Question

Refer to Exercise 55. Suppose that Gallup wanted to cut the margin of error in half from $3$percentage points to $1.5$percentage points. How should they adjust their sample size?

a. Multiply the sample size by $4$.

b. Multiply the sample size by $2$.

c. Multiply the sample size by $1/2$.

d. Multiply the sample size by $1/4$.

e. There is not enough information to answer this question.

Short answer

Option (a) is correct.

Step-by-step solution

Given Information

It is given that margin of error is reduced from $3$percentage points to$1.5$percentage points.

Concept Used 

Formula to be used is

$E=Z_{\alpha /2}\times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Calculation

Hence, margin of error

$\frac{E}{2}=\frac{1}{2}\times Z_{\alpha /2}\times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

$\frac{E}{2}=Z_{\alpha /2}\times \sqrt{\frac{1}{4}\times \frac{\hat{p}(1-\hat{p})}{n}}$

$\frac{E}{2}=Z_{\alpha /2}\times \sqrt{\frac{\hat{p}(1-\hat{p})}{4n}}$

So, sample size needs to be replaced by$4n$