Your company has asked you to estimate the proportion of people who prefer the color red over other primary colors for manufacturing purposes. If they want the estimate to be within $.01$ of the true proportion with $95\%$ confidence, how many people should you survey?  

• A. 1825
• B. 6766
• C. 9604
• D. 97

Answer: (C)

Verified step by step guidance

1. Identify the formula for sample size estimation in proportion problems: \( n = \frac{{Z^2 \, p \, (1-p)}}{{E^2}} \), where \( n \) is the sample size, \( Z \) is the Z-score corresponding to the desired confidence level, \( p \) is the estimated proportion, and \( E \) is the margin of error.
2. Determine the Z-score for a 95% confidence level. The Z-score for 95% confidence is typically 1.96.
3. Assume an estimated proportion \( p \). If no prior estimate is available, use \( p = 0.5 \) as it maximizes the sample size.
4. Set the margin of error \( E \) to 0.01, as specified in the problem.
5. Substitute the values into the formula: \( n = \frac{{(1.96)^2 \, 0.5 \, (1-0.5)}}{{(0.01)^2}} \) and solve for \( n \) to find the required sample size.