Testing a Single Mean

Prerequisites
Logic of Hypothesis Testing, Areas Under Normal Distributions, Sampling Distribution of the Mean, Introduction to Sampling Distributions, t Distribution

Learning Objectives

  1. Compute the probability of a sample mean being at least as high as a specified value when σ is known
  2. Compute a two-tailed probability
  3. Compute the probability of a sample mean being at least as high as a specified value when σ is estimated
  4. State the assumptions required for item 3 above.

This section shows how to test the null hypothesis that the population mean is equal to some hypothesized value.

The significance test consists of computing the probability of a sample mean differing from μ by one or more. The first step is to determine the sampling distribution of the mean. As shown in a previous section, the mean and standard deviation of the sampling distribution of the mean are

μM = μ

and

respectively.

Before normal calculators were widely available, probability calculations were made based on the standard normal distribution. This was done by computing Z based on the formula


where Z is the value on the standardized normal distribution, M is the sample mean, μ is the hypothesized value of the mean, and μM is the standard error of the mean.

In real-world data analysis it is very rare that you would know σ and wish to estimate μ.

To test this null hypothesis for difference scores, we compute t using a special case of the following formula

The special case of this formula applicable to testing a single mean is:

where t is the value of t we compute for the significance test, M is the sample mean, μ is the hypothesized value of the population mean, and sM is estimated standard error of the mean. Notice the similarity of this formula to the formula for Z.

In the previous example, we assumed the distribution of scores was normally distributed. In this case, it is the population of difference scores that we assume to be normally distributed.


Review of Assumptions

  1. Each value is sampled independently from each other value.
  2. The values are sampled from a normal distribution.