Review the following case and answer the provided questions.

Marketing companies are interested in knowing the population percent of women who make the majority of household purchasing decisions.

1. 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 .05?
2. If it were later determined that it was important to be more than 90% confident and a new survey were commissioned, how would it affect the minimum number you need to survey and why?

Answer and show your work for the two questions above.

To address the questions, we can use the formula for estimating the sample size required for a proportion in a population. The formula for calculating the minimum sample size \( n \) is given by:

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

where:
- \( Z \) is the Z-score corresponding to the desired confidence level,
- \( p \) is the estimated proportion of the population,
- \( E \) is the margin of error.

### Question 1: Minimum Number for 90% Confidence

1. **Determine the Z-score for 90% confidence:**
   - For a 90% confidence level, the Z-score is approximately \( 1.645 \).

2. **Estimate the proportion \( p \):**
   - If no prior information is available, it is common to use \( p = 0.5 \) for maximum variability.

3. **Set the margin of error \( E \):**
   - The margin of error is \( 0.05 \).

Now we can substitute these values into the formula:

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

Calculating step by step:

- \( Z^2 = (1.645)^2 = 2.706025 \)
- \( p(1-p) = 0.5 \times 0.5 = 0.25 \)
- \( E^2 = (0.05)^2 = 0.0025 \)

Putting it all together:

\[
n = \left( \frac{2.706025 \cdot 0.25}{0.0025} \right) = \frac{0.67650625}{0.0025} = 270.6025
\]

Since the sample size must be a whole number, we round up to the nearest whole number:

\[
n \approx 271
\]

### Question 2: Impact of Higher Confidence Level

If we want to be more than 90% confident, we typically look at 95% confidence, which corresponds to a Z-score of approximately \( 1.96 \).

1. **Using the new Z-score:**
   - For 95% confidence, \( Z \approx 1.96 \).

Substituting this into the sample size formula:

\[
n = \left( \frac{(1.96)^2 \cdot 0.5(1-0.5)}{(0.05)^2} \right)
\]

Calculating step by step:

- \( Z^2 = (1.96)^2 = 3.8416 \)
- \( p(1-p) = 0.5 \times 0.5 = 0.25 \)
- \( E^2 = (0.05)^2 = 0.0025 \)

Putting it all together:

\[
n = \left( \frac{3.8416 \cdot 0.25}{0.0025} \right) = \frac{0.9604}{0.0025} = 384.16
\]

Rounding up:

\[
n \approx 385
\]

### Summary of Results
1. Minimum sample size for 90% confidence: **271**
2. Minimum sample size for more than 90% confidence (assuming 95%): **385**

### Explanation of Changes
Increasing the confidence level increases the Z-score, which in turn increases the required sample size to maintain the same margin of error. This is because a higher confidence level means we need a wider range to capture the true population proportion.

### Questions for Further Exploration
1. How does changing the margin of error affect sample size?
2. What other factors should be considered when designing a survey?
3. How would you calculate the sample size if the population size is finite?
4. What is the role of the confidence interval in interpreting survey results?
5. How does the choice of \( p \) influence the required sample size?

### Tip
Always use the most conservative estimate for \( p \) (usually 0.5) if you lack prior data, as it will yield the largest sample size, ensuring your results are robust.

Review the following case and answer the provided questions. Marketing companies are interested in knowing the population percent of women who make the majority of household purchasing decisions. 1. 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 .05? 2. If it were later determined that it was important to be more than 90% confident and a new survey were commissioned, how would it affect the minimum number you need to survey and why?

Explore how to calculate the minimum sample size required to estimate the population proportion of women making household purchasing decisions with 90% confidence. Learn about the impact of increasing the confidence level on sample size requirements, including specific calculations and results.

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