Attachment K - Census Calculations on Power

CPS ASEC Sample Size for Test Questions
Finding True Differences
DSMD’s Sample Size and True Difference Calculations
DSMD was asked to find the difference one would be able to detect for a test questionnaire given
certain known population proportions. The two proportions involving income and health insurance were
0.07189 and 0.03700, respectively. DSMD used the following formula to calculate sample size given a
margin of error – to find the margin of error, or true difference detectable, we used Excel’s Analytic
Solver:
√𝑝 ∗ (1 − 𝑝) ∗ 𝑧𝛼⁄2
𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒 = (
)
𝑒

2

𝑛
𝑆
𝑒 = 𝑚𝑎𝑟𝑔𝑖𝑛 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟 = 𝑧𝛼⁄2 √(1 − ) ∗
𝑁 √𝑛
𝑊ℎ𝑒𝑟𝑒 𝑆 = √𝑝 ∗ (1 − 𝑝)
Using the formulas laid out above, we are able to use the same means and sample sizes in the
table produced by Ben Sommers and Colleen Carey to show our true differences (margin of error, e); the
column True Difference is the true difference detectable with the given sample size:
Mean
3.7%
3.7%
15.0%
15.0%
15.0%
15.0%
65.0%
65.0%
65.0%
65.0%

Given Sample
5,000
13,600
5,000
8,200
13,600
39,000
5,000
13,600
15,200
39,000

True Difference (e)
0.439%
0.266%
0.831%
0.649%
0.504%
0.297%
1.110%
0.673%
0.636%
0.397%