Lecture: One Sample Z Interval for p

Overview

• Focus on a specific type of confidence interval: one sample Z interval for p.
• Builds on previous lessons about confidence intervals.
• Method involves constructing and interpreting confidence intervals for a true proportion.

Key Concepts

• Procedure Name: One sample Z interval for p.
• Parameter Definition: Define the true proportion in the context of the problem.
• Conditions to Verify:
  • Random
  • 10%
  • Large Counts

Steps to Construct and Interpret

1. Name the Procedure: Identify the method (One sample Z interval for p).
2. Define the Parameter:
   • Use context, refer to the true proportion not the sample proportion.
3. Check Conditions:
   • Random selection verification (e.g., 738 randomly selected users).
   • 10% condition (sample should be less than 10% of population).
   • Large counts (at least 10 successes and failures).
4. Perform Calculations:
   • Calculate sample proportion (e.g., 170/738 = 0.23 or 23%).
   • Determine critical value (e.g., for 95% confidence, Z = 1.96).
   • Use the formula: [ \text{Confidence Interval} = \hat{p} \pm Z \times \text{Standard Error} ].
   • Standard Error formula: [ \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} ].
   • Example calculation: 0.23 ± 1.96 × 0.0155 → interval of 20% to 26%.
5. Conclusion:
   • Template: "We are X% confident that the interval from A to B captures the true parameter in context."
   • Example conclusion for cell phone users walking into objects.

Example Problem

• Context: Survey of cell phone users walking into objects.
• Data: 738 participants, 170 admitted.
• Calculate and Interpret:
  • Procedures as described above.

Calculator Usage

• Calculator Steps:
  • STAT > TESTS > 1-PropZInterval for ease.
  • Input successes and sample size.
  • Default confidence level 95%.

Interpretation of Confidence Levels

• Confidence Level vs. Interval:
  • Confidence level template: "If we were to sample many times, X% of intervals would capture the true parameter."

Additional Practice

• Problem: National Sleep Foundation's poll on sleep habits.
• Steps:
  • Follow procedure: name, parameter, conditions, calculations, conclusion.
  • Example calculation for 90% confidence interval.
• Interpret Results:
  • Evaluate if interval suggests fewer than 50% of adults get enough sleep.

Key Takeaways

• Understand the steps from procedure naming to conclusion.
• Practice interpreting both the interval and level of confidence.
• Utilize calculator for precise interval calculations.