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) $n=$ (Round up to the nearest integer.)

Step 1 ：The researcher wants to estimate the percentage of adults who support abolishing the penny with a margin of error of 3 percentage points and a confidence level of 99%.

Step 2 ：In statistics, the sample size needed for a proportion can be calculated using the formula: \[n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2}\] where: n is the sample size, Z is the z-score, which corresponds to the desired confidence level (for a confidence level of 99%, the z-score is approximately 2.576), p is the estimated proportion of the population (in this case, the previous estimate of 24% or 0.24), E is the desired margin of error (in this case, 3 percentage points or 0.03).

Step 3 ：We can use this formula to calculate the sample size needed for the researcher's study.

Step 4 ：Given: Z = 2.576, p = 0.24, E = 0.03

Step 5 ：Substitute the given values into the formula: n = \(\frac{(2.576)^2 \cdot 0.24 \cdot (1-0.24)}{(0.03)^2}\)

Step 6 ：Solving the above expression gives n = 1345

Step 7 ：Final Answer: The sample size needed for the researcher's study, if he uses a previous estimate of 24%, is \(\boxed{1345}\).