The sample size needed to estimate the difference between two population proportions to within a margin of error E with a confidence level of 1- alpha can be found by using the equation shown below. Replace n1 and n2 by n and solve for n.
E= z alpha/2 square root p1q1/n1 + p2q2/n2
Solve for n and state the result. Show your work.
To solve for n, let's first simplify the equation:
E = z(alpha/2) * sqrt((p1q1/n1) + (p2q2/n2))
Where:
- E is the margin of error,
- z(alpha/2) is the z-score corresponding to the confidence level (1 - alpha),
- p1 and p2 are the proportions of the two populations,
- q1 and q2 are (1 - p1) and (1 - p2) respectively.
Now, let's square both sides of the equation:
E^2 = (z(alpha/2))^2 * ((p1q1/n1) + (p2q2/n2))
Rearrange the equation to isolate n:
n1 * ((p1q1/n1) + (p2q2/n2)) = n1 * E^2 / (z(alpha/2))^2
Simplify further by expanding the left side:
p1q1 + p2q2 = n1 * E^2 / (z(alpha/2))^2
Now, isolate n1:
n1 = (p1q1 + p2q2) * (z(alpha/2))^2 / E^2
Finally, substitute n1 with n:
n = (p1q1 + p2q2) * (z(alpha/2))^2 / E^2
This is the equation to find the required sample size (n) to estimate the difference between two population proportions within a margin of error (E) with a confidence level of (1 - alpha).