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Means: paired samples

Читайте также:
  1. Confidence interval for : small samples
  2. Confidence interval for the difference between the population means: unknown population variances that are assumed to be equal
  3. Confidence intervals for population proportion: Large samples
  4. Population variance unknown. Small samples
  5. Tests based on independent samples
  6. Tests based on independent 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              
After              

 

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
    -2  
   

 

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.

 


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Читайте в этой же книге: Confidence interval and confidence level | Normally distributed: population variance known | Exercises | Normally distributed: large sample size | Exercises | Answers | Student’s t distribution | Confidence interval for : small samples | Exercises | Confidence intervals for population proportion: Large samples |
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