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semesters > winter 2021 > mth208 > week 8

Introductory Statistics (openstax.org)

Section 8.3 #117

Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car.

(a) When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 95% confident that the population proportion is estimated to within \(0.03\)?

(b) If it were later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Why?

Solution

(a)
The Error Bound for Proportions: \[ EBP=z_{\alpha/2}\cdot\left(\sqrt{\frac{p'q'}{n}}\right) \] where

\(z_{\alpha/2}\) is the critical value associated with the confidence level \(1-\alpha\),
\(p'\) is the sample proportion of successes, and
\(q'=1-p'\) is the sample proportion of failures,

can be solved for \(n\): \[ n=z_{\alpha/2}^{2}\cdot\left(\frac{p'q'}{EBP^{2}}\right) \] Using the maximum value of \(p'q'\) which is \(\displaystyle{\frac{1}{4} \mbox{ when } p'=q'=\frac{1}{2}}\), we can identify an \(n\) that will give us an appropriate confidence level: \begin{equation} \begin{split} n & = z_{\alpha/2}^{2}\cdot\left(\frac{p'q'}{EBP^{2}}\right)\\ & \\ & \ge \frac{1}{4}\left(\frac{z_{\alpha/2}^{2}}{EBP^{2}}\right)\\ & \\ & = \left(\frac{z_{\alpha/2}}{2\,EBP}\right)^{2} \end{split} \end{equation} If we want a \(95\%\)-confidence interval of with a population proportion estimate within \(0.03\), then we will need \( z_{\alpha/2}=z_{0.025}=1.9600 \) and \( EBP=0.03 \). Now we can find an appropriate value for \( n \): \begin{equation} \begin{split} n & \ge \left(\frac{z_{\alpha/2}}{2\,EBP}\right)^{2} \\ & \\ & = \left(\frac{1.9600}{2(0.03)}\right)^{2} \\ & \\ & \approx 1067.11 \ldots \end{split} \end{equation} and \(n=1068\) is an appropriate choice.

(b)
If we want to be more than \(95\%\) confident (\(\alpha=0.05\)), we will need to lower \(\alpha\), which will increase \(z_{\alpha/2}\). If we want the population proportion to still be estimated to within \(0.03\), we would need to increase the sample size \(n\) to compensate for the increase in \(z_{\alpha/2}\).