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.

Astrology

A sociologist plans to conduct a survey to estimate the percentage of adults who believe in astrology. How many people must be surveyed if we want a confidence level of 99% and a margin of error of four percentage points?

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

b. Use the information from a previous Harris survey in which 26% of respondents said that they believed in astrology.

Short answer

a. The required sample size when the value of the sample proportion is not known is equal to 1037.

b. The required sample size when the value of the sample proportion is equal to 0.26 (26%) is equal to 798.

Step-by-step solution

Given information

The percentage of adults who believe in astrology is to be estimated.

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

The margin of error is equal to 0.04.

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 adults who believe in astrology.

Let $\hat{q}$denote the sample proportion of adults who do not believe in astrology.

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:

$$\begin{gathered} n=\frac{{\left(2.5758\right)}^2\times 0.25}{{\left(0.04\right)}^2} \\ =1036.68 \\ \approx 1037 \end{gathered}$$

Hence, the required sample size is equal to 1037.

Finding the sample size when the sample proportion is known

b.

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

$$\begin{gathered} \hat{p}=26\% \\ =\frac{26}{100} \\ =0.26 \end{gathered}$$

Thus, the value of $\hat{q}$is computed below:

$$\begin{gathered} \hat{q}=1-\hat{p} \\ =1-0.26 \\ =0.74 \end{gathered}$$

The formula for finding the sample size is as follows:

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

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

$$\begin{gathered} n=\frac{{\left(2.5758\right)}^2\times 0.26\times 0.74}{{\left(0.04\right)}^2} \\ =797.83 \\ \approx 798 \end{gathered}$$

Hence, the required sample size is equal to 798.