A researcher wishes to estimate the percentage of adults who support abolishing the penny. What size sample should be obtained if he wishes the estimate to be within 3 percentage points with $99 \%$ confidence if
(a) he uses a previous estimate of $24 \%$ ?
(b) he does not use any prior estimates?
Click here to view the standard normal distribution table (page 1). Click here to view the standard normal distribution table (page 2).
(a) $n=1345$ (Round up to the nearest integer.)
(b) $n=\square$ (Round up to the nearest integer.)
Answer
Final Answer: (a) The required sample size when using a previous estimate of 24% is \(\boxed{1345}\). (b) The required sample size when not using any prior estimates is \(\boxed{1844}\).
Step 1 ：Given that the researcher wishes to estimate the percentage of adults who support abolishing the penny within 3 percentage points with 99% confidence, we can use the formula for sample size estimation for proportions: \(n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2}\), where n is the sample size, Z is the z-score, p is the estimated proportion, and E is the margin of error.
Step 2 ：For a confidence level of 99%, the z-score is approximately 2.576. The margin of error, E, is 3% or 0.03.
Step 3 ：For part (a), we use the previous estimate of 24% or 0.24 for p. Substituting these values into the formula, we get \(n = \frac{(2.576)^2 \cdot 0.24 \cdot (1-0.24)}{(0.03)^2}\). Calculating this gives us a sample size of approximately 1345.
Step 4 ：For part (b), if no prior estimate is given, we use 0.5 for p as it provides the maximum variability and thus the largest sample size. Substituting these values into the formula, we get \(n = \frac{(2.576)^2 \cdot 0.5 \cdot (1-0.5)}{(0.03)^2}\). Calculating this gives us a sample size of approximately 1844.
Step 5 ：Final Answer: (a) The required sample size when using a previous estimate of 24% is \(\boxed{1345}\). (b) The required sample size when not using any prior estimates is \(\boxed{1844}\).
