by what factor should the sample size in a poll be increased in order to cut the margin of error in half
To understand how to calculate the increase in sample size to halve the margin of error in a poll, let's start by looking at the formula for the margin of error (MOE) in a sample proportion.
The margin of error can be calculated using the formula:
MOE = Z * sqrt((p * (1 - p)) / n)
In this formula:
- Z represents the z-score that corresponds to the desired level of confidence. Typically, a z-score of 1.96 is used for a 95% confidence level.
- p represents the estimated proportion or percentage of the population you are trying to measure.
- n represents the sample size.
Now, let's assume that you have a current sample size of n and you want to reduce the margin of error (MOE) by half. We'll call the new sample size N.
To calculate the increase in sample size, we need to rearrange the formula to solve for the new sample size N:
MOE_1/2 = Z * sqrt((p * (1 - p)) / N)
Since we want to halve the MOE, we can rewrite the equation as:
(MOE / 2) = Z * sqrt((p * (1 - p)) / N)
To simplify the equation further, recall that the square root of 1/2 is approximately 0.7071. Thus, we can rewrite the equation as:
(MOE / 2) = Z * (0.7071) * sqrt((p * (1 - p)) / N)
Now, we can solve for N:
N = ((Z * sqrt((p * (1 - p))) / (MOE / 2))^2
By substituting the desired values for the level of confidence (Z), the estimated proportion (p), and the current margin of error (MOE) into the equation, you can calculate the required increase in sample size (N) to halve the margin of error.
Remember that increasing the sample size will require more resources and time, so it's important to weigh the benefits of reducing the margin of error against the practical constraints of conducting a larger poll.