( )( ) χ = where Ei = npi. χ = where. : At least one proportion differs significantly. H0 : The two classifications are independent H

Transcription

1 Formula Card for Exam 4 ST33 Testig Categorical Probabilities: Oe Way Table Steps Express the Claim: : p = p = = pk ypotheses: * : t least oe proportio differs sigificatly 3 Get Data ad determie your alpha level: 4 Calculate the Test Stat: 5 Get Your Critical Value: χ ( O E ) i i χ = where Ei = pi E α, k 6 Form Your Iitial Coclusio: 7 Fial Coclusio: *Note: hypotheses will vary depedig upo the problem Testig Categorical Probabilities: Two-Way Table Steps Express the Claim: : The two classificatios are idepedet ypotheses: : The two classificatios are depedet 3 Calculate Expected Cell Values for each cell: 4 Calculate the Test Stat: ( O ˆ ) ij Eij χ = where Eˆ i ij ( )( ) row total colum total Eˆij = 5 Get Your Critical Value: χαwhich has (r )(c ) degrees of freedom 6 Form Your Iitial Coclusio: Reject the Null if χ > χ α 7 Fial Coclusio: The Sig Test Right tailed case: Step Claim: η > η* : η η Step ypotheses: : η > η Step 3 Test Stat: S = Number of measuremets greater tha η Step 4 Determie : = sample size mius ay sample measuremets which have a value equal toη P X S p = 5, ad fid ( ) p value = P( X S ) < α reject

2 The Sig Test (Left tailed case): Step Claim: η < η* : η η Step ypotheses: : η < η Step 3 Test Stat: S = Number of measuremets less tha η Step 4 Determie : = sample size mius ay sample measuremets which have a value equal toη P X S p = 5, ad fid ( ) p value = P( X S ) < α reject The Sig Test (Two tailed case): Step Claim: η η* : η = η Step ypotheses: : η η Step 3 Test Stat: S = Larger of S (# of measuremets less tha η ) ad S (# of measuremets more tha η ) S B Step 4 Determie : = sample size mius ay sample measuremets which have a value equal toη P X S p = 5, ad fid ( ) p value = P( X S ) < α reject *ote: The claim ca vary depedig upo the wordig of the problem Large sample case: Most biomial tables do ot exceed = 5, so whe we ecouter problems that have a sample size greater tha 5 we ca use a large sample approximatio The test stat will become: ( Smi + 5) z =, Where Smi is the umber of times the less frequet sig occurs The Wilcoxo Rak Sum Test (two-tailed case) Step Claim: D (distributio for pop ) ad D (distributio for pop ) are idetical Step : D ad Dare idetical, : D is shifted either to the left or to the right of D

3 Step 3 Rak the data as if it is all oe set of values (If ties exist give the tied values the average of the raks they would have gotte if they were i successive order) Step 4 Calculate T =sum of the raks for pop ad T = sum of the raks for pop ( + ) (s a check make suret + T = ) If <, T =T = Test stat, If >, T =T = Test stat (if both sample sizes are equal use either sum as your test stat) Step 5 The rejectio regio is determied by lookig up alpha i the Wilcoxo Rak-Sum table (Reject the ull if T TL ort TU ) Step 6 Form your iitial coclusio Step 7 Word fial coclusio Wilcoxo Rak Sum Test Oe-tailed test : D ad D are idetical a : D is shifted right of D or [ a : D is shifted left of D ] Test statistic: T, if < T, if > Either if = Rejectio regio: T : T T U or [T T L ] T : T T L or [T T U ] Two-tailed test : D ad D are idetical a : D is shifted right or left of D Test statistic: T, if < T, if > Either if = Rejectio regio: T T L or T T U Wilcoxo Sig-Rak Test (two-tailed case) Step : State the claim Step : List your hypotheses : = : η d

4 Step 3: Get your raks a) For each pair of data, fid the differece (d) by subtractig the secod value from the first Discard ay pairs for which d = b) Take the absolute values of d ad rak them from low (= ) to high If ay differeces are tied give the tied differeces the average of the raks they would have if they were i successive order Step 4: dd up all the raks of the egative differeces (T ) ad do the same for the raks of the positive differeces (T + ) Let T = mi( T, T ) + Step 5: Let = umber of pairs that have a ozero differece, the fid your Critical Value T from the Wilcoxo Sig-Rak Test table Step 6: Form your iitial coclusio We reject if our test stat is below our critical value (ie Reject whe T T ) Step 7: Word your fial coclusio Wilcoxo Sig-Rak Test (oe-tailed case) Step : State the claim Step : List your hypotheses : : > or : : < Step 3: Get your raks a) For each pair of data, fid the differece (d) by subtractig the secod value from the first Discard ay pairs for which d = b) Take the absolute values of d ad rak them from low (= ) to high If ay differeces are tied give the tied differeces the average of the raks they would have if they were i successive order Step 4: dd up all the raks of the egative differeces (T ) [dd up the raks of the positive differeces (T + )] Step 5: Let = umber of pairs that have a ozero differece, the fid your Critical Value T from the Wilcoxo Sig-Rak Test table Step 6: Form your iitial coclusio We reject if our test stat T is below our critical value T [or Reject whet + T ] Step 7: Word your fial coclusio The Kruskal-Wallis -test (Noparametric CRD) Step Express the Claim that all of the treatmets produce the same media Step ypotheses : η = η = = η k : t least two medias differ from each other Step 3 Temporarily view the etire data set as a whole ad rak the data values (verage the raks i case of a tie as usual) The for each treatmet add up its raks

5 Step 4 Calculate the test statistic R R R = ( N + ) N( N ) k + k Step 5 Get Critical Value: The statistic ca be approximated by a Chi-Squared distributio with k degrees of freedom, so we will look up alpha o the Chi-Squared table The test is always a right tailed test Step 6 State your iitial coclusio We will reject the ull whe Step 7 Word fial coclusio > χ α, k Friedma Fr Test (Noparametric RBD) Step Express the claim that all treatmets have the same probability distributio Step ypotheses : The probability distributios for the treatmets are idetical : t least two of the distributios differ i locatio Step 3 Rak the data values i each block (verage the raks i case of a tie as usual) The for each treatmet add up its raks Step 4 Calculate the test statistic Fr = R j 3 b( k + ) bk( k + ) Step 5 Get Critical Value: The F r statistic ca be approximated by a Chi-Squared distributio with k degrees of freedom, so we will look up alpha o the Chi-square table The test is always a right tailed test Step 6 State your iitial coclusio We will reject the ull whe Step 7 Word fial coclusio F r > χ α, k