Download Lecture 3: Confidence Intervals in Statistical Inference and more Study notes Statistics in PDF only on Docsity! Statistics 431: Statistical Inference Lecture 3: Confidence intervals Introduction • A point estimate (eg, sample mean estimating population mean ) could be very precise, or not at all. Can’t tell from just the number. • Instead of reporting single estimate of , can report a range of plausible values based on data: a confidence interval for . • Each CI has an associated confidence level, like 90%, 95%, ... - the higher the confidence level, the more likely the CI is to contain • A wide interval implies we don’t have a good handle on ; a narrow interval implies is known precisely. • To find the CI for a given confidence level, we need assumptions plus a probability calculation. 2 X̄ µ µ µ µ µ µ • Before the data are observed: - the CI is a random interval (in this case, centered at ) - there is probability that the observed CI will cover - note: center is random but width is not • After the data are observed: - the CI is a fixed interval, determined by - this fixed interval either covers or it doesn’t (no probability statement applies) 5 X̄ ! µ x1, . . . , xn µ • After the data is observed and a 95% CI is computed, nothing is random. • In particular, .95 is not the probability that the observed interval contains : the interval is now fixed, not random, and is an unknown constant, not a random variable. • Meaning of .95: if I build 95% CIs from many independent samples of size , then in the long run, 95% of those intervals will cover , and 5% will not. 6 More on interpretation 09/09/2005 04:23 PMconexa.gif 383!287 pixels Page 1 of 1http://ewr.cee.vt.edu/environmental/teach/smprimer/intervals/conexa.gif µ µ n µ Confidence vs. width • Higher confidence (good) = wider interval (bad) • The only way to get higher confidence and a narrower interval is to increase the sample size . • For confidence 100 % and width we need (Again, we don’t know : we’ll come back to this.) • Example: Fisher’s iris data had , , 95% CI CI width . To achieve on a new sample, 7 n ! w n(w) = ! 2z ! 2 · " w "2 ! w = 0.5 n = (2 · 1.96 · 3.5/0.5)2 ! 753 n = 50 ! = 3.5 5.0± 1.96 · 3.5/ ! 50 = 5.0± 0.97 = (4.03, 5.97) w = 2 · 0.97 = 1.94 Derivation of CI: example • ; . • Can show has chi-square distribution with degrees of freedom, . Since this is a known distribution (in particular, it doesn’t depend on ), is a pivot. 10 09/09/2005 05:22 PMchspdftb.gif 380!280 pixels Page 1 of 1http://www.itl.nist.gov/div898/handbook/eda/section3/gif/chspdftb.gif a b X1, . . . , Xn ! Exp(!) p(x) = !e!!x , x > 0, ! > 0 h(X1:n, !) = 2n!X̄ 2n !22n ! h(X1:n, !) • So implies • We pivoted to get the 100(1- )% CI for . 11 P ! a 2n X̄ < ! < b 2n X̄ " = 1! " P(a < 2n!X̄ < b) = 1! " h(X1:n, !) ! ! a 2n X̄ , b 2n X̄ " ! Large-sample CIs • Up to now, popn distrn was , , and was known. • When is large, we can get rid of both assumptions, and our previous CI for the popn mean is still approximately correct. • Let be a sample from any distrn with unknown mean and unknown variance (both finite). - Central Limit Theorem says - so • We are back to our previous CI derivation. 12 N (µ, ! 2) ! 2 n µ X1, . . . , Xn µ ! 2 X̄ ! N (µ, ! 2/n) Z = (X̄ ! µ)/(!/ " n) # N (0, 1)
