Differences between Means

Prerequisites
Sampling Distribution of Difference between Means, Confidence Intervals, Confidence Interval on the Mean

Learning Objectives

  1. State the assumptions for computing a confidence interval on the difference between means
  2. Compute a confidence interval on the difference between means
  3. Format data for computer analysis

It is much more common for a researcher to be interested in the difference between means than in the specific values of the means themselves.

The difference in sample means is used to estimate the difference in population means. The precision of the estimate is revealed by a confidence interval.

In order to construct a confidence interval, we are going to make three assumptions:

 

  1. The two populations have the same variance. This assumption is called the assumption of homogeneity of variance.
  2. The populations are normally distributed.
  3. Each value is sampled independently from each other value.

 

The consequences of violating these assumptions are discussed in a later section. For now, suffice it to say that small-to-moderate violations of assumptions 1 and 2 do not make much difference.

A confidence interval on the difference between means is computed using the following formula:

Lower Limit = M1 - M2 -(tCL)()
Upper Limit = M1 - M2 +(tCL)()

where M1 - M2 is the difference between sample means, tCL is the t for the desired level of confidence, and is the estimated standard error of the difference between sample means.

The first step is to compute the estimate of the standard error of the difference between means (). Recall from the relevant section in the chapter on sampling distributions that the formula for the standard error of the difference in means in the population is:


In order to estimate this quantity, we estimate σ2 and use that estimate in place of σ2. Since we are assuming the population variances are the same, we estimate this variance by averaging our two sample variances. Thus, the estimate of variance is computed using the following formula:

where MSE is the estimate of σ2.

=

The next step is to find the t to use for the confidence interval (tCL). To calculate tCL, we need to know the degrees of freedom. The degrees of freedom is the number of independent estimates of variance on which MSE is based. This is equal to (n1 -1) + (n2 -1) where n1 is the sample size for the first group and n2 is the sample size of the second group.When n1= n2, it is conventional to use "n" to refer to the sample size of each group.