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.