2017, Study Session # 3, Reading # 12 ∝ = Level of Significance TS = Test Statistics TV = Table Value Hypothesis Testing Procedure Hypothesis Statement about one or more populations Null Hypothesis H0 Tested for possible rejection Always includes ‘=’ sign SS = Sample Statistic CV = Critical Value SE = Standard Error “HYPOTHESIS TESTING” Two Types Alternative Hypothesis Ha Hypothesis is accepted when the null hypothesis is rejected It is based on sample statistics & probability theory It is used to determine whether a hypothesis is a reasonable statement or not Steps in Hypothesis Testing One Tailed Test Alternative hypothesis having one side Upper Tail H0:µ ≤ µ0 vs Ha: µ > µ0 Decision rule Reject H0 if TS > TV Lower Tail H0:µ ≥ µ0 vs Ha: µ < µ0 Decision rule Reject H0 if TS < TV Two Tailed Test Alternative hypothesis having two sides H0: µ = µ0 vs Ha µ ≠ µ0 Reject H0 if TS > TV or TS < –TV State the hypothesis Identify the appropriate test statistic and its probability distribution Specify the significance level State the decision rule Collect the data & calculate the test statistic Make the statistical decision Make the economic or investment decision (Source: Wayne W Daniel and James C Terrell, Business Statistics, Basic Concepts and Methodology, Houghton Mifflin, Boston, 1997.) Test Statistics Statistical Significance vs Economical Significance Statistically significant results may not necessarily be economically significant A very large sample size may result in highly statistically significant results that may be quite small in absolute terms Hypothesis testing involves two statistics: TS calculated from sample data Critical values of TS Two Types of Errors Type I Error Rejecting a true null hypothesis Type II Error Failing to reject a false null hypothesis Decision Rule Based on comparison of TS to specified value(s) It is specific & quantitative Copyright © FinQuiz.com All rights reserved Significance Level (α ) Probability of making a type I error Denoted by Greek letter alpha (α ) Used to identify critical values 2017, Study Session # 3, Reading # 12 σ = Population Variance N.Dist = Normally Distributed N.N.Dist = Non Normally Distributed Relationship b/w Confidence Intervals & Hypothesis Tests Related because of critical value C.I [(SS)- (CV)(SE)] ≤ parameter ≤ [(SS) + (CV)(SE)] It gives the range within which parameter value is believed to lie given a level of confidence Hypothesis Test -C V ≤ TS ≤ + CV Range within which we fail to reject null hypothesis of two tailed test at given level of significance Testing • Population Mean • • • • • σ2 known N dist ݖ n ≥30 unknown Decision Rule x z= n ௫̅ ିఓ = ௦ బ ൗ √ σ2 unknown n TV or TS < – TV ( x1 − x2 ) − ( µ1 − µ ) 1 sP + n1 n2 • ( n1 − 1) s1 + ( n2 − 1) s n1 + n2 − 2 Ho:µ1 - µ2 ≤ vs Ha: µ1 -µ2 > Reject H0 if TS > TV df = n1+n2 - Distributed Ho:µ > µ0 vs Ha: µ µ0 Reject H0 if TS > TV ௫̅ ିఓ ∗ or ݐିଵ = ௌ/ బ √ *(more conservative) Means of Two Samples Power of a Test – P(type II error) Probability of correctly rejecting a false null hypothesis Test Statistics t( Equality of the p- value The smallest level of significance at which null hypothesis can be rejected Reject H0 if p-value < α Conditions • df = Degree of Freedom n ≥ 30 = Large Sample n< 30 = Small Sample n = Sample Size ( x1 − x ) − ( µ1 − µ ) • Reject H0 if TS < -TV s12 s 22 + n1 n 2 s12 s22 + n n d f = 12 s12 s22 n1 + n2 n1 n2 Copyright â FinQuiz.com All rights reserved Ho:à1 - µ2 > vs Ha: µ1 -µ2 < • Ho:µ1 - µ2 = vs Ha: µ1 - µ2 ≠ Reject H0 if TS > TV or TS < – TV 2017, Study Session # 3, Reading # 12 Paired Comparisons Test TS t(n-1 )= ௗതିఓబ ௦ ഥ TS ଶ ߯ሺିଵሻ ݀̅ = ݀ ݊ ୀଵ Sୢഥ = ݏௗ =ඨ Testing Variance of a N.dist Population Sୢ √n ∑ୀଵ(݀ − ݀̅ )ଶ ݊−1 = ሺ݊ − 1ሻ ݏଶ ߪଶ Decision Rule Reject H0 if TS > TS Chi-Square Distribution Asymmetrical Bounded from below by zero Chi-Square values can never be –ve Testing Equality of Two Variances from N.dist Population TS =ܨ ௌభమ ; ܵଵଶ ௌమమ > ܵଶଶ Parametric Test Specific to population parameter Relies on assumptions regarding the distribution of the population Non-Parametric Test Decision Rule Reject H0 if TS > TV F- Distribution Right skewed Bounded by zero Decision Rule H0: µd ≤ µd0 vs Ha: µd > µd0 Reject H0 if TS > TV H0: µd ≥ µd0 vs Ha:µd < µd0 Reject H0 if TS TV Copyright © FinQuiz.com All rights reserved Do not consider a particular population parameter Or Have few assumptions regarding population ... Sample n = Sample Size ( x1 − x ) − ( 1 ) Reject H0 if TS < -TV s12 s 22 + n1 n 2 s12 s22 + n n d f = 12 s12 s22 n1 + n2 n1 n2 Copyright © FinQuiz.com... TS > TV or TS < – TV ( x1 − x2 ) − ( 1 − µ ) 1 sP + n1 n2 • ( n1 − 1) s1 + ( n2 − 1) s n1 + n2 − 2 Ho: 1 - µ2 ≤ vs Ha: 1 -µ2 > Reject H0 if TS > TV df = n1+n2 - Distributed Ho:µ > µ0 vs Ha: µ... rights reserved Ho: 1 - µ2 > vs Ha: 1 -à2 < Ho: 1 - à2 = vs Ha: 1 - µ2 ≠ Reject H0 if TS > TV or TS < – TV 2 017 , Study Session # 3, Reading # 12 Paired Comparisons Test TS t(n -1 )= ௗതିఓబ ௦ ഥ