Confidence Interval Procedures - Supplemental Review Sheet | STT 351, Study notes of Probability and Statistics

Download Confidence Interval Procedures - Supplemental Review Sheet | STT 351 and more Study notes Probability and Statistics in PDF only on Docsity! Supplemental review of CI and testing. 12-01-08 We've studied a variety of confidence interval (CI) procedures. Their objective, when used at 95 % confidence, is to establish a margin of error for a statistical estimator: q ` ± (t or z) Var ` Hq`L . In this notation q` denotes an estimator of some parameter q and Var ` Hq`L denotes the estimator of the standard deviation of estimator q`. When using bootstrap we typically never see (t or z) or Var ` Hq`L since the bootstrap method deliv- ers an estimator of the entire package (t or z) Var ` Hq`L . CI Here is a brief summary of most CI studied in this course (ex indicates that the CI is exact with perfect calculations, otherwise the CI is approximate as n -> ¶, and other assumptions including N - n -> ¶ when sampling finite populations). q ` ± (t or z) Var ` Hq`L caveat ___________________________________________ x ± t s n ex eq-pr w/r sam of n from normal pop x ± z s n eq-pr w/r sample of n x ± z s n N-n N-1 eq-pr without/r sample of n p̀ ± z p̀ H1- p̀L n eq-pr w/r sample of n p̀ ± z p̀ H1- p̀L n N-n N-1 eq-pr without/r sample of n x-y ± z sx 2 nx + sy2 ny indep x, y sam, eq-pr, w/r px` - py` ± z px ` H1-px` L nx + py ` H1-py` L ny same as just above q ` ± 95-th (or other) percentile of †q*̀ - q† as approp x ± z ⁄i=1 K Wi si2 n prop'l eq-pr w/r strat sam p̀ ± z ⁄i=1 K Wi pi ` H1-pi` L n prop'l eq-pr w/r strat sam (y + (x-mx) r s̀y s̀x ) ± z sy n 1 - r2 eq-pr w/r prs (x, y) b ` i ± z Hi, iL entry of Hxtr xL-1 n-1n-d s@residD2 with { ei} ind N(0, s2) b ` = PseudoInverse[x].y ỳ = x.b ` resid = y - ỳ q ` ± (t or z) Var ` Hq`L caveat ___________________________________________ x ± t s n ex eq-pr w/r sam of n from normal pop x ± z s n eq-pr w/r sample of n x ± z s n N-n N-1 eq-pr without/r sample of n p̀ ± z p̀ H1- p̀L n eq-pr w/r sample of n p̀ ± z p̀ H1- p̀L n N-n N-1 eq-pr without/r sample of n x-y ± z sx 2 nx + sy2 ny indep x, y sam, eq-pr, w/r px` - py` ± z px ` H1-px` L nx + py ` H1-py` L ny same as just above q ` ± 95-th (or other) percentile of †q*̀ - q† as approp x ± z ⁄i=1 K Wi si2 n prop'l eq-pr w/r strat sam p̀ ± z ⁄i=1 K Wi pi ` H1-pi` L n prop'l eq-pr w/r strat sam (y + (x-mx) r s̀y s̀x ) ± z sy n 1 - r2 eq-pr w/r prs (x, y) b ` i ± z Hi, iL entry of Hxtr xL-1 n-1n-d s@residD2 with { ei} ind N(0, s2) b ` = PseudoInverse[x].y ỳ = x.b ` resid = y - ỳ Tests from CI Here is a brief description of how CI may be used to test hypotheses of the kind stud- ied in tis course. The idea is a simple one. A CI is trying to locate (cover) a parame- ter q. If it is a 95% CI then P(CI covers q) ~ 0.95. So P(CI misses q) ~ 1 - 0.95 = 0.05. If could devise a test of (for example) the null hypothesis that q = 17 versus the two-sided alternative hypothesis q ≠ 17 which rejects H0 : q = 17 if CI fails to cover 17. If truly q = 17 such a test commits type I error precisely when CI fails to cover 17. This has probability 0.05 as above. So a = 0.05 for such a use of CI to test. If, instead, we wish to perform a one-sided test H0: q = 17 versus Ha: q > 17 we could harness the CI in the following way: reject H0 if CI falls entirely to the right of 17. For this test, a = (1-0.95)/2 = 0.025 since the 0.05 probability of having the CI fail to cover 17 is about equally divided between missing to the left or missing to the right. Testing H0: q = 17 versus Ha: q < 17 we would reject H0 if CI falls entirely to the left of 17. Once again, a = 0.025. 2 review12-01-08.nb