lecture07 notes

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

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Stat 312: Lecture 06 Quantile-quantile plots Moo K. Chung [email protected] September 23, 2004 1. In order to compute 100(1 - α )% confidence interval, it is required to find z α/ 2 that satisfies P ( Z > z α/ 2 ) = α/ 2 for given α . We will study how to find z α/ 2 and more. This lecture is based on Chapter 4.6. 2. The p -th quantile point q for random variable X is the point such that F ( q ) = P ( X q ) = p. The textbook represent it in terms of percentile . Note that p -th quantile = 100 × p -th percentile. So given p , q = F - 1 ( p ) . For X N (0 , 1) , it is easy to find the p -th qun- tile using > qnorm(1) [1] Inf > qnorm(0.5) [1] 0 > qnorm(0) [1] -Inf > qnorm(0.5) [1] 0 > qnorm(0.95) [1] 1.644854 > qnorm(0.05) [1] -1.644854 In order to find z α , we use command qnorm ( 1 - α ) . 3. Given n observations x 1 , · · · , x n , we order them from the smallest to the largest and we have x (1) , · · · , x ( n ) . The i -th smallest observation is defined as the ( i - 0 . 5) /n -th sample quantile point or 100( i - 0 . 5) /n sample percentile point.

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