Chapter 10 Basics of Confidence Intervals October 20

Chapter 10: Basics of Confidence Intervals October 20

In Chapter 10: 10. 1 Introduction to Estimation 10. 2 Confidence Interval for μ when σ is known 10. 3 Sample Size Requirements 10. 4 Relationship Between Hypothesis Testing and Confidence Intervals

§ 10. 1: Introduction to Estimation Two forms of estimation • Point estimation ≡ single best estimate of parameter (e. g. , x-bar is the point estimate of μ) • Interval estimation ≡ surrounding the point estimate with a margin of error to create a range of values that seeks to capture the parameter; a confidence interval

Reasoning Behind a 95% Confidence Interval • A schematic (next slide) of a sampling distribution of means based on repeated independent SRSs of n = 712 is taken from a population with unknown μ and σ = 40. • Each sample derives a different point estimate and 95% confidence interval • 95% of the confidence intervals will capture the value of μ

Confidence Intervals • To create a 95% confidence interval for μ, surround each sample mean with a margin of error m that is equal to 2 standard errors of the mean: m ≈ 2×SE = 2×(σ/√n) • The 95% confidence interval for μ is now

This figure shows a sampling distribution of means. Below the sampling distribution are five confidence intervals. In this instance, all but the third confidence captured μ

Example: Rough Confidence Interval Suppose body weights of 20 -29 -year-old males has unknown μ and σ = 40. I take an SRS of n = 712 from this population and calculate x-bar =183. Thus:

Confidence Interval Formula Here is a better formula for a (1−α)100% confidence interval for μ when σ is known: Note that σ/√n is the SE of the mean

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

90% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183

95% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183

99% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183

Confidence Level and CI Length ↑ confidence costs ↑ confidence interval length Confidence level Illustrative CI CI length = UCL – LCL 90% 95% 99% 180. 5 to 185. 5 180. 1 to 185. 9 179. 1 to 186. 9 185. 5 – 180. 5 = 5. 0 185. 9 – 180. 1 = 5. 8 186. 9 – 179. 1 = 7. 8

10. 3 Sample Size Requirements To derive a confidence interval for μ with margin of error m, study this many individuals:

Examples: Sample Size Requirements Suppose we have a variable with s = 15 and want a 95% confidence interval. Note, α =. 05 z 1–. 05/2 = z. 975 = 1. 96 round up to ensure precision Smaller margins of error require larger sample sizes

10. 4 Relationship Between Hypothesis Testing and Confidence Intervals A two-sided test will reject the null hypothesis at the α level of significance when the value of μ 0 falls outside the (1−α)100% confidence interval This illustration rejects H 0: μ = 180 at α =. 05 because 180 falls outside the 95% confidence interval. It retains H 0: μ = 180 at α =. 01 because the 99% confidence interval captures 180.