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:
- No preliminary estimate available (part A).
- 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:
Where:
- is the critical value from the standard normal distribution corresponding to the confidence level.
- is the estimated population proportion (if no estimate is available, we use , which maximizes the variability).
- 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 , because it maximizes the required sample size (most conservative assumption).
- Confidence level: 90%, which corresponds to a value of approximately 1.645.
- Margin of error: 3% or 0.03.
Substituting into the formula:
Part B: Prior estimate available ()
For this scenario, we use (from the prior study), and the same confidence level and margin of error.
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:
- How does changing the confidence level affect the required sample size?
- What if the margin of error is reduced to 2%? How would the sample size change?
- How does using a prior estimate influence the calculation of the sample size?
- What is the interpretation of the critical value in the sample size formula?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Calculate Minimum Sample Size for Population Proportion at 90% Confidence
Calculate Sample Size for Population Proportions at 90% Confidence
Calculate Sample Size for Population Proportion with 90% Confidence and 2% Margin of Error
Sample Size Determination for Estimating Population Proportion at 92.08% Confidence Level
Sample Size Calculation for Population Proportion with 90% Confidence