6.7.6. Two Proportions
6.7.6.1. Sample Size for Two Proportions
One sample size (n_{1}) can be estimated assuming that the ratio of the two sample sizes (r = n_{2}/n_{1}) is known. The algorithm for power of the test for two proportions (see next section) is also used here, employing a halfinterval search method.
The user is asked to enter:
· Proportion 1
· Proportion 2
· Power of the test
· Confidence Level
· 1 or 2 tailed test
· n_{2}/n_{1}
and the program will output the estimated sample size (n_{1}).
Example
Example 24.26 on p. 561 from Zar, J. H. (1999). Select Statistics 1 → Sample Size and Power Estimation → Two Proportions and the Sample Size option. Enter the following data at the next dialogue:
Sample Size and Power Estimation: Two Proportions
Sample Size
Proportion 1 = 
0.4500 
Proportion 2 = 
0.2500 
Power of the Test = 
0.9000 
Confidence Level = 
0.9500 
1 or 2 Tailed Test = 
2.0000 
N2 / N1 = 
1.0000 
Estimated Sample Size = 
127.2343 
Next, click [Last Procedure Dialogue], enter n_{2} / n_{1} = 3 and click [Finish] again:
Sample Size and Power Estimation: Two Proportions
Sample Size
Proportion 1 = 
0.4500 
Proportion 2 = 
0.2500 
Power of the Test = 
0.9000 
Confidence Level = 
0.9500 
1 or 2 Tailed Test = 
2.0000 
N2 / N1 = 
3.0000 
Estimated Sample Size = 
83.1527 
n_{2} is found as 3 x 83.1527 = 249.4581.
6.7.6.2. Power of the Test for Two Proportions
Power of the test in comparing two proportions is given by the following formula:
where:
· is the (one or) twotailed critical value from the standard normal distribution for Type I error probability α.
The user is expected to enter:
· Sample Size 1
· Sample Size 2
· Proportion 1
· Proportion 2
· Confidence Level
· 1 or 2 tailed test
and the program will output the two Zstatistics A and B and the power of the test computed from the above equation. For a onetailed test the power of the test is computed from:
Example
Example 24.16 on p. 554 from Zar, J. H. (2010). Select Statistics 1 → Sample Size and Power Estimation → Two Proportions and the Power of the Test option. Enter the following data at the next dialogue:
Sample Size and Power Estimation: Two Proportions
Power of the Test
Sample Size 1 = 
50.0000 
Sample Size 2 = 
45.0000 
Proportion 1 = 
0.7500 
Proportion 2 = 
0.5000 
Confidence Level = 
0.9500 
1 or 2 Tailed Test = 
2.0000 
Power of the Test: 

2Tail Probability = 
0.6402 