Question

Vlad the Impaler promotes many of his soldiers into generals (don't ask what happened to the past generals). Haley thinks the proportion is about 64%, but wants to estimate the proportion better by sampling soldiers until her margin of error for a 94% Confidence Interval is less than 0.008. How many soldiers should Haley sample?

Answer 1

Using the formula for the margin of error, it is found that Haley should sample 12,724 soldiers.

In a sample with a number n of people surveyed with a probability of a success of $\pi$, and a confidence level of $\alpha$, we have the following confidence interval of proportions.

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which

z is the z-score that has a p-value of $\frac{1+\alpha}{2}$.

The margin of error is of:

$M = z\sqrt{\frac{\pi(1-\pi)}{n}}$

Estimate of the proportion of 64%, hence $\pi = 0.64$.

94% confidence level

So $\alpha = 0.94$, z is the value of Z that has a p-value of $\frac{1+0.94}{2} = 0.97$, so $z = 1.88$.  

Margin of error of less than 0.008 is wanted, hence, we have to find n when M = 0.008.

$M = z\sqrt{\frac{\pi(1-\pi)}{n}}$

$0.008 = 1.88\sqrt{\frac{0.64(0.36)}{n}}$

$0.008\sqrt{n} = 1.88\sqrt{0.64(0.36)}$

$\sqrt{n} = \frac{1.88\sqrt{0.64(0.36)}}{0.008}$

$(\sqrt{n})^2 = (\frac{1.88\sqrt{0.64(0.36)}}{0.008})^2$

$n = 12723.84$

Rounding up, Haley should sample 12,724 soldiers.

A similar problem is given at brainly.com/question/25404151