Question

Determining Sample Size. In Exercises 31–38, use the given data to find the minimum sample size required to estimate a population proportion or percentage.

Women who give birth An epidemiologist plans to conduct a survey to estimate the percentage of women who give birth. How many women must be surveyed in order to be 99% confident that the estimated percentage is in error by no more than two percentage points?

a. Assume that nothing is known about the percentage to be estimated.

b. Assume that a prior study conducted by the U.S Census Bureau showed that 82% of women give birth.

c. What is wrong with surveying randomly selected adult women?

Short answer

a. Assuming that nothing is known about the percentage of women who give birth, the required sample size is 4147.

b. Assuming that a sample proportion of 82% women give birth, the required sample size is 2448.

c. If women are randomly selected, the sample will include women of a very young age who won’t give birth until a couple of years and older women who have passed the age of reproducing. Thus, to correctly understand the proportion of women who give birth, the sample should be representative of adult women who can bear children.

Step-by-step solution

Given information

The percentage of women who give birth is to be estimated.

The sample size needs to be determined. The following values are given:

The margin of error is equal to 0.02 (two percentage points).

The confidence level is equal to 99%.

Finding the sample size when the sample proportion is not known

a.

Let$\hat{p}$ denote the sample proportion of women who give birth.

Let $\hat{q}$denote the sample proportion of women who do not give birth.

Here, nothing is known about the sample proportions.

The formula for finding the sample size is as follows:

$n=\frac{{\left[z_{\frac{\alpha }{2}}\right]}^{2}0.25}{E^2}$

The confidence level is equal to 99%. Thus, the level of significance is equal to 0.01.

The value of $z_{\frac{\alpha }{2}}$for $\alpha =0.01$from the standard normal table is equal to 2.5758.

Substituting the required values, the following value of the sample size is obtained:

$$\begin{gathered} n=\frac{{\left(2.5758\right)}^2\times 0.25}{{\left(0.02\right)}^2} \\ =4146.72 \\ \approx 4147 \end{gathered}$$

Hence, the required sample size is equal to 4147.

Finding the sample size when the sample proportion is known

b.

The value of$\hat{p}$ is given to be equal to:

$$\begin{gathered} \hat{p}=82\% \\ =\frac{82}{100} \\ =0.82 \end{gathered}$$

Thus, the value of is computed below:

$$\begin{gathered} \hat{q}=1-\hat{p} \\ =1-0.82 \\ =0.18 \end{gathered}$$

The formula for finding the sample size is as follows:

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

By substituting the required values, the following value of the sample size is obtained:

$$\begin{gathered} n=\frac{{\left(2.5758\right)}^2\times 0.82\times 0.18}{{\left(0.02\right)}^2} \\ =2448.22 \\ \approx 2448 \end{gathered}$$

Hence, the required sample size is equal to 2448.

Comparison

c.

It is known that women bear children after a certain age and until their menopause.

A randomly selected sample of women will also include younger women who won’t go for a baby until several years and women who have passed the age of reproduction.

Therefore, to accurately estimate the proportion of women who give birth, the population should include only adult women.