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:

p^±Z×p^(1p^)n\hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}

where:

  • p^=xn\hat{p} = \frac{x}{n} is the sample proportion,
  • xx is the number of successes (people who did volunteer work),
  • nn is the sample size,
  • ZZ is the Z-score corresponding to the desired confidence level.

Step 1: Calculate p^\hat{p}

Given:

  • x=530x = 530
  • n=1297n = 1297

p^=53012970.409\hat{p} = \frac{530}{1297} \approx 0.409

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)

SE=p^(1p^)n=0.409(10.409)12970.409×0.59112970.24112970.014SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.409(1 - 0.409)}{1297}} \approx \sqrt{\frac{0.409 \times 0.591}{1297}} \approx \sqrt{\frac{0.241}{1297}} \approx 0.014

Step 4: Calculate the margin of error (ME)

ME=Z×SE=1.96×0.0140.0274ME = Z \times SE = 1.96 \times 0.014 \approx 0.0274

Step 5: Construct the confidence interval

Lower limit=p^ME0.4090.02740.3816\text{Lower limit} = \hat{p} - ME \approx 0.409 - 0.0274 \approx 0.3816 Upper limit=p^+ME0.409+0.02740.4364\text{Upper limit} = \hat{p} + ME \approx 0.409 + 0.0274 \approx 0.4364

Thus, the 95% confidence interval for the proportion of people who performed volunteer work during the past year is approximately:

(0.382,0.436)(0.382, 0.436)

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:

  1. What does the confidence interval indicate about the population proportion?
  2. How does changing the confidence level affect the width of the interval?
  3. What are some potential sources of error in this survey?
  4. How would you calculate a confidence interval for a different sample size?
  5. 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