Question

use the given data to find the minimum sample size required to estimate the population proportion. margin of error: 0.028, confidence level: 99%; p and q unknown

a. 1116
b. 2223
c. 2115
d. 1939

Explanation:

Step1: Determine the z - score

For a 99% confidence level, the z - score \(z_{\alpha/2}\) is 2.576.

Step2: Assume \(p = q=0.5\)

When \(p\) and \(q\) are unknown, we use \(p = q = 0.5\) (this gives the maximum value of \(pq\)).

Step3: Use the formula for sample size

The formula for the sample size \(n\) to estimate a population proportion is \(n=\frac{z_{\alpha/2}^{2}\times pq}{E^{2}}\).
Substitute \(z_{\alpha/2} = 2.576\), \(p = 0.5\), \(q = 0.5\), and \(E=0.028\) into the formula:

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Since we can't have a fraction of a sample, we round up to \(n = 2115\) (due to the nature of sample - size calculation, when the decimal is non - zero, we round up to the next whole number).

Answer:

C. 2115