Chapter 10: Basics of Confidence Intervals February 19 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

10: Intro to Confidence Intervals Chapter 10 2/18/2019 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 2/18/2019Basic Biostat 10: Intro to Confidence Intervals Basic Biostat

§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) 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

10: Intro to Confidence Intervals 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 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

10: Intro to Confidence Intervals 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

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: 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

10: Intro to Confidence Intervals Sampling distribution of a mean (curve). Below the curve are five CIs. In this example, all but the third CI captured μ 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

10: Intro to Confidence Intervals “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 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

Confidence Interval Formula Here is a more accurate and flexible formula 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

Common Levels of Confidence Confidence level 1 – α Alpha level α Z value z1–(α/2) .90 .10 1.645 .95 .05 1.960 .99 .01 2.576 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

90% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

95% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

99% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

Confidence Level and CI Length UCL ≡ Upper Confidence Limit; LCL ≡ Lower Limit; Confidence level Body weight example CI length = UCL – LCL 90% 180.5 to 185.5 185.5 – 180.5 = 5.0 95% 180.1 to 185.9 185.9 – 180.1 = 5.8 99% 179.1 to 186.9 186.9 – 179.1 = 7.8 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

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) 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

Illustrative Examples: Sample Size Round up to ensure precision Smaller m require larger n 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

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 2/18/2019Basic Biostat 10: Intro to Confidence Intervals

Example: Testing and CIs Illustration: Test H0: μ = 180 This CI excludes 180 Reject H0 at α =.05 Retain H0 at α =.01 This CI includes 180 2/18/2019Basic Biostat 10: Intro to Confidence Intervals