1 / 17

Chapter 10: Basics of Confidence Intervals

Chapter 10: Basics of Confidence Intervals. 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.

lois
Download Presentation

Chapter 10: Basics of Confidence Intervals

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Chapter 10: Basics of Confidence Intervals

  2. 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

  3. §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

  4. 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

  5. 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

  6. 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

  7. 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:

  8. 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

  9. Common Levels of Confidence

  10. 90% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183

  11. 95% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183

  12. 99% Confidence Interval for μ Data: SRS, n = 712, σ = 40, x-bar = 183

  13. Confidence Level and CI Length ↑ confidence costs  ↑ confidence interval length

  14. 10.3 Sample Size Requirements To derive a confidence interval for μ with margin of error m, study this many individuals:

  15. 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

  16. 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.

More Related