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 pp represents the fraction of voters who are against the school bond measures. It is calculated as:

p=Number of voters against the measureTotal number of voters in the sample=4241000=0.424p = \frac{\text{Number of voters against the measure}}{\text{Total number of voters in the sample}} = \frac{424}{1000} = 0.424

So, the sample proportion pp 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:

m=Zα/2×p(1p)nm = Z_{\alpha/2} \times \sqrt{\frac{p(1-p)}{n}}

Where:

  • Zα/2Z_{\alpha/2} is the critical value from the standard normal distribution for the desired confidence level (for 90% confidence, Zα/21.645Z_{\alpha/2} \approx 1.645),
  • pp is the sample proportion,
  • nn is the sample size (1000 voters in this case).

Now, let's calculate:

  1. p(1p)=0.424×(10.424)=0.424×0.576=0.244224p(1 - p) = 0.424 \times (1 - 0.424) = 0.424 \times 0.576 = 0.244224
  2. p(1p)n=0.2442241000=0.000244224\frac{p(1 - p)}{n} = \frac{0.244224}{1000} = 0.000244224
  3. 0.000244224=0.01563\sqrt{0.000244224} = 0.01563
  4. m=1.645×0.01563=0.0257m = 1.645 \times 0.01563 = 0.0257

Thus, the margin of error mm is approximately 0.0257 or 2.57%.

Summary:

  • Sample Proportion pp: 0.424
  • Margin of Error mm 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:

  1. How does the margin of error change with different confidence levels (e.g., 95%)?
  2. What would the sample proportion be if 476 voters were against the bond measure instead?
  3. How would increasing the sample size affect the margin of error?
  4. How do you calculate the critical value for a 95% or 99% confidence interval?
  5. What happens to the margin of error if the sample proportion pp 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