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Chapter 10: Basics of Confidence Intervals

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## Chapter 10: Basics of Confidence Intervals

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**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 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**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**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.