A recent study indicated that 25% of the 117 women over age 55 in the study were widows. Round up your answers to the next whole number for the following questions.How large a sample must you take to be 90% confident that the estimate is within 0.02 of the true proportion of women over age 55 who are widows?

To calculate the required sample size to estimate the true proportion of women over age 55 who are widows, given a confidence level and margin of error, we use the formula for the 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-score corresponding to the confidence level (for 90% confidence, \( Z = 1.645 \))
- \( p \) = estimated proportion of the population who are widows (from the study, \( p = 0.25 \))
- \( E \) = margin of error (0.02 in this case)

Let's calculate the sample size.

### Step 1: Substitute the known values into the formula:

\[
n = \left( \frac{(1.645)^2 \cdot (0.25) \cdot (1 - 0.25)}{(0.02)^2} \right)
\]

I'll now calculate this step by step.The required sample size to be 90% confident that the estimate is within 0.02 of the true proportion of women over age 55 who are widows is **1,268**.

Do you have any further questions or need more details?

Here are 5 related questions you might find useful:
1. How does changing the confidence level affect the required sample size?
2. What happens to the sample size if we change the margin of error to 0.05?
3. How is the Z-score determined for different confidence levels?
4. Why is it necessary to round up the sample size to the next whole number?
5. How does the estimated proportion (\(p\)) impact the sample size calculation?

**Tip:** The smaller the margin of error you want, the larger your sample size will need to be.

A recent study indicated that 25% of the 117 women over age 55 in the study were widows. Round up your answers to the next whole number for the following questions. How large a sample must you take to be 90% confident that the estimate is within 0.02 of the true proportion of women over age 55 who are widows?

Find out how to calculate the required sample size for estimating the proportion of women over age 55 who are widows. This problem demonstrates the use of statistical formulas for proportion estimation with a 90% confidence level and a 0.02 margin of error. Suitable for students in Grades 10-12.

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