Finding Confidence Intervals with the Normal Distribution

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  • 0:01 Standard Deviation
  • 0:36 Cases Where This Applies
  • 1:25 Calculating the…
  • 2:05 Example
  • 4:20 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 to construct a confidence interval when the population's standard deviation is known and the population is normally distributed.

Standard Deviation

Reality often differs from theory; in the real world we seldom know what the true population standard deviation is. The standard deviation refers to the variability of individual observations around their mean. But for this lesson we are going to pretend that the population standard deviation denoted by the symbol sigma, is known to us and we are going to use that to help us construct the confidence interval for the population mean, which itself is denoted by the symbol mu.

A confidence interval is a range of values that expresses the uncertainty associated with a parameter, like the population mean.

Cases Where This Applies

There are three possible cases where this can be applied.

Case 1: the population standard deviation is known. The sample size is small (n<30). In other words, n, the sample size, is less than 30. And, the population is normally distributed.

Case II: Again, the population standard deviation is known. But, this time around, the sample size is large (n>=30). This means that n is greater than or equal to 30.

In Case III: Again, the population standard deviation is known. The sample size is small (n<30). And, the population is not normally distributed or we don't know its distribution.

The third case uses nonparametric methods to find the confidence interval for mu, meaning we use inferential methods that are not concerned with parameters- such as the population mean or population standard deviation.

Calculating the Confidence Interval

In this lesson, we are going to focus on the first two cases where we use the normal distribution to make the confidence interval for mu. In the first two cases, we would calculate the confidence interval for mu using the following equations:

  • Where x bar denotes the value of the sample mean
  • Sigma refers to the population standard deviation
  • And n refers to the sample size

The value for z is found from standard normal distribution tables for a given confidence level right here. The quantity of z times sigma x bar is the margin of error and it is denoted by the symbol E. In other words, E = z times (x) sigma x bar


Simply put, the margin of error (E) is the quantity we subtract or add to x bar to obtain a confidence interval for mu. Let's build on this to solidify your knowledge of all this crazy terminology with an actual example.

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