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
 Compute the probability of a sample mean being at least as high as
a specified value when σ is known
 Compute a twotailed probability
 Compute the probability of a sample mean being at least as high as
a specified value when σ is estimated
 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 realworld 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
 Each value is sampled independently from each
other value.
 The values are sampled from a normal distribution.
