Math Problem Statement
The General Social Survey asked 1297 people whether they performed any volunteer work during the past year. A total of 530 people said they did. A 95% confidence interval for the proportion of people who performed volunteer work during the past year is
Solution
To calculate a 95% confidence interval for the proportion of people who performed volunteer work, we can use the formula for a confidence interval for a proportion:
where:
- is the sample proportion,
- is the number of successes (people who did volunteer work),
- is the sample size,
- is the Z-score corresponding to the desired confidence level.
Step 1: Calculate
Given:
Step 2: Find the Z-score
For a 95% confidence interval, the Z-score is approximately 1.96.
Step 3: Calculate the standard error (SE)
Step 4: Calculate the margin of error (ME)
Step 5: Construct the confidence interval
Thus, the 95% confidence interval for the proportion of people who performed volunteer work during the past year is approximately:
Summary
The 95% confidence interval for the proportion of people who performed volunteer work is approximately (0.382, 0.436).
Do you want details on any of the steps, or do you have any questions? Here are some follow-up questions you might consider:
- What does the confidence interval indicate about the population proportion?
- How does changing the confidence level affect the width of the interval?
- What are some potential sources of error in this survey?
- How would you calculate a confidence interval for a different sample size?
- What assumptions must be met to use this method for calculating the confidence interval?
Tip: Always ensure the sample size is sufficiently large for the normal approximation to be valid when calculating confidence intervals for proportions.
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Math Problem Analysis
Mathematical Concepts
Statistics
Confidence Intervals
Proportions
Formulas
Confidence interval formula: \hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
Sample proportion: \hat{p} = \frac{x}{n}
Standard error: SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
Margin of error: ME = Z \times SE
Theorems
Central Limit Theorem
Law of Large Numbers
Suitable Grade Level
Grades 11-12
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