Question

In a hypothetical group, it is given that \( d = 0.05 \), \( p=0.5\alpha \) and \( t = 2 \). If N is large, then the sample size \( n_0 \), is

• A. 250
• B. 325
• C. 400
• D. 550

Hint:

When a question asks for sample size for a proportion but doesn't provide a preliminary estimate for \(p\), always use \(p=0.5\). This is the "worst-case scenario" as it yields the maximum possible sample size needed to achieve the desired margin of error and confidence level.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question is about calculating the required sample size for estimating a population proportion. The notation, though slightly unconventional, maps to the standard formula for sample size calculation where the population size (N) is large enough to be considered infinite. The term \( p=0.5\alpha \) is likely a typo in the OCR and should be \(p=0.5\). This value of \(p\) is used to get the maximum possible sample size when no prior estimate of the proportion is available.

Step 2: Key Formula or Approach:
The formula for the sample size (\(n_0\)) required to estimate a population proportion with a given margin of error is: \[ n_0 = \frac{Z^2 p(1-p)}{d^2} \] Where: - \( Z \) is the Z-score corresponding to the desired confidence level (here represented by \(t\)).
- \( p \) is the estimated population proportion.
- \( d \) is the desired margin of error.

Step 3: Detailed Explanation:
Let's interpret the given values based on the standard formula: - Margin of error, \( d = 0.05 \).
- Z-score, \( Z = t = 2 \). A Z-score of 2 corresponds to approximately a 95.45% confidence level.
- Estimated proportion, \( p = 0.5 \). Using \(p=0.5\) maximizes the product \(p(1-p)\), ensuring a sufficiently large sample size for any true proportion.
Now, substitute these values into the formula: \[ n_0 = \frac{(2)^2 \times 0.5 \times (1-0.5)}{(0.05)^2} \] \[ n_0 = \frac{4 \times 0.5 \times 0.5}{0.0025} \] \[ n_0 = \frac{4 \times 0.25}{0.0025} \] \[ n_0 = \frac{1}{0.0025} \] Since \( 0.0025 = \frac{1}{400} \), \[ n_0 = 1 \div \frac{1}{400} = 400 \]
Step 4: Final Answer:
The required sample size \(n_0\) is 400.