Math Problem Statement

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.

Solution

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=(Zα/22p(1p)E2)n = \left( \frac{Z_{\alpha/2}^2 \cdot p \cdot (1 - p)}{E^2} \right)

Where:

  • Zα/2Z_{\alpha/2} is the critical value from the standard normal distribution corresponding to the confidence level.
  • pp is the estimated population proportion (if no estimate is available, we use p=0.5p = 0.5, which maximizes the variability).
  • EE 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=0.5p = 0.5, because it maximizes the required sample size (most conservative assumption).

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

Substituting into the formula:

n=(1.64520.5(10.5)0.032)n = \left( \frac{1.645^2 \cdot 0.5 \cdot (1 - 0.5)}{0.03^2} \right)

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

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

n=(1.64520.36(10.36)0.032)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 Zα/2Z_{\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.

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Math Problem Analysis

Mathematical Concepts

Statistics
Population Proportion Estimation
Confidence Intervals
Sample Size Determination

Formulas

n = (Z^2 * p * (1 - p)) / E^2

Theorems

Central Limit Theorem

Suitable Grade Level

Undergraduate Statistics or Advanced High School Statistics