This file is part of a program based on the Bio 4835 Biostatistics class taught at
Daniel, W. W. 1999. Biostatistics: a foundation for analysis in the health sciences.
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.
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, ( / ) + ( / ).
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?
(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.