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May 14

Chapter 10: Basics of Confidence Intervals. May 14. In Chapter 10:. 10.1 Introduction to Estimation 10.2 Confidence Interval for μ (σ known) 10.3 Sample Size Requirements 10.4 Relationship Between Hypothesis Testing and Confidence Intervals. §10.1: Introduction to Estimation.

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May 14

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  1. Chapter 10: Basics of Confidence Intervals May 14 10: Intro to Confidence Intervals

  2. In Chapter 10: 10.1 Introduction to Estimation 10.2 Confidence Interval for μ (σ known) 10.3 Sample Size Requirements 10.4 Relationship Between Hypothesis Testing and Confidence Intervals 10: Intro to Confidence Intervals

  3. §10.1: Introduction to Estimation Two forms of estimation • Point estimation ≡ most likely value of parameter (e.g., x-bar is point estimator of µ) • Interval estimation ≡ range of values with known likelihood of capturing the parameter, i.e., a confidence interval (CI) 10: Intro to Confidence Intervals

  4. Reasoning Behind a 95% CI • The next slide demonstrates how CIs are based on sampling distributions • If we take multiple samples from the sample population, each sample will derive a different 95% CI • 95% of the CIs will capture μ & 5% will not 10: Intro to Confidence Intervals

  5. 10: Intro to Confidence Intervals

  6. Confidence Interval for μ • To create a 95% confidence interval for μ, surround each sample mean with margin of error m:m ≈ 2×SE = 2×(σ/√n) • The 95% confidence interval for μ is: 10: Intro to Confidence Intervals

  7. Sampling distribution of a mean (curve). Below the curve are five CIs. In this example, all but the third CI captured μ 10: Intro to Confidence Intervals

  8. “Body Weight” Example • Body weights of 20-29-year-old males have unknown μ and σ = 40 • Take an SRS of n = 712 from population • Calculate: x-bar =183 10: Intro to Confidence Intervals

  9. Confidence Interval Formula Here is a more accurate and flexible formula 10: Intro to Confidence Intervals

  10. Common Levels of Confidence 10: Intro to Confidence Intervals

  11. 90% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183 10: Intro to Confidence Intervals

  12. 95% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183 10: Intro to Confidence Intervals

  13. 99% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183 10: Intro to Confidence Intervals

  14. Confidence Level and CI Length UCL ≡ Upper Confidence Limit; LCL ≡ Lower Limit; 10: Intro to Confidence Intervals

  15. 10.3 Sample Size Requirements Ask: How large a sample is need to determine a (1 – α)100% CI with margin of error m? Illustrative example: Recall that WAIS has σ= 15. Suppose we want a 95% CI for μ For 95% confidence, α = .05, z1–.05/2 = z.975= 1.96 (Continued on next slide) 10: Intro to Confidence Intervals

  16. Illustrative Examples: Sample Size • Round up to ensure precision • Smaller m require larger n 10: Intro to Confidence Intervals

  17. 10.4 Relation Between Testing and Confidence Intervals Rule: Rejects H0 at α level of significance when μ0 falls outside the (1−α)100% CI. Illustration: Next slide 10: Intro to Confidence Intervals

  18. Reject H0 at α =.05 Retain H0 at α =.01 This CI excludes 180 This CI includes 180 Example: Testing and CIs Illustration: Test H0: μ = 180 10: Intro to Confidence Intervals

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