A researcher wishes to estimate, with 90% confidence, the population proportion of likely US voters who think congress is doing a good or excellent job. Her estimate must be accurate within 3% of the true proportion.
A. No preliminary estimate is available. Find the minimum sample size needed.
B. Find the minimum sample size needed, using a prior study that found that 36% of the responders said they think that congress is doing a good or excellent job.

### Problem Analysis:

We are tasked with finding the minimum sample size for a population proportion estimate, using two different scenarios:

1. **No preliminary estimate available** (part A).
2. **A preliminary estimate from a prior study is available** (part B).

To solve both, we use the formula for determining the minimum sample size for estimating a population proportion:

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

Where:
- \( Z_{\alpha/2} \) is the critical value from the standard normal distribution corresponding to the confidence level.
- \( p \) is the estimated population proportion (if no estimate is available, we use \( p = 0.5 \), which maximizes the variability).
- \( E \) is the margin of error (in this case, 3% or 0.03).

#### Step-by-step solution:

### Part A: No preliminary estimate available

In this case, we use \( p = 0.5 \), because it maximizes the required sample size (most conservative assumption).

- **Confidence level**: 90%, which corresponds to a \( Z_{\alpha/2} \) value of approximately **1.645**.
- **Margin of error**: 3% or **0.03**.

Substituting into the formula:

\[
n = \left( \frac{1.645^2 \cdot 0.5 \cdot (1 - 0.5)}{0.03^2} \right)
\]

### Part B: Prior estimate available (\( p = 0.36 \))

For this scenario, we use \( p = 0.36 \) (from the prior study), and the same confidence level and margin of error.

\[
n = \left( \frac{1.645^2 \cdot 0.36 \cdot (1 - 0.36)}{0.03^2} \right)
\]

Let me compute the values for both cases.### Solutions:

- **Part A**: When no preliminary estimate is available, the minimum sample size needed is **752**.
- **Part B**: When using the prior estimate that 36% of respondents think Congress is doing a good or excellent job, the minimum sample size needed is **693**.

Would you like further details or have any questions?

Here are five related questions:
1. How does changing the confidence level affect the required sample size?
2. What if the margin of error is reduced to 2%? How would the sample size change?
3. How does using a prior estimate influence the calculation of the sample size?
4. What is the interpretation of the critical value \( Z_{\alpha/2} \) in the sample size formula?
5. What factors could impact the accuracy of the sample size estimation in real-world studies?

**Tip**: The sample size increases as the margin of error decreases or the confidence level increases, reflecting a higher demand for precision or certainty.

A researcher wishes to estimate, with 90% confidence, the population proportion of likely US voters who think Congress is doing a good or excellent job. Her estimate must be accurate within 3% of the true proportion. A. No preliminary estimate is available. Find the minimum sample size needed. B. Find the minimum sample size needed, using a prior study that found that 36% of the responders said they think that Congress is doing a good or excellent job.

Discover how to calculate the minimum sample size for estimating a population proportion with 90% confidence, both with and without a prior estimate. Learn to apply key statistical concepts, including the formula n = (Z^2 * p * (1 - p)) / E^2, and understand the importance of margin of error and confidence levels in sample size determination.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation