This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Lecture 13: Additional Confidence Intervals’ Related Topics Devore: Section 7.37.4 March, 2011 Page 1 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 tconfidence intervals • Largesample confidence intervals are based on the fact that, for n large enough, Z = ¯ X μ S/ √ n is approximately normally distributed • But what if n < 40 ? • For small n , this test statistic is denoted T = ¯ X μ S/ √ n to stress the fact it is no longer normally distributed March, 2011 Page 2 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 t Distribution • A t distribution is governed by one parameter ν which is called the number of degrees of freedom (df) • Properties: 1. t ν curve is bellshaped and centered at 2. It has heavier tails than normal distribution (more spread out) 3. As ν → ∞ , the t ν density curve approaches the normal curve March, 2011 Page 3 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 • Let t α,ν be the number on the horizontal axis such that the area to the left of it under t ν curve is α ; t α,ν is a t critical value . • For fixed ν , t α,ν increases as α decreases • For fixed α , as ν increases, the value t α,ν decreases. The process slows down as ν increases; that is why the table values are shown in increments of 2 between 30 df and 40 df, but then jump to ν = 50 , ν = 60 etc. • z α is the last row of the table since t ∞ is the standard normal distribution March, 2011 Page 4 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Figure 1: March, 2011 Page 5 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Onesample t confidence interval • The number of df for T is n 1 since S is based on deviations X 1 ¯ X,...,X n ¯ X that add up to zero • By definition of t critical value, we have P ( t α/ 2 ,n 1 < T < t α/ 2 ,n 1 ) = 1 α • It is easy to show that 100(1 α )% confidence interval for μ is ¯ x t α/ 2 ,n 1 · s √ n , ¯ x + t α/ 2 ,n 1 · s √ n • The alternative, more compact notation is ¯ x ± t α/ 2 ,n 1 · s √ n March, 2011 Page 6 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Example • Sweetgum lumber is quite valuable but there’s a general shortage of highquality sweetgum today. Because of this, composite beams that are designed to add value to lowgrade...
View
Full
Document
This note was uploaded on 02/20/2012 for the course STAT 511 taught by Professor Bud during the Fall '08 term at Purdue.
 Fall '08
 BUD
 Statistics

Click to edit the document details