Sampling Distributions for Proportions Question 6

The manager of a car company will select a random sample of its customers to create a 90 percent confidence interval to estimate the proportion of its customers who have children. Of the following, which is the smallest sample size that will result in a margin of error of no more than 6 percentage points?

100

150

200

275

Approach

\text{Confidence Interval} : \text{statistic} \pm (\text{critical value})(\text{standard error of statistic})

To be within 6 percentage points, we need to find the critical value and standard error. The critical value can be found using the z table. A 90% confidence interval corresponds to a 0.05 tail probability, which correspoinds to a z-score of 1.645.

The standard error of the the sample statistic for population proportion can be found in the formula sheet. Putting it all together:

0.06 \geq (1.645)\left(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\right)

We assume a probability of 0.5 to anticipate the maximum margin of error.

0.06 \geq (1.645)\left(\sqrt{\frac{(0.5)(0.5)}{n}}\right)

\frac{0.06}{1.645}\geq \frac{0.5}{\sqrt{n}}

\sqrt{n}\geq \frac{(0.5)(1.645)}{0.06}

n \geq \left(\frac{(0.5)(1.645)}{0.06}\right)

n \geq 187.918

n= \boxed{200}