Math Problem Statement

How many voters should be sampled 95 % for a confidence interval? Round up to a whole number. A poll reported a approval rating 36% for a politician with a margin of error of 1 percentage point.

Solution

To determine the required sample size for a confidence interval, you can use the following formula for estimating proportions:

n=(Z2p(1p)E2)n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \right)

Where:

  • ZZ is the Z-score for a 95% confidence level (which is 1.96),
  • pp is the estimated proportion of the population, which is 36% or 0.36,
  • EE is the margin of error, which is 1% or 0.01.

Let's calculate the sample size:

n=(1.9620.36(10.36)0.012)n = \left( \frac{1.96^2 \cdot 0.36 \cdot (1-0.36)}{0.01^2} \right)

I'll compute that for you.The required sample size for the poll to achieve a 95% confidence interval with a 1 percentage point margin of error is approximately 8,852 voters.

Would you like more details or have any further questions?

Here are 5 related questions to expand on this concept:

  1. How does the margin of error affect the required sample size?
  2. What is the significance of the 95% confidence level in polls?
  3. How would the sample size change with a different margin of error?
  4. Why is it necessary to round up the sample size in such calculations?
  5. How does the estimated proportion (36% in this case) influence the sample size?

Tip: A larger sample size reduces the margin of error, increasing the accuracy of poll results.

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Math Problem Analysis

Mathematical Concepts

Statistics
Confidence Interval
Sample Size Estimation
Proportion

Formulas

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

Theorems

Normal Distribution
Central Limit Theorem

Suitable Grade Level

College-Level Statistics