XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON
XI-2 (1075) STATISTICAL DECISION MAKING Advaced Statistics Advaced Statistics is preseted i the followig topic areas: C Statistical decisio makig C Aalysis of variace (ANOVA) C Relatioships betwee variables C Desig ad aalysis of experimets Statistical Decisio Makig Statistical Decisio Makig is preseted i the followig topic areas: C Poit estimates C Cofidece itervals C Hypothesis testig C Paired-compariso tests C Goodess-of-fit tests C Cotigecy tables
XI-2 (1076) STATISTICAL DECISION MAKING Statistical Iferece The objective of statistical iferece is to draw coclusios about populatio characteristics based o the iformatio cotaied i a sample. The steps ivolved i statistical iferece are: C Precisely defie the problem objective C Formulate a ull ad a alterate hypothesis C Decide if the problem will be evaluated by a oe-tail or two-tail test C Select a test distributio ad a critical value for the test statistic C Calculate a test statistic from the sample C Make a iferece by comparig the calculated ad the critical values C Report the fidigs
XI-3 (1077) STATISTICAL DECISION MAKING / POINT ESTIMATES Poit Estimate for Populatio Mea I aalyzig sample values to arrive at populatio probabilities, two major estimators are used: poit estimates ad cofidece itervals. A poit estimate of the populatio mea, μ, is the sample mea, X. Xi i = 1 X = Example: Give the followig tesile stregth readigs from 4 piao wire segmets: 28.7, 27.9, 29.2, ad 26.5 psi, calculate the poit estimatio of the populatio mea. X i i = 1 X = = 28.7 + 27.9 + 29.2 + 26.5 4 X = 28.08 psi 28.08 psi is the poit estimate for the populatio mea.
XI-3 (1078) STATISTICAL DECISION MAKING / POINT ESTIMATES Poit Estimate for Populatio Variace The sample variace, s 2, is the best poit estimate of the populatio variace, σ 2. The sample stadard deviatio, s, is the best poit estimate of the populatio stadard deviatio, σ. X i - X X i - s = = - 1 2 2 2 i = 1 2 i = 1 X i - X X i - s = = - 1 2 2 i = 1 i = 1 N N
XI-4 (1079) STATISTICAL DECISION MAKING / CONFIDENCE INTERVALS Cofidece Iterval for the Mea Cotiuous Data - σ Kow The cofidece iterval of the populatio mea, μ, whe the populatio stadard deviatio, σ, is kow, is calculated usig the sample mea, X, the populatio stadard deviatio, σ, the sample size,, ad the ormal distributio. X - Z X + Z X /2 /2 From sample data oe ca calculate the iterval withi which the populatio mea, μ, is predicted to fall. Cofidece itervals are always estimated for populatio parameters. A cofidece iterval is a twotail evet ad requires critical values based o a alpha/2 risk i each tail. X
XI-4 (1080) STATISTICAL DECISION MAKING / CONFIDENCE INTERVALS Cotiuous Data - σ Ukow The cofidece iterval of the populatio mea, μ, whe the populatio stadard deviatio, σ, is ukow, is calculated usig the sample mea, X, the sample stadard deviatio, s, the sample size,, ad the t distributio. s s X - t /2, -1 X + t /2, -1 Example: The average of 25 samples is 18 with a sample stadard deviatio of 6. Calculate the 95% cofidece iterval for the populatio mea. s X - t X + t /2, -1 /2, -1 6 6 18-2.064 18 + 2.064 25 25 15.52 20.48 s
XI-5 (1081) STATISTICAL DECISION MAKING / CONFIDENCE INTERVALS Cofidece Iterval for Proportio For large sample sizes, with p ad (1-p) greater tha or equal to 5, the biomial distributio ca be approximated by the ormal distributio to calculate a cofidece iterval for populatio proportio. p 1 - p p 1 - p p - Z p p + Z s s s s s /2 s /2 p s = sample proportio p = populatio proportio = sample size
XI-6 (1082) STATISTICAL DECISION MAKING / CONFIDENCE INTERVALS Cofidece Iterval for Variace The cofidece iterval or iterval estimate for the populatio variace, σ 2, is give by: - 1 s 2 2 X 2 X 2 2 /2, - 1 1 - /2, - 1 s 2 = sample variace = sample size - 1 = degrees of freedom - 1s - 1 s Cofidece Iterval for Stadard Deviatio The cofidece iterval for the populatio stadard deviatio, σ, is give by: - 1s 2 2 X X 2 2 /2, - 1 1 - /2, - 1
XI-7 (1083) STATISTICAL DECISION MAKING / HYPOTHESIS TESTING Hypothesis Testig Hypothesis testig is a type of statistical iferece i which a ull hypothesis ad alterative hypothesis are stated. The ull hypothesis is a statemet about the value of a populatio parameter such as the mea, ad must cotai the coditio of equality. The alterative hypothesis is a statemet that must be true if the ull hypothesis is false. A ull hypothesis ca oly be rejected, or fail to be rejected, it caot be accepted because of a lack of evidece to reject it. As a example of hypothesis tests for a populatio mea, there are oly three possible forms, where μ is the populatio mea ad μ 0 is a specified value: H 0 : μ = μ 0 H 0 : μ <_ μ 0 H 0 : μ >_ μ 0 H 1 : μ =/ μ 0 H 1 : μ > μ 0 H 1 : μ < μ 0
XI-7 (1084) STATISTICAL DECISION MAKING / HYPOTHESIS TESTING Hypothesis Testig (Cotiued) The steps of hypothesis testig are: C State the ull ad alterative hypothesis C Specify the level of sigificace, α C Determie the critical values separatig the reject ad orejectio areas C Determie the samplig distributio ad test statistic C Determie if the test statistic is i the reject or orejectio area C Coclude if the ull hypothesis is rejected or failed to be rejected C State the statistical decisio i terms of the origial problem
XI-8 (1085) STATISTICAL DECISION MAKING / HYPOTHESIS TESTING Types of Errors Whe formulatig a coclusio regardig a populatio based o observatios from a small sample, two types of errors are possible: C Type I error: This error results whe the ull hypothesis is rejected whe it is, i fact, true. The probability of makig a type I error is called α (alpha) or producer s risk. C Type II error: This error results whe the ull hypothesis is ot rejected whe it should be rejected. This error is called the cosumer s risk ad is deoted by the symbol β (beta). The degree of risk (α) is ormally chose by the cocered parties (α is ofte take as 5%) i arrivig at the critical value of the test statistic. Icreasig the sample size ca reduce both the α ad β risks.
XI-8 (1086) STATISTICAL DECISION MAKING / HYPOTHESIS TESTING Types of Errors (Cotiued) The types of errors are show i the Figure below: Null Hypothesis True False The Decisio Made Fail to Reject H 0 Reject H 0 p = 1 - α Correct Decisio p = α Type I Error p = β Type II Error p = 1- β Correct Decisio Error Matrix
XI-9 (1087) STATISTICAL DECISION MAKING / HYPOTHESIS TESTING Oe-Tail Test If a ull hypothesis is established to test whether a sample value is smaller or larger tha a populatio value, the the etire α risk is placed o oe ed of a distributio curve. This costitutes a oe tail-test. H 0 : ew <_ to preset H 1 : ew > preset ENTIRE " = 5% 0 : 0 = 35 HOURS Determie if the true mea is withi the α critical regio.
XI-10 (1088) STATISTICAL DECISION MAKING / HYPOTHESIS TESTING Two-Tail Test If a ull hypothesis is established to test whether a populatio shift has occurred, i either directio, the a two-tail test is required. The allowable α error is geerally divided ito two equal parts. H 0 : levels are = H 1 : levels are =/ = 0.025 2 = 0.025 2-1.96 0 0 +1.96 Determie if the true mea is withi either the upper or lower α critical regios.