Question

In measuring reaction times, a psychologist estimates that the standard deviation is 0.05 seconds. How large a sample of measurements should be taken in order to be 95% confident that the error of the estimate will not exceed 0.01 seconds?

• A. \( n \ge 80 \)
• B. \( n \ge 72 \)
• C. \( n \ge 96 \)
• D. \( n \ge 69 \)

Hint:

When calculating sample size, always round the result up to the next integer. Rounding down would result in a sample size that is slightly too small to guarantee the desired margin of error. For 95% confidence, use \(Z=1.96\); sometimes problems will approximate this with \(Z=2\) for simpler calculations, which would have resulted in \(n=100\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires calculating the minimum sample size needed to estimate a population mean (the true mean reaction time) with a specified margin of error and confidence level. We assume the population is large enough to ignore the finite population correction.

Step 2: Key Formula or Approach:
The margin of error (E) in estimating a population mean is given by: \[ E = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \] We need to solve this formula for the sample size, \(n\). \[ n = \left( \frac{Z_{\alpha/2} \sigma}{E} \right)^2 \]
Step 3: Detailed Explanation:
We are given the following values: - Confidence Level: 95%. The corresponding critical Z-value is \(Z_{\alpha/2} = Z_{0.025} = 1.96\). - Population Standard Deviation (\(\sigma\)): The estimate is given as 0.05 seconds. - Maximum Margin of Error (E): The error should not exceed 0.01 seconds, so \(E = 0.01\). 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 the sample size must be an integer, and we need to ensure the error does not exceed the specified amount, we must always round the result up to the next whole number. Therefore, the required sample size is \(n=97\). The question asks how large the sample should be, which means we need \(n \ge 96.04\). The closest option that satisfies this condition is \(n \ge 96\), although strictly \(n \ge 97\) is required. Given the options, 96 is the intended numerical answer. The format of the options in the OCR as \(n \le k\) is likely incorrect; it should be \(n \ge k\) or \(n=k\). Based on our calculation, the required size is at least 96.04, so we choose the closest appropriate option.
Step 4: Final Answer:
A sample size of at least 97 is required. The closest option is \(n \ge 96\).