Question

In a study to find the proportion of children with asthma in a certain population, we desire a 95% confidence interval with standard error no more than 0.05. Without any prior knowledge of the proportion, what should be our minimum sample size

Answer 1

Answer:

The minimum sample size should be of 381.

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 zscore that has a pvalue of $1 - \frac{\alpha}{2}$.

The margin of error is given by:

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

95% confidence level

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

Without any prior knowledge of the proportion, what should be our minimum sample size

We dont know the population proportion, so we use $\pi = 0.5$, which is when the largest sample size is needed. We have to find n for which $M = 0.05$. So

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

$0.05 = 1.95\sqrt{\frac{0.5*0.5}{n}}$

$0.05\sqrt{n} = 1.95*0.5$

$\sqrt{n} = \frac{1.95*0.5}{0.05}$

$(\sqrt{n})^2 = (\frac{1.95*0.5}{0.05})^2$

$n = 380.25$

Rounding up

The minimum sample size should be of 381.