  • The population standard deviation is not known
  • The sample size is small; ergo n < 30
  • The population is not normally distributed, or we don't know its distribution

In case III, we would need to use nonparametric methods to figure out the confidence interval for the population mean, mu. In cases I and II, we would use the t distribution to construct the confidence interval for mu. It is these two cases that will be the focus of this lesson.

The T Distribution

The t distribution, aka the Student's t distribution is a kind of symmetric, bell-shaped distribution curve that has a lower height but a wider spread than the standard normal distribution curve. We can basically say that because the t distribution is flatter than the standard normal distribution curve it has a larger standard deviation. By the way, just in case you were wondering, the name comes from Student, a pseudonym used by W.S. Gosset, the man who developed the t-distribution.

Anyway, the actual shape of a t distribution curve really depends on something known as the degrees of freedom (df), which are calculated as n - 1, where n is our sample size. As the sample size in question becomes larger and larger, the t distribution will actually approach the standard normal distribution.

There are several things you should know about the t distribution:

  • The total area under its curve is 1.0 (or 100%)
  • The curve never touches the horizontal axis
  • The mean of the t distribution is 0
  • The standard deviation is equal to the square root of df / (df - 2)

Calculating Confidence Intervals

When we don't know the value of sigma, the population standard deviation, we use the sample standard deviation (s) instead. This means we use the following equation to find the standard deviation of x-bar (the sample mean):

sx-bar = s / √ n

The confidence interval for mu is found using:

x-bar +/- t * sx-bar

The value of t is found from a t distribution table using n - 1 degrees of freedom and the appropriate confidence level in this table.

t * sx-bar is really something known as the margin of error, and is otherwise labeled as E. Meaning:

E = t * sx-bar = margin of error


Let's work through an example together.

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