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

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Stat 312: Lecture 09 Confidence intervals from normal population Moo K. Chung [email protected] September 30, 2004 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 16 18 20 22 24 26 28 30 Normal Q-Q Plot Theoretical Quantiles Sample Quantiles Figure 1: Normal probability plot showing normality of data. 1. Fat content of 10 randomly selected hotdogs is given by > x<-c(25.2,21.3,22.8,17.0,29.8, 21.0,25.5,16.0,20.9,19.5) We are interested in constructing a 95 % CI for the population mean fat content. Since the sample size is small, we can not apply large sample size results. Let us check the normality of data first using qqnorm(x) . So we can assume our data to follow a normal distribution but the variance is unknown. 2. If X i N ( μ, σ 2 ) , T ¯ X - μ S/ n t n - 1 , a t distribution with n - 1 degrees of freedom. As n → ∞ , t n N (0 , 1) so for large sample size n , we can approximate t distribution with N (0 , 1) . Let us plot t distributions and compare them with N (0 , 1) . -4 -2 0 2 4 0.00 0.15 0.30 x dt(x, 1) -4 -2 0 2 4 0.0 0.2 0.4 x dt(x, 100) -4 -2 0 2 4 0.0 0.2 0.4 x dt(x, 1e+08) -4 -2 0 2 4 0.0 0.2 0.4 x dnorm(x, 0, 1) Figure 2: From left to right t 1 , t 100 , t 10000000 and N (0 , 1) . > y<- -50:50/10 > par(mfrow=c(2,2)) > plot(y,dt(y,1),type=’l’) > plot(y,dt(y,100),type=’l’)
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