Concepts Module 7: Comparing Datasets and Comparing a Dataset with a Standard

Transcription

1 Cocepts Module 7: Comparig Datasets ad Comparig a Dataset with a Stadard Idepedece of each data poit Test statistics Cetral Limit Theorem Stadard error of the mea Cofidece iterval for a mea Sigificace levels How to apply i Excel How differet is eough? Test Statistics Idepedet Measuremets Some umber calculated based o data I studet s t test, for example, t If t is >= 1.96 ad Each measuremet must be idepedet (shake up basket of tickets) Example of o-idepedet measuremets populatio ormally distributed, you re to right of curve, where 9 of data is i ier portio, symmetrically betwee right ad left (t=1.96 o right, o left) Public resposes to questios (oe result affects ext perso s aswer) Samplers too close together, so air flows affected 3 Test statistics correspod to sigificace levels 4 Two Major Types of Questios P stads for percetile Pth percetile is where p of data falls below, ad 1-p fall above Comparig mea agaist a stadard Comparig two datasets 2 Does air quality here meet NAAQS? Is air quality differet i 26 tha 2? Better? Worse? 6

2 Comparig Mea to a Stadard Cetral Limit Theorem (magic) Did air quality meet CARB aual stadard of 12 microg/m3? Ft Ft Smith Ft Smith N_Fort Smith avg Max Smith Mi year 7 Magic Cocept #2: Stadard Error of the Mea Represets ucertaity aroud mea As sample size N gets bigger, error gets smaller The bigger the N, the more tightly you ca estimate mea LIKE stadard deviatio for a populatio, but this is for YOUR sample Eve if uderlyig populatio is ot ormally distributed If we repeatedly take datasets These differet datasets have meas that cluster aroud true mea Distributio of these meas is ormally distributed 8 For a large sample (N > 6), or whe very close to a ormal distributio Cofidece iterval for populatio mea is: & s # = N Choice of z determies 9, 9, etc. 9 1 Studet s t Distributio vs. Normal Z Distributio For a Small Sample Replace Z value with a t value to get T-distributio ad Stadard Normal Z distributio s x ± t$ ' # &.4 Z distributio desity.3.2 T with d.f. where t comes from Studet s t distributio, ad depeds o sample size Value 12

3 What happes as sample gets larger? Compare t ad Z Values Cofidece t value with Z value level d.f T-distributio ad Stadard Normal Z distributio.4 Z distributio desity.3 T with 6 d.f What happes to CI as sample gets larger? For large samples Z ad t values become almost idetical, so CIs are almost idetical & & x ± t$ 14 First, graph ad review data s # s # Use box plot add-i Evaluate spread Evaluate how far apart mea ad media are (assume samplig desig ad QC are good) 1 Excel Summary Stats 2. Calculate summary stats Use the box-plot add-i 4 3 Value module 7 13 Ft Smith N=77 Mi 2th Media 7th Max Mea SD

4 The mea is what? Our Questio We Ca we be 9, 9, or how cofidet that this mea of is really greater tha stadard of 12? We saw that N = 77, ad mea ad media ot too differet Use z (ormal) rather tha t kow equatio for CI is & s # 19 Width of cofidece iterval represets how sure we wat to be that this CI icludes true mea Now, decide how cofidet we wat to be 2 Excel ca also calculate a cofidece iterval aroud the mea CI Calculatio For 9, z = 1.96 (ofte rouded to 2) Std error (sigma/n) = (8.66/square root of 77) =.98 CI aroud mea = 2 x.98 We ca be 9 sure that mea is icluded i (mea +- 2), or at low ed, to at high ed This does NOT iclude 12 Mea, plus ad mius 1.93, is a 9 cofidece iterval that does NOT iclude We kow we are more tha 9 cofidet, but how cofidet ca we be that Ft Smith mea > 12? To fid where we are o the curve, calc the test statistic Calculate where o curve our mea of 14.8 is, i terms of z (ormal) score or if N small, use t score 23 Ft Smith mea = 14.8, sigma =8.66, N =77 Calculate test statistic, i this case the z factor z= (we decided we ca use the z rather tha the t distributio) 22 If N was < 6, test stat is t, but calculated the same way (x µ) N Data s mea Stadard of 12 24

5 Calculate z Easily Where o the curve? Our mea 14.8 mius stadard of 12 (treat real mea µ (mu) as stadard) is umerator (= 2.8) Stadard error is sigma/square root of N =.98 (same as for CI) so z = (2.8)/.98 = z = 2.84 So where is this z o the curve? Remember, at z = 3 we are to the right of ~ 99 2 Z = 2 Z = 3 So betwee 9 ad 99 probable that the true mea will ot iclude You ca calculate exactly where o the curve, usig Excel Use Normsdist fuctio, with z If z (or t) = 2.84, i Excel Yields 99.8 probability that the true mea does NOT iclude 12 27