Question

a researcher wishes to estimate, with 99% confidence, the population proportion of motor vehicle fatalities that were caused by alcohol - impaired driving. his estimate must be accurate within 4% of the population proportion.
(a) no preliminary estimate is available. find the minimum sample size needed.
(b) find the minimum sample size needed, using a prior study that found that 28% of motor vehicle fatalities that were caused by alcohol - impaired driving.
(c) compare the results from parts (a) and (b).
(a) what is the minimum sample size needed assuming that no prior information is available?
n = (round up to the nearest whole number as needed.)

Explanation:

Step1: Determine the z - value

For a 99% confidence level, the z - value \(z_{\alpha/2}\) is 2.576. The margin of error \(E = 0.04\). When no prior estimate is available, we use \(p=0.5\) (this gives the maximum value of \(p(1 - p)\)).

Step2: Use the formula for sample size

The formula for sample size when estimating a proportion is \(n=\frac{z_{\alpha/2}^{2}\times p(1 - p)}{E^{2}}\).
Substitute \(z_{\alpha/2}=2.576\), \(p = 0.5\), \(1-p=0.5\), and \(E = 0.04\) into the formula:

$$
LATEXBLOCK0
$$

Answer:

\(n = 1037\)