Question

You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable estimate for the population proportion. You would like to be 99.9% confident that your estimate is within 1% of the true population proportion. How large of a sample size is required?

n =




Do not round mid-calculation. However, use of a critical value rounded to three decimal places will suffice.

Answer 1

To determine the required sample size \( n \) to estimate a population proportion with a specified confidence level and margin of error, we can use the formula for the sample size for proportions when no prior estimate is available:

\[
n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right)
\]

where:
- \( Z \) is the critical value from the Z-distribution corresponding to the desired confidence level,
- \( p \) is the estimated population proportion (when no prior estimate is available, we can use \( p = 0.5 \) for maximum variability),
- \( E \) is the margin of error.

### Step 1: Determine the values
1. **Critical Value \( Z \) for 99.9% Confidence Level**:
- The confidence level is 99.9%, which means the significance level \( \alpha = 1 - 0.999 = 0.001 \).
- We need \( Z \) such that \( P(Z < z) = 0.999 \).
- Looking up in the Z-table or using a calculator, we find that \( Z \approx 3.291 \) (this might slightly vary based on the source, but it should be around this value).

2. **Estimated Population Proportion \( p \)**:
- Since we have no prior estimate, we use \( p = 0.5 \) (which maximizes the overall sample size).

3. **Margin of Error \( E \)**:
- Given in the problem, \( E = 0.01 \) (which is 1%).

### Step 2: Substitute into the formula

We plug in the values into the formula:

\[
n = \frac{(3.291)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.01)^2}
\]

### Step 3: Calculate

Now we perform the calculations:

1. Calculate \( Z^2 \):
\[
Z^2 = (3.291)^2 \approx 10.814581
\]

2. Calculate \( p(1 - p) \):
\[
p(1 - p) = 0.5 \times 0.5 = 0.25
\]

3. Calculate \( E^2 \):
\[
E^2 = (0.01)^2 = 0.0001
\]

4. Substitute into the formula:
\[
n = \frac{10.814581 \cdot 0.25}{0.0001} = \frac{2.70364525}{0.0001} = 27036.4525
\]

5. Round up (sample size must be a whole number):
\[
n \approx 27037
\]

### Final Answer
Thus, the required sample size is:

\[
\boxed{27037}
\]