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Lecture11standard - Statistics 511 Statistical Methods Dr...

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Lecture 13: Additional Confidence Intervals’ Related Topics Devore: Section 7.3-7.4 March, 2011 Page 1
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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 t-confidence intervals Large-sample 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
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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 bell-shaped and centered at 0 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
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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
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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Figure 1: March, 2011 Page 5
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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 One-sample 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
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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Example Sweetgum lumber is quite valuable but there’s a general shortage of high-quality sweetgum today. Because of this, composite beams that are designed to add value to low-grade sweetgum lumber are commonly used.
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