You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable preliminary estimation for the population proportion. You would like to be 90% confident that you estimate is within 2% of the true population proportion. How large of a sample size is required? **Do not round mid-calculation.**

*n*=

To determine the required sample size (\( n \)) for estimating a population proportion with a given margin of error, confidence level, and no preliminary estimate for the proportion, we use the following formula:

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

Where:
- \( Z \) is the Z-score corresponding to the confidence level.
- \( p \) is the estimated population proportion (since there is no prior estimate, we use \( p = 0.5 \), which maximizes the sample size).
- \( E \) is the margin of error.

### Step 1: Identify the values
- Confidence level = 90% → \( Z \)-score for 90% confidence is **1.645**.
- Margin of error \( E \) = 2% or 0.02.
- Since we have no preliminary estimate for the population proportion, we use \( p = 0.5 \).

### Step 2: Apply the formula

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

\[
n = \frac{{(2.706025) \cdot (0.25)}}{{0.0004}}
\]

\[
n = \frac{{0.67650625}}{{0.0004}}
\]

\[
n = 1691.265625
\]

### Step 3: Final Answer
Since the sample size must be a whole number, round up to the nearest whole number:

\[
n = 1692
\]

Therefore, you would need a sample size of **1,692**.

Would you like any further explanation or more details? Here are some related questions:

1. How does increasing the confidence level affect the sample size?
2. What happens to the required sample size if we have a prior estimate for \( p \)?
3. How do we calculate the Z-score for different confidence levels?
4. What impact does changing the margin of error have on sample size?
5. How is this formula derived from the margin of error formula?

**Tip:** Always round the sample size up to ensure the margin of error stays within the desired range.

You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable preliminary estimation 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?

Learn how to calculate the required sample size to estimate a population proportion with 90% confidence and a 2% margin of error. This detailed step-by-step solution covers key statistical concepts, including confidence intervals and the formula for population proportion estimation, using a Z-score of 1.645 for 90% confidence.

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