Determine Sample Size: Population Proportion - More on Gender Equality

Before we conduct and experiment to collect data for estimating parameters we consider the accuracy of our estimate.   We can change the accuracy via two variables:  confidence level and sample size.  Thus, for a level of accuracy we want a specific sample size.  Recall the Margin of Error for the  estimate of a population proportion:

$E=z_{\frac{\alpha}{2}}\cdot\sqrt{\frac{\hat{p}\hat{q}}{n}}$.  

For a selected level of confidence and desired margin of error we can find the needed sample size.

Let's derive the formula for n.

$E=z_{\frac{\alpha}{2}}\cdot\sqrt{\frac{\hat{p}\hat{q}}{n}}$

$\frac{E}{z_{\frac{\alpha}{2}}}=\sqrt{\frac{\hat{p}\hat{q}}{n}}$

$\left(\frac{z_{\frac{\alpha}{2}}}{E}\right)^2=\frac{n}{\hat{p}\hat{q}}$

$n=\hat{p}\hat{q}\left(\frac{z_{\frac{\alpha}{2}}}{E}\right)^2$

Round UP the resulting n.

In the absence of a previous estimate of $\hat{p}$ use $\hat{p}=\frac{1}{2}$.

Example

The UN (United Nations) routinely conduct studies relating to issues of gender equality.  This is accomplished by measuring several social characteristics of a nation.  Access to family planning by a women 15 to 49 years old is one such characteristic.  Suppose a goal of a study is to estimate the proportion of women with access to such resources in a country.  The goal of the study to estimate this proportion to within 3% of the true value.  The confidence level is at 90%.  A similar study from several years ago was conducted, and the proportion women with access to family planning from that study is 77.7%.  What is the necessary sample size to achieve the above margin of error?

Solution

We will use the equation for n derived above.

$n=\hat{p}\hat{q}\left(\frac{z_{\frac{\alpha}{2}}}{E}\right)^2$

$\hat{p}=0.777, \hat{q}=1-\hat{p}=1-0.777=0.223, z_{\frac{\alpha}{2}}=1.645, E=0.03$

$n=0.777\cdot0.223\cdot\left(\frac{1.645}{0.03}\right)^2$

$n=520.97$

Rounded up n=521.  We need a sample of size 521 to achieve the desired level of accuracy.