Chapter 10 Lecture: Confidence Intervals

Introduction to Confidence Intervals

• Focus on in-class activity 10a.
• Using technology to calculate confidence intervals.

Review of Key Concepts

• Point Estimate (p-hat): Best guess for true population proportion (p), used to estimate the unknown true proportion.
• Margin of Error: Includes standard error, the estimated standard deviation if p had no hats.
  • Standard Error (SE): Approximates the standard deviation.
• Critical Value (z-star): Associated with a given confidence level, important for determining the margin of error.

Understanding Sampling Distribution

• Center of distribution is the true population proportion.
• Shape: Bell-shaped if sample size n is large enough.
• Standard Deviation: σ = sqrt(p(1-p)/n)

Exploring Confidence Intervals with Technology

• Use of software tools for confidence intervals.
• Example of sleep deprivation in college students:
  • Claim: 20% of students pull all-nighters.
  • Personal Opinion: Belief that the true proportion could be higher based on experience.
  • Sample Survey Method: Use surveys to estimate true proportion with considerations for bias.

Confidence Interval Calculation

• Interval Notation: Lower bound and upper bound to estimate population proportion.
• Increased confidence level (90%, 95%, 99%) results in wider intervals.

Example Calculation

• Sample Data: 34 out of 253 students pulled all-nighters.
  • Calculate p-hat = number of successes / total sample.
  • Use decimal format for proportions.

Creating a Confidence Interval

• Plus-Minus Format: p-hat ± margin of error.
• Interval Notation Format: [low bound, high bound].
• Example for 95% confidence: Calculate using software, yielding [0.0924, 0.1764].

Conditions for Valid Confidence Intervals

• Sample must be random.
• Sample size n should be less than 10% of the population.
• Check conditions:
  • n * p-hat ≥ 10
  • n * (1 - p-hat) ≥ 10

Technology Assisted Calculation

• Use online tools for calculating confidence intervals.
• Verify critical z-value and margin of error values align with manual calculations.

Analysis and Interpretation

• Comparison with Claims: Mattress claim of 20% may exceed data estimation.
  • Point estimate and confidence interval suggest lower true proportion than claimed.

Conclusion on Confidence Levels

• Higher confidence levels lead to wider intervals and vice versa.
• Importance of balancing confidence level with precision of the interval.
• Applying confidence intervals in practical research for accurate data insights.

Summary

• Confidence intervals provide a range of estimates for true population proportions.
• Essential conditions for accurate confidence interval calculation.
• Use of technology simplifies calculations and provides reliable estimates for statistical analysis.