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 90% confident that your estimate is within 2% 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 sample size needed to estimate a population proportion with a specified confidence level and margin of error, you can use the formula for sample size in proportion estimation:

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

Where:
- \( n \) = required sample size
- \( Z \) = Z-value corresponding to the desired confidence level
- \( p \) = estimated population proportion (since we have no reasonable estimate, we use \( p = 0.5 \) for maximum variability)
- \( E \) = margin of error

### Given:
- Confidence level = 90%
- Margin of error (E) = 0.02 (2%)

### Step 1: Find the Z-value
For a 90% confidence level, the Z-value can be found in a Z-table or using statistical software. The critical value for a two-tailed test at 90% confidence is:

\[
Z \approx 1.645
\]

### Step 2: Plug values into the formula
We will use \( p = 0.5 \) to ensure the sample size is sufficient (since it provides the largest sample size estimate).

Now substitute the values into the formula:

\[
n = \left( \frac{(1.645)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.02)^2} \right)
\]

Calculating each component step-by-step:

1. Calculate \( Z^2 \):
\[
Z^2 = (1.645)^2 \approx 2.706
\]

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

3. Calculate \( E^2 \):
\[
E^2 = (0.02)^2 = 0.0004
\]

Now substitute these values into the formula:
\[
n = \frac{2.706 \cdot 0.25}{0.0004}
\]

Calculating that:
\[
n = \frac{0.6765}{0.0004} = 1691.25
\]

### Step 3: Round up to the next whole number
Since sample size must be a whole number, we round up:

\[
n = 1692
\]

### Conclusion
The required sample size to estimate the population proportion with 90% confidence and a margin of error of 2% is:

\[
\boxed{1692}
\]