## Distribution
of the Difference between Two Means

*This file is part of a program based on the
Bio 4835 Biostatistics class taught at Kean
University in Union, New Jersey.
The course uses the following text:*

Daniel, W. W. 1999. Biostatistics: a foundation for analysis in the
health sciences. New York:
John Wiley and Sons.

The file follows this text very closely and readers are encouraged to consult
the text for further information.

**(B) Distribution of the difference
between two means**

It often becomes important to compare two population means. Knowledge of
the sampling distribution of the difference between two means is useful in
studies of this type. It is generally assumed that the two populations
are normally distributed.

*Sampling distribution of *

Plotting sample differences against frequency gives a normal distribution with
mean equal to

which is the
difference between the two population means.

*Variance*

The variance of the distribution of the sample differences is equal to ( / ) + ( / ).

Therefore, the standard error of the differences between two means would be
equal to .

*Converting to a z score*

To convert to the standard normal distribution, we use the formula, . We find the z score by assuming that there is no difference between the
population means.

*Sampling from normal populations*

This procedure is valid even when the population
variances are different or when the sample sizes are different. Given two
normally distributed populations with means, and , and
variances, and , respectively, the sampling distribution of the difference, , between the means of independent samples
of size and drawn from
these populations is normally distributed with mean, , and variance, ( / ) + ( / ).

**Example**

In a study of annual family expenditures for general health care, two
populations were surveyed with the following results:

Population 1: =
40, = $346

Population 2: =
35, = $300

If the variances of the populations are = 2800 and = 3250, what is the probability of obtaining
sample results ( - ) as large
as those shown if there is no difference in the means of the two populations?

Solution

(1) Write the given information

= 40, = $346, = 2800

= 35, =
$300, = 3250

(2) Sketch a normal curve

(3) Find the z score

(4) Find the appropriate value(s) in the table

A value of z = 3.6 gives an area of .9998. This is
subtracted from 1 to give the probability

P (z > 3.6) = .0002

(5) Complete the answer

The probability that - is as
large as given is .0002.