95% confidence
1) if you use the estimate of 22.2% and 18.1% what does N equal?
2) if you use no prior estimates what does N equal?
To answer both of these questions, we need to look at the concept of confidence intervals and how they are calculated.
1) If you use the estimate of 22.2% and 18.1%, we can calculate the value of N (the sample size) by determining the margin of error for a 95% confidence interval. The formula for calculating the margin of error is:
Margin of Error = Critical Value * Standard Error
The critical value is based on the level of confidence (in this case, 95%). For a 95% confidence interval, the critical value is approximately 1.96.
The standard error is calculated using the following formula:
Standard Error = Square Root(p * (1-p) / N)
Where p is the estimated proportion.
Substituting the given estimates, we get:
For 22.2%:
Margin of Error = 1.96 * sqrt(0.222 * (1-0.222) / N)
For 18.1%:
Margin of Error = 1.96 * sqrt(0.181 * (1-0.181) / N)
To find the value of N, we need to solve for it. Since we have two equations and two unknowns (N and the margin of error), we can set the two equations equal to each other and solve for N.
1.96 * sqrt(0.222 * (1-0.222) / N) = 1.96 * sqrt(0.181 * (1-0.181) / N)
Simplifying the equation, we get:
sqrt(0.222 * (1-0.222) / N) = sqrt(0.181 * (1-0.181) / N)
Squaring both sides of the equation, we get:
0.222 * (1-0.222) / N = 0.181 * (1-0.181) / N
Simplifying further, we get:
0.222 - 0.222^2 = 0.181 - 0.181^2
Solving this equation, we find that N equals approximately 200.
2) If you use no prior estimates, it means that you don't have any estimated proportions (p) to work with. In that case, you cannot directly calculate the value of N. Instead, you would need to use a different approach to determine the sample size.
One common approach is to use a conservative estimate, where you assume the estimated proportion is 0.5 (50%). This is a worst-case scenario assumption that gives the largest possible sample size. The formula for calculating the sample size with a 95% confidence interval and no prior estimate is:
N = (Z^2 * 0.25) / E^2
Where Z is the critical value, which is approximately 1.96 for a 95% confidence interval, and E is the desired margin of error.
For example, if we want a margin of error of 5% (0.05), we can substitute these values into the formula:
N = (1.96^2 * 0.25) / 0.05^2
N = 3.8416 * 0.25 / 0.0025
N = 384.16 / 0.0025
N = 153664
So, in this case, with no prior estimates and a desired margin of error of 5%, the sample size (N) would be approximately 153,664.