Activity 21

MATH 216: Statistical Thinking

Confidence Intervals for Proportions

Time Allocation: 15 minutes total (3 min reading, 12 min individual work)

Part 1: Conceptual Understanding (3 minutes)

Instructions: Answer the following questions based on the lecture content:

  1. What are the three essential conditions for valid large-sample confidence intervals for proportions?
  1. Explain the Central Limit Theorem for proportions and why it’s important for confidence intervals.
  1. How does the success-failure condition (\(n\hat{p} \geq 15\) and \(n(1-\hat{p}) \geq 15\)) ensure the validity of our confidence intervals?

Part 2: Real-World Applications (4 minutes)

Calculate confidence intervals for real-world scenarios:

  1. Political Polling: In a survey of 1200 voters, 48% support Candidate A. Calculate the 95% confidence interval for the true proportion of voters who support Candidate A.

    CI =

  2. Quality Control: A manufacturer tests 500 products and finds a 2% defect rate. Calculate the 95% confidence interval for the true defect rate.

    CI =

  3. Medical Research: In a clinical trial with 800 patients, 65% show treatment success. Calculate the 99% confidence interval for the true success rate.

    CI =

Show your work for one calculation:

Part 3: Critical Thinking and Interpretation (3 minutes)

Interpret and analyze confidence intervals:

  1. For the political polling scenario (48% support, 95% CI: 45.2% to 50.8%), what does this confidence interval tell us about the election outcome?
  1. For the quality control scenario (2% defect rate, 95% CI: 0.8% to 3.2%), if the target defect rate is 1%, what decision should management make about the manufacturing process?
  1. Critical Thinking: Why is it important to check the success-failure condition before calculating confidence intervals for proportions?

Part 4: Sample Size Calculations (5 minutes)

Sample Size Formula: \(n = \left(\frac{z_{\alpha/2}}{ME}\right)^2 \cdot p(1-p)\)

Calculate required sample sizes for research planning:

  1. Political Polling: A polling organization wants to estimate voter support with a margin of error of 3% at 95% confidence level. Using the conservative approach (p = 0.5), calculate the required sample size.

    Required sample size =

  2. Market Research: A company wants to estimate customer preference for a new product with a margin of error of 5% at 90% confidence level. Using a prior estimate of p = 0.30, calculate the required sample size.

    Required sample size =

  3. Medical Study: A research team needs to estimate treatment success rate with a margin of error of 2% at 99% confidence level. Using the conservative approach, calculate the required sample size.

    Required sample size =

Show your work for one calculation:

Critical Thinking Questions:

  1. Why do we use p = 0.5 in the conservative approach for sample size calculations?
  1. How does increasing the confidence level affect the required sample size for the same margin of error?