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 = 1037 (round up to the nearest whole number as needed.)
(b) what is the minimum sample size needed using a prior study that found that 28% of motor vehicle fatalities that were caused by alcohol - impaired driving?
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 \(E = 0.04\).

Step2: Use the formula for sample size when a prior estimate is available

The formula for sample size \(n\) when estimating a proportion is \(n=\dfrac{z_{\alpha/2}^{2}\cdot p(1 - p)}{E^{2}}\). Here, \(p = 0.28\) (from the prior study), \(1-p=1 - 0.28=0.72\), \(z_{\alpha/2}=2.576\), and \(E = 0.04\).

Substitute the values into the formula:

$$
LATEXBLOCK0
$$

Answer:

\(n = 837\) (Since we round up to the nearest whole number)