Final answer:
To determine the minimum sample size to estimate a population proportion with an unknown p and q, a 98% confidence level, and a 0.008 margin of error, we use a conservative p' and q' of 0.5. With a calculated sample size of 21,176, the closest matching provided answer is option A) 21,207.
Step-by-step explanation:
To find the minimum sample size required to estimate the population proportion with a given margin of error, a specified confidence level, and unknown p and q, we use the following formula:
Solving for n: n = (Z₂ᵢ / 2EBP²) (p'q')
When p and q are unknown, it's common to use the most conservative estimate, which is p' = q' = 0.5. For a 98% confidence level, the Z-score associated (Z₂ᵢ / 2) is approximately 2.33. Using the given margin of error EBP of 0.008, the formula becomes:
n = (2.33²)(0.5)(0.5) / 0.008²
Calculating this gives us a sample size of approximately:
n = (5.4289)(0.25) / 0.000064n = 1.357225 / 0.000064n = 21,175.39
We always round up because we can't have a fraction of a sample, so the minimum sample size is 21,176.
However, this calculated number doesn't perfectly match any of the options provided (A) 21,207 (B) 22,184 (C) 10,384 (D) 20,308, but it is closest to option (A) 21,207, which may suggest a rounding difference in the Z-score used or other factors not accounted for in the question details.