Question

Appendix D: Estimation and Hypothesis Testing. 1. Briefly, in your own words, explain what it means for an estimator to be BLUE. 2. Briefly, in your own words, explain a type I error. 3. Briefly, in your own words, explain a type II error. 4. Use the table below to answer this question: what is the 90% confidence interval for the mean fx given a sample X with 24 degrees of freedom? (a) Beginning with the formula P(-terit < t-value < +terit) = Pci, compute the lower and upper limits - in terms of X, Sx and n - on the population mean. (b) Replace X, Sx and n based on a sample of 25 points with a sample mean of 18 and a sample variance of 36. d.f. 22 23 24 25 26 0.80 1.321 1.319 1.318 1.316 1.315 confidence 0.90 0.95 1.717 2.074 1.714 2.069 1.711 2.064 1.708 2.060 1.706 2.056 www 0.98 2.508 2.500 2.492 2.485 2.479

Answer 1

Sair. #Best Lineart Intiased Estimator - [ BLUE] an orod let (x, X2, ..., Xu) be a random sample from a population with means and Variance or (<00). Then an ertimator T(x) = Žaixi, a linear function, u Xi's is callech a linear estimator ... [the estimators To Eau xe 2, Tz = zaieti are non-linear estimators]. A lineare estimators T = arxt is unbiased for my ()y 4 cm Duzy (3 arxada ya 1 s (aixida er ai E (xi)= Consider the class of all linear I žaikat. Untiased estimation of nie (= {T: Ta Žaixo, & ai= 1} We wish to find the best estimator in the Claro e in the sense of having minimum MSE. / BLUE- A linear subiased estimates To Saixi, Sairs, of the popth mean gr, that has the minimam Variance among all linear Vie's of gra å called the Blue of M. 21 -

Example - let a zen an de an ol 3, H... an de an . Considen a hypothesis observed sample from 1:0=17, one possible O 10 test is as follows: Reject tiff nx 17 + 5/5m 1. Critical region & lest - tet & be collection of all porrible samples of size n. Here it is called the sample space. clearly & ER. A lest procedure assigns to each possible sample sex, one of todo decisions; accept thi or reject H (ow) it is accepted. The w ex in kn is known as the citical region o & we is called the critical region of acceplance.

46% Performance of a Test - While performing a test one may arrive at the correct decision or may commit one of two errors. (o rejecting null hypothesis Ho when it is really lone [type II (in accepting Ho when it is really fase [type II Reject. Ho to Accept to | Tone State Ho is true correct Decision froma l sample Wrong I type I error] Ho is false l correct (Hi is trine) LOW

Then there are ludo erron probatinties : by the proc. of rejecting to when it is habe fmea P[ xew10], 06 20 probe of mjpe - I.error. ut the proc. of accepting to when it is fasoe. - P[ XE w/o], deel - 1- P(x EW|0] , der s prob. of type 1 error.

4 90% CI level let x = level for. Mx of Confidence = 0.90 of significance 2 1-0.90 0.10. - 1-2 Is pl-tair a't-value at teir) - Pel. 4 PL - Halaguici s top;:1) x= sample mean S = Sample standard deviation na sauple não plus/anta/2; n-1s X-M 2 I to/2n-1) it P(-x- /ta/25n-1 L-M <-x + s tappine) 9 PE (84 santa finale = 1-2 is taljn-1 E-M2 (8-5/intars no 1-11

a toros; 24 from t table [-(** su twf 3 und 5-M < (- Sun tx/131-137 (x - vega thean al-d. pl its taheinis > Sh change to x-s tape}n-i] = 1-2 a pl ²-5 tasign-1 L MG e me D_1414 internal - (x - I to/2; 4-1, x + 2 talin- - (1- xi-all, 18+ 4+1-71) ^ (15.9468, 20.05 32). Aus #tian tera;-u) sind ( *- Fun tolazury s** Santay-31) a mleñal kneen limit upper liunt (6) Sx=36,5x26 M = 18, n=25