Chapter 10 Basics of Confidence Intervals June 21

Chapter 10: Basics of Confidence Intervals June 21 10: Intro to Confidence Intervals 1

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 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 2

§ 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) 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 3

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 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 4

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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: 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 6

Sampling distribution of a mean (curve). Below the curve are five CIs. In this example, all but the third CI captured μ 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 7

“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 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 8

Confidence Interval Formula Here is a more accurate and flexible formula 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 9

Common Levels of Confidence level 1–α. 90 Alpha level α. 10 Z value z 1–(α/2) 1. 645 . 95 . 05 1. 960 . 99 . 01 2. 576 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 10

90% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 11

95% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 12

99% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 13

Confidence Level and CI Length UCL ≡ Upper Confidence Limit; LCL ≡ Lower Limit; Confidence Body weight level example 90% 180. 5 to 185. 5 95% 180. 1 to 185. 9 99% 179. 1 to 186. 9 6/6/2021 Basic Biostat CI length = UCL – LCL 185. 5 – 180. 5 = 5. 0 185. 9 – 180. 1 = 5. 8 186. 9 – 179. 1 = 7. 8 10: Intro to Confidence Intervals 14

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, z 1–. 05/2 = z. 975 = 1. 96 (Continued 6/6/2021 Basic Biostat on next 10: slide) Intro to Confidence Intervals 15

Illustrative Examples: Sample Size (1) Round up to ensure precision 6/6/2021 Basic Biostat (2) Smaller require 10: Intro tom Confidence Intervalslarger n 16

10. 4 Relation Between Testing and Confidence Intervals Rule: Rejects H 0 at α level of significance when μ 0 falls outside the (1−α)100% CI. Illustration: Next slide 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 17

Example: Testing and CIs Illustration: Test H 0: μ = 180 Reject H 0 at α =. 05 Retain H 0 at α =. 01 This CI excludes 180 This CI includes 180 6/6/2021 Basic Biostat 10: Intro to Confidence Intervals 18