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 difference between two sample means

Plotting sample differences against frequency gives a normal distribution with mean equal to
difference between two population means which is the difference between the two population means.

Variance

The variance of the distribution of the sample differences is equal to (sigma1-squared /n1 ) + (sigma2-squared /n2 ).  
Therefore, the standard error of the differences between two means would be equal to standard error formula.

Converting to a z score
To convert to the standard normal distribution, we use the formula,   z score 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, mu1 and , and variances, sigma1-squared and sigma2-squared, respectively, the sampling distribution of the difference, x-bar1 - x-bar2, between the means of independent samples of size n1 and n2 drawn from these populations is normally distributed with mean, mu1 - mu2, and variance, (sigma1-squared /n1 ) + (sigma2-squared /n2 ).

Example

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

Population 1:  n1 = 40, x1 = $346
Population 2:  n2 = 35, x2 = $300

If the variances of the populations are sigma1 squared = 2800 and sigma2-squared = 3250, what is the probability of obtaining sample results (x1 - x2) as large as those shown if there is no difference in the means of the two populations?

Solution

(1) Write the given information

        n1 = 40,  = $346, sigma1 squared = 2800
        n2 = 35, x2 = $300, sigma2 squared = 3250

(2) Sketch a normal curve

       normal curve 
(3) Find the z score

        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 x1x2 is as large as given is .0002.