Math Problem Statement
In a poll of 1000 randomly selected voters in a local election, 424 voters were against school bond measures. What is the margin of error m for the 90% confidence level? What is the sample proportion p?
Solution
Let's break this down step by step:
1. Sample Proportion (p):
The sample proportion represents the fraction of voters who are against the school bond measures. It is calculated as:
So, the sample proportion is 0.424.
2. Margin of Error (m) for the 90% Confidence Level:
The margin of error for a sample proportion is given by the formula:
Where:
- is the critical value from the standard normal distribution for the desired confidence level (for 90% confidence, ),
- is the sample proportion,
- is the sample size (1000 voters in this case).
Now, let's calculate:
Thus, the margin of error is approximately 0.0257 or 2.57%.
Summary:
- Sample Proportion : 0.424
- Margin of Error at 90% Confidence Level: 0.0257 or 2.57%
Would you like to see further details on any of these steps, or have any other questions?
Related questions:
- How does the margin of error change with different confidence levels (e.g., 95%)?
- What would the sample proportion be if 476 voters were against the bond measure instead?
- How would increasing the sample size affect the margin of error?
- How do you calculate the critical value for a 95% or 99% confidence interval?
- What happens to the margin of error if the sample proportion is very close to 0 or 1?
Tip: The margin of error decreases as the sample size increases, making your estimate more precise.
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Math Problem Analysis
Mathematical Concepts
Statistics
Confidence Intervals
Sample Proportion
Margin of Error
Formulas
Sample Proportion p = Number of voters against the measure / Total number of voters in the sample
Margin of Error m = Z_{α/2} * √(p(1-p) / n)
Theorems
Central Limit Theorem
Normal Distribution
Suitable Grade Level
Grades 10-12
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