Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Lecture 13: Additional Confidence Intervals’ Related TopicsDevore: Section 7.3-7.4March, 2011Page 1Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011t-confidence intervals• Large-sample confidence intervals are based on the fact that,for n large enough,Z =¯X − µS/√nis approximately normally distributed• But what if n < 40?• For small n, this test statistic is denotedT =¯X − µS/√nto stress the fact it is no longer normally distributedMarch, 2011Page 2Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011t Distribution• A t distribution is governed by one parameter ν which is calledthe number of degrees of freedom (df)• Properties:1. tνcurve is bell-shaped and centered at 02. It has heavier tails than normal distribution (more spread out)3. As ν → ∞, the tνdensity curve approaches the normalcurveMarch, 2011Page 3Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011• Let tα,νbe the number on the horizontal axis such that the areato the left of it under tνcurve is α; tα,νis a t critical value.• For fixed ν, tα,νincreases as α decreases• For fixed α, as ν increases, the value tα,νdecreases. Theprocess slows down as ν increases; that is why the table valuesare shown in increments of 2 between 30 df and 40 df, but thenjump to ν = 50, ν = 60 etc.• zαis the last row of the table since t∞is the standard normaldistributionMarch, 2011Page 4Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Figure 1:March, 2011Page 5Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011One-sample t confidence interval• The number of df for T is n − 1 since S is based on deviationsX1−¯X, . . . , Xn−¯X that add up to zero• By definition of t critical value, we haveP (−tα/2,n−1< T < tα/2,n−1) = 1 − α• It is easy to show that 100(1 −α)% confidence interval for µ is¯x − tα/2,n−1·s√n, ¯x + tα/2,n−1·s√n• The alternative, more compact notation is¯x ± tα/2,n−1·s√nMarch, 2011Page 6Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Example• Sweetgum lumber is quite valuable but there’s a generalshortage of high-quality sweetgum today. Because of this,composite beams that are designed to add value to low-gradesweetgum lumber are commonly used.• The sample consists of 30 observations on the modulus ofrapture in psi• Checking normal probability plot first - the data looks normal!• R Code:1. sweetgum < − as.vector(as.matrix(swgum))2. mean(sweetgum)+qt(.025,29)*sd(sweetgum)/sqrt(length(sweetgum))March, 2011Page 7Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Prediction interval• Consider a random sample X1, . . . , Xnfrom a normalpopulation distribution. Suppose you want to predict Xn+1.• A point predictor is¯X; clearly, E(¯X − Xn+1) = µ − µ = 0andV (¯X−Xn+1) = V (¯X)+V (Xn+1) = σ2+σ2n= σ21 +1n• The prediction error is normally distributed and, therefore,Z =¯X − Xn+1qσ21 +1nhas a standard normal distributionMarch, 2011Page 8Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011• It is possible to show thatT =¯X − Xn+1Sq1 +1nhas t distribution with n − 1 df• Consequently, the prediction interval for Xn+1is¯x ± tα/2,n−1· sr1 +1n• Note the obvious difference with the t confidence interval for themean µ...Why is the prediction interval wider?March, 2011Page 9Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011• Note that the estimation error¯X − µ is the deviation from thefixed value while the prediction error¯X − Xn+1is a differencebetween two random variables. The second has much morevariability in it than the first...• Even when n → ∞, the PI approaches µ ± zα/2· σ. Thismeans that there is uncertainty about the true value X evenwhen the infinite amount of information is available.March, 2011Page 10Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Example• A meat inspector has randomly measured 30 packs of 95%lean beef. The sample resulted in the mean 96.2% with thesample standard deviation of 0.8%. Find a 99% predictioninterval for a new pack. Assume normality• For ν = 29 df, we have the critical value t0.005= 2.756.Hence a 99% prediction interval for a new observation x0is96.2−(2.756)(0.8)r1 +130< x0< 96.2−(2.756)(0.8)r1 +130which reduces to (93.96, 98.44).March, 2011Page 11Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Bootstrap:The Introduction• Suppose you have some distribution with the density f(x; θ)where θ is an unknown parameter• Given a sample x1, . . . , xnfrom this distribution, you canobtain a point estimateˆθ; as an example, if you have normaldistribution with mean µ, you can always estimate it by ¯x.• If θ is the only unknown parameter, you can say that the(unknown) pdf f(x; θ) can be estimated by f(x;ˆθ). Now youcan generate multiple samples from f(x;ˆθ) distribution to getx∗1, x∗2, . . . , x∗n(1)March, 2011Page 12Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011• With B bootstrap samples at our disposal, we can have thebootstrap estimate of θˆθ∗. For example, if the parameter inquestion is the mean µ, we have ˆµ∗= B−1Px∗i.• Why do we need bootstrap? An important issue is estimatingthe precision of the estimatorˆθ; if θ = σ2, it is difficult toestimate the variance σˆθ.• Using the bootstrap samples, we can estimate it asSˆθ=r1B − 1X(ˆθ∗i−¯θ∗)2March, 2011Page 13Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011Example• Let X be the time to breakdown of an insulating fluid betweenelectrodes at some voltage and assume it is exponentiallydistributed f(x) = λe−λx• A random sample of n = 1 breakdown times (min) is 41.53,18,73, 2.99, 30.34, 12.33, 117.52, 73.02, 223.63, 4.00, 26.78• A reasonable estimate of the distribution parameter isλ =1¯x= 1/55.087 = 0.018153• Generate B = 100 samples, each of size 10, fromf(x; 0.018153)• Determine the value ofˆλ∗ifor each i = 1, . . . , B and find¯λ∗= 0.02153 and sˆλ= 0.0091; that last value can be usedMarch, 2011Page 14Statistics 511: Statistical MethodsDr. LevinePurdue UniversitySpring 2011to construct a confidence