Question

Calculate the minimum number of subjects needed for a research study regarding the proportion of respondents who reported a history of diabetes using the following criteria:
95% confidence, within 5 percentage points, and a previous estimate is not known.

Answer 1

Answer:

The minimum number of subjects needed is 385.

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of $\pi$, and a confidence level of $1-\alpha$, 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 $1 - \frac{\alpha}{2}$.

The margin of error is:

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

95% confidence level

So $\alpha = 0.05$, z is the value of Z that has a p-value of $1 - \frac{0.05}{2} = 0.975$, so $Z = 1.96$.

95% confidence, within 5 percentage points, and a previous estimate is not known.

The sample size is n for which M = 0.05. We don't know the true proportion, so we use $\pi = 0.5$

Then

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

$0.05 = 1.96\sqrt{\frac{0.5*0.5}{n}}$

$0.05\sqrt{n} = 1.96*0.5$

$\sqrt{n} = \frac{1.96*0.5}{0.05}$

$(\sqrt{n})^2 = (\frac{1.96*0.5}{0.05})^2$

$n = 384.16$

Rounding up:

The minimum number of subjects needed is 385.