Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 8 Student’s t -distribution Instructor’s valentine gift to students :) Moo K. Chung [email protected] February 13, 2003 Concepts 1. When n large, Z ¯ X - μ S/ n N (0 , 1) . 2. What if n is small? T ¯ X - μ S/ n t n - 1 , a t distribution with n - 1 degrees of freedom (Review STAT311). 3. As n → ∞ , t n N (0 , 1). 4. Critical values: P ( - t α/ 2 ,n - 1 < T < t α/ 2 ,n - 1 ) = 1 - α. 5. Suppose X 1 , · · · , X n N ( μ, σ 2 ), where μ and σ are unknown. 100(1 - α ) CI for μ is ¯ x ± t α/ 2 s/ n. In-class problems > x<- -50:50/10 > plot(x,dt(x,1000),type=’l’) > qt(.975, df = c(1:10,20,100,1000,10000)) [1] 12.706205 4.302653 3.182449 [4] 2.776445 2.570582 2.446912 [7] 2.364624 2.306004 2.262157 [10] 2.228139 2.085963 1.983972 [13] 1.962339 1.960201 > qnorm(0.975) [1] 1.959964 > pt(2.228,10) [1] 0.9749941 Example 7.12. Fat content of 10 randomly se- lected hot dogs. Assuming that observations come
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Unformatted text preview: from normal distribution, ±nd a 95% CI for the pop-ulation mean fat content. > x<-c(25.2,21.3,22.8,17.0,29.8,21.0, 25.5,16.0,20.9,19.5) > sd(x) [1] 4.13414 > mean(x)+qt(0.975,9)*sd(x)/sqrt(10) [1] 24.85739 > mean(x)-qt(0.975,9)*sd(x)/sqrt(10) [1] 18.94261 Assuming X i ∼ N ( μ, σ 2 = 4 . 13 2 ), ±nd a 95% CI for μ . > mean(x)+qnorm(0.975)*4.13/sqrt(10) [1] 24.45975 > mean(x)-qnorm(0.975)*4.13/sqrt(10) [1] 19.34025 Self-study problems Exercise 7.29., 7.33. Assignment III. Due Feb 27, 11:00am. Exercise 7.32, 7.34., 7.42., 7.44., 8.10. The ±rst midterm will cover chapters 6,7 and 8.1. Sample midterm problems will be posted on Feb. 25....
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