Step 1: Concept Explanation:
This problem involves determining the minimum sample size to accurately estimate a population mean (true reaction time) within a given margin of error and confidence level. We assume a large population, negating the need for finite population correction.
Step 2: Core Formula:
The margin of error (E) for estimating a population mean is:
\[ E = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \]
Solve for the sample size, \(n\):
\[ n = \left( \frac{Z_{\alpha/2} \sigma}{E} \right)^2 \]
Step 3: Detailed Solution:
Given values:
- Confidence Level: 95%, with a Z-value of \(Z_{\alpha/2} = Z_{0.025} = 1.96\).
- Population Standard Deviation (\(\sigma\)): Estimated at 0.05 seconds.
- Maximum Margin of Error (E): Should not exceed 0.01 seconds, thus \(E = 0.01\).
Substitute these values into the sample size formula:
\[ n = \left( \frac{1.96 \times 0.05}{0.01} \right)^2 \]
\[ n = \left( \frac{0.098}{0.01} \right)^2 \]
\[ n = (9.8)^2 = 96.04 \]
Since sample size must be a whole number, always round up to ensure the error does not exceed the limit.
Therefore, the minimum required sample size is \(n=97\).
The question seeks the required sample size, implying \(n \ge 96.04\). While strictly \(n \ge 97\) is necessary, 96 is the closest available option. The OCR's \(n \le k\) format is likely incorrect; it should be \(n \ge k\) or \(n=k\). Based on our calculation, a sample size of at least 96.04 is needed; therefore, the closest appropriate option is chosen.
Step 4: Final Answer:
A minimum sample size of 97 is required. The nearest option is \(n \ge 96\).