6.7.3. Variance
6.7.3.1. Sample Size for Variance
The relationship between the estimates of sample and population variances is given as follows:
Here:
· is the estimate of the population variance,
· is the sample variance with ν degrees of freedom,
· α is the probability of committing a Type I error and 1 α is the confidence level,
· β is the probability of committing a Type II error and 1 – β is the power of the test,
Since critical values from chisquare distribution are both dependent on the sample size n, an iterational algorithm should be employed.
The user is expected to enter:
· Sample Variance
· Population Variance
· Power of the test
· Confidence Level
and the program will output the estimated sample size.
Example
Example 7.12 on p. 124 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → Variance and the Sample Size option. Enter the following data at the next dialogue:
Sample Size and Power Estimation: Variance
Sample Size
Sample Variance = 
2.6898 
Population Variance = 
1.5000 
Power of the Test = 
0.9000 
Confidence Level = 
0.9500 
Estimated Sample Size = 
50.7813 
6.7.3.2. Power of the Test for Variance
Power of the test is defined as the following probability:
Since the first chisquare value on the right hand side of this equation depends on β, an iterational algorithm should be employed.
The user is expected to enter:
· Sample Size
· Sample Variance
· Population Variance
· Confidence Level
and the program will output the estimated chisquare statistic and its pvalue, β.
Example
The example on p. 124 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → Variance and the Power of the Test option. Enter the following data at the next dialogue:
Sample Size and Power Estimation: Variance
Power of the Test
Sample Size = 
8.0000 
Sample Variance = 
2.6898 
Population Variance = 
1.5000 
Confidence Level = 
0.9500 
Power of the Test: 

ChiSquare Statistic = 
7.8447 
RightTail Probability = 
0.3465 