Question

use the given data to find the minimum sample size required to estimate the population proportion. margin of error: 0.008; confidence level: 98%; \\( \hat { p } \\) and \\( \hat { q } \\) unknown

a. 10,384
b. 20,308
c. 22,184
d. 21,207

Explanation:

Step1: Find the z - value

For a 98% confidence level, the significance level \(\alpha=1 - 0.98 = 0.02\). Then \(\alpha/2=0.01\).
Using the standard normal distribution table or a calculator, \(z_{\alpha/2}=z_{0.01}\approx2.33\).

Step2: Assume \(\hat{p}=\hat{q} = 0.5\)

When \(\hat{p}\) and \(\hat{q}\) are unknown, we use the formula \(n=\frac{z_{\alpha/2}^{2}\hat{p}\hat{q}}{E^{2}}\). Substituting \(\hat{p}=\hat{q} = 0.5\), \(z_{\alpha/2}=2.33\), and \(E = 0.008\) into the formula.
We get \(n=\frac{(2.33)^{2}\times0.5\times0.5}{(0.008)^{2}}\).
First, calculate \((2.33)^{2}=5.4289\), then \(n=\frac{5.4289\times0.25}{0.000064}\).
\(5.4289\times0.25 = 1.357225\).
Then \(n=\frac{1.357225}{0.000064}\approx21207\).

Answer:

D. 21,207