Math Problem Statement
The candidate in a municipal election asks you what the sample size should be to gather voting
intentions in his municipality with a margin of error of 10%.
-
What should the sample size be if he wishes to double the precision of his estimate?
-
The poll indicates that 53% of the population (sample of 400) is ready to vote for Mr. X. Can we
say (with certainty) that Mr. X will be elected?
- On election day, Mr. X receives only 48% of the vote and criticizes the reliability of your poll. How
do you respond?
Solution
To answer these questions, let's go step by step using statistical formulas related to sample size, margin of error, and confidence intervals.
1. What should the sample size be to gather voting intentions with a 10% margin of error?
The formula for the sample size in a proportion estimate is:
where:
- is the critical value for the desired confidence level (typically 1.96 for 95% confidence),
- is the estimated proportion (if unknown, 0.5 is used for maximum variability),
- is the margin of error (10% = 0.10).
Using for a conservative estimate:
Thus, the required sample size is about 96 respondents.
2. What should the sample size be if he wishes to double the precision of his estimate?
"Doubling the precision" means halving the margin of error, so . Using the same formula:
Thus, the required sample size is about 385 respondents.
3. Can we say (with certainty) that Mr. X will be elected if 53% of a sample of 400 support him?
The confidence interval (CI) for a proportion is given by:
where:
- ,
- ,
- for 95% confidence.
\approx 0.0489 \text{ (or 4.89%)}
Thus, the confidence interval is:
Since this interval includes values below 50%, we cannot say with certainty that Mr. X will be elected.
4. Mr. X actually gets 48% on election day. How do you respond?
The pre-election poll indicated support between 48.1% and 57.9% with 95% confidence. Since 48% is just below the lower bound (48.1%), it suggests the poll was fairly accurate but close to the edge of error.
Possible explanations:
- Sampling Variability – Polls have inherent uncertainty, and a slight difference is expected.
- Voter Behavior Changes – People's opinions can shift between polling and election day.
- Turnout Differences – The sample may not have fully represented actual voters.
- Margin of Error – The poll predicted a result within a close range, but real-world events can affect outcomes.
Thus, the poll was not "wrong," but rather subject to statistical uncertainty and real-world dynamics.
Do you want more details or explanations?
Here are 5 related questions:
- How does increasing confidence level affect sample size?
- What factors besides margin of error influence poll accuracy?
- How do non-response biases impact survey results?
- What are the effects of using a smaller or larger sample size?
- How can weighting techniques improve poll accuracy?
Tip: Always check how recent the poll was conducted—voter preferences can change rapidly!
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Math Problem Analysis
Mathematical Concepts
Statistics
Sample Size Calculation
Confidence Intervals
Margin of Error
Formulas
Sample size formula: n = (Z^2 * p * (1 - p)) / E^2
Confidence interval for proportion: p ± Z * sqrt(p * (1 - p) / n)
Theorems
Central Limit Theorem
Law of Large Numbers
Suitable Grade Level
Grades 10-12
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