Chapter 10: Basics of Confidence Intervals April 17

In Chapter 10: 10.1 Introduction to Estimation 4/24/2017 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 Basic Biostat

§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 2standard 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 Confidence level 1 – α Alpha level α Z value z1–(α/2) .90 .10 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% 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

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  z1–.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 μ0 falls outside the (1−α)100% confidence interval This illustration rejects H0: μ = 180 at α =.05 because 180 falls outside the 95% confidence interval. It retains H0: μ = 180 at α = .01 because the 99% confidence interval captures 180.