Marketing companies are interested in knowing the population proportion of
women who make the majority of household purchasing decisions. When
designing a study to determine this population proportion, what is the
minimum number you would need to survey to be 90% confident that the
population proportion is estimated to within 0.05?

To determine the minimum sample size needed to estimate a population proportion with a specified confidence level and margin of error, we 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-score corresponding to the desired confidence level
- \( p \) = estimated proportion (if unknown, use \( p = 0.5 \) for maximum variability)
- \( E \) = margin of error

### Step 1: Identify the values
- **Confidence Level:** 90% (Z-score ≈ 1.645)
- **Margin of Error (E):** 0.05
- **Estimated Proportion (p):** If unknown, we use 0.5.

### Step 2: Substitute into the formula
Using \( p = 0.5 \):

\[
n = \left( \frac{(1.645)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.05)^2} \right)
\]

Calculating:

\[
n = \left( \frac{2.706025 \cdot 0.5 \cdot 0.5}{0.0025} \right) = \left( \frac{0.67650625}{0.0025} \right) = 270.6025
\]

### Step 3: Round up
Since we cannot survey a fraction of a person, we round up to the next whole number:

\[
n = 271
\]

### Conclusion
The minimum number of respondents needed to be 90% confident that the population proportion is estimated within 0.05 is **271**.

Would you like more details or have any questions? Here are some related questions you might consider:

1. What is the significance of choosing \( p = 0.5 \) for maximum variability?
2. How does the confidence level affect the required sample size?
3. What are the potential consequences of using a sample size that is too small?
4. Can this sample size calculation be applied to other populations or studies?
5. What are some methods to ensure a representative sample in surveys?

**Tip:** Always check if the sample is randomly selected to minimize bias in your results.

Marketing companies are interested in knowing the population proportion of women who make the majority of household purchasing decisions. When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 90% confident that the population proportion is estimated to within 0.05?

Discover how to calculate the minimum sample size needed to estimate the population proportion of women making household purchasing decisions with 90% confidence within a margin of error of 0.05. This problem illustrates key statistical concepts including Z-scores and sample size estimation.

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