We are asked to suppose that a random sample of size n is required to produce a margin of error of ± E. Let's recall that a margin of error is one over the square root of the sample size n.
± E = ± 1/sqrt(n) ⇕ E=1/sqrt(n)Now, if we want to reduce the margin of error to ± 12E we should change the sample size n. To determine how, let's use the definition that we recalled.
± 1/2E = ± 1/21/sqrt(n) ⇕
1/2E = 1/2* 1/sqrt(n)
Let's simplify the right-hand side of the equation. Our goal is to express the margin of error as one fraction where the numerator will be 1 and the denominator will be the square root of the new sample size.
This means that if we want to reduce the margin of error 2 times, we need to increase the sample size 4 times. Remember that the more accurately we want our results to represent the population, the greater the sample should be.
