6.7. Confidence intervals for the difference between

means of two normal populations

Let be the mean of the first population and be the mean of the second population. Suppose we want to make a confidence interval for the difference between these two population means, that is, .

Let be the mean of a sample from the first population and be the mean

of a sample taken from the second population. Then is the sample statistic that is used to make an interval estimate. We will consider several cases.

6.7.1. Confidence intervals for the difference between

means: paired samples

In the case of two dependent samples, two data values-one in each sample- are collected from the same source and these are called paired or matched pairs.

Suppose that n matched pairs of observations, denoted by

, are selected from two populations with means and .

Our aim is to find confidence interval for .

To find interval estimation we apply following steps:

1. Find n differences

2. Find

3. Calculate

If the population distribution of differences is assumed to be normal, then

confidence interval for the difference between means is given by

where is the number for which

The random variable has a Student’s t distribution with ( n -1 ) degrees of freedom.

Example:

A company claims that its special exercise program significantly reduces weight. A random sample of seven persons were put on exercise program. The following table gives the weights (in kg) of those seven persons before and after the program

Before	68	81	98	86	110	92	80
After	62	76	86	79	103	87	82

Make a 95 % confidence interval for the mean of the population paired differences. Assume that the population of paired differences is (approximately) normally distributed.

Solution:

Let d be the difference between the weights before and after the program.

The necessary calculations are shown in the following table

Before	After	Difference
68 81 98 86 110 92 80	62 76 86 79 103 87 82	6 5 12 7 7 5 -2	36 25 144 49 49 25 4

The values of and are calculated as follows:

.

Then

and

.

In the end, 90 % confidence interval for is

Thus, we can state with 90 % confidence that the mean difference between the weights before and after exercise program is between 2.6 and 8.82 kg.

Exercises

1. Find the following confidence interval for the difference between two population means assuming that the populations of paired differences are normally distributed

a) n =10 ; ; confidence level = 99 %

b) n =26 ; ; confidence level = 95 %

c) n =14 ; ; confidence level = 90 %

2. A company attempts to evaluate the potential for a new bonus plan by selecting a random sample of 5 salespersons to use the bonus plan for a week period. The weekly sales volumes before and after the bonus plan implementation shown below

Weekly sales Before	Weekly sales After
15 12 18 15 16	18 14 19 18 18

Construct a 90 % confidence estimate for the mean increase in weekly sales that can be expected if a new bonus plan is implemented.

3. A company claims that the course they offer significantly increases the writing speed of secretaries. The following table gives the scores of eight secretaries before and after they attended this course.

Before	81	75	89	91	65	70	90	69
After	97	72	93	110	78	69	115	75

Make a 90 % confidence interval for the mean of the population paired differences, where a paired differences is equal to the score before attending the course minus the score after attending the course.

4. A company sent 7 of its employees to attend a course in building self-confidence. The following table gives the scores of these employees before and after attending the course

Before	8	5	4	9	6	8	5
After	10	7	5	11	6	7	9

Construct a 95 % confidence interval for the mean of population paired differences where a paired difference is equal to the score of an employee before attending the course minus after attending this course.

Answers

1. a) (10.65; 36.55); b) (11.26; 15.14); c) (39.76; 52.64); 2. 1.40 to 3.00;

3. (-16.54; -3.21); 4.(-2.93; 0.07).