Confidence Interval for the Mean

Prerequisites
Areas Under Normal Distributions, Sampling Distribution of the Mean, Introduction to Estimation, Introduction to Confidence Intervals

Learning Objectives

  1. Use the normal distribution calculator to find the value of z to use for a confidence interval
  2. Compute a confidence interval on the mean when σ is known
  3. Determine whether to use a t distribution or a normal distribution
  4. Compute a confidence interval on the mean when σ is estimated

When you compute a confidence interval, you compute the mean of a sample in order to estimate the mean of the population. Clearly, if you already knew the population mean, there would be no need for a confidence interval.

Recall from the section on the sampling distribution of the mean that the mean of the sampling distribution is μ and the standard error of the mean is

In general, you compute the 95% confidence interval for the mean with the following formula:

Lower limit = M - Z.95σm
Upper limit = M + Z.95σm

where Z.95 is the number of standard deviations extending from the mean of a normal distribution required to contain 0.95 of the area and σm is the standard error of the mean.

If you look closely at this formula for a confidence interval, you will notice that you need to know the standard deviation (σ) in order to estimate the mean. This may sound unrealistic, and it is. However, computing a confidence interval when σ is known is easier than when σ has to be estimated, and serves a pedagogical purpose. Later in this section we will show how to compute a confidence interval for the mean when σ has to be estimated.

When the variance is not known but has to be estimated from sample data you should use the t distribution rather than the normal distribution. When the sample size is large, say 100 or above, the t distribution is very similar to the standard normal distribution. However, with smaller sample sizes, the t distribution is leptokurtic, which means it has relatively more scores in its tails than does the normal distribution. As a result, you have to extend farther from the mean to contain a given proportion of the area. Recall that with a normal distribution, 95% of the distribution is within 1.96 standard deviations of the mean. If you have a sample size of only 5, 95% of the area is within 2.78 standard deviations of the mean. Therefore, the standard error of the mean would be multiplied by 2.78 rather than 1.96.

The values of t to be used in a confidence interval can be looked up in a table of the t distribution.

You can also use the "inverse t distribution" calculator to find the t values to use in confidence intervals. You will learn more about the t distribution in the next section.

More generally, the formula for the 95% confidence interval on the mean is:

Lower limit = M - (tCL)(sm)
Upper limit = M + (tCL)(sm)

where M is the sample mean, tCL is the t for the confidence level desired (0.95 in the above example), and sm is the estimated standard error of the mean.