Question

You want to create a $95\%$ confidence interval with a margin of error of no more than $0.05$ for a population proportion.

The historical data indicates that the population has remained constant at about $0.55$.

What is the minimum size random sample you need to construct this interval?

• A. $223$
• B. $324$
• C. $378$
• D. $381$

Answer 1

The correct option is D

$381$

Explanation for the correct option:

Determining the minimum size of the random sample:

Given that, Population or $p=0.55$, Error or $E=0.05$.

We know that $z$ score, for the given error is $1.96$

Since, We know that $E=z\times \sqrt{\frac{p(1-p)}{n}}$, where $n$ is the sample size.

Substituting the values we have,

$\Rightarrow 0.05=1.96\times \sqrt{\frac{0.55\times (1-0.55)}{n}}$

$\Rightarrow \frac{0.05}{1.96}=\sqrt{\frac{0.55\times (1-0.55)}{n}}$

Squaring both sides,

$\Rightarrow {\left(\frac{0.05}{1.96}\right)}^2={\left(\sqrt{\frac{0.55\times (1-0.55)}{n}}\right)}^2$

$\Rightarrow {\left(\frac{0.05}{1.96}\right)}^2=\frac{0.55\times (1-0.55)}{n}$

$\Rightarrow n=\left(0.55\times 0.45\right)\times {\left(\frac{1.96}{0.05}\right)}^2$

$$\begin{gathered} \Rightarrow n=380.31 \\ \Rightarrow n\cong 381 \end{gathered}$$

Hence, the correct answer is option $\left(D\right)$