5.2.6 COMPARISON OF QUALITY CONTROL AND VERIFICATION TESTS

Transcription

1 5..6 COMPARISON OF QUALITY CONTROL AND VERIFICATION TESTS Thi proedure i arried out to ompare two different et of multiple tet reult for finding the ame parameter. Typial example would be omparing ontrator QC tet reult and KDOT erifiation tet reult to determine if the material under tet ame from the ame population. The tatitial tet whih would be ued to ompare two mean would be popularly known a Student' t-tet or imply t-tet for teting a null hypothei (H 0 ) with ertain onfidene (e.g. 99%) or leel of ignifiane (rik of rejeting a null hypothei when it i true, e.g., 1%) i a follow: H o : There i no differene in the ample mean, ie. the mean are tatitially equal If the tet reult do not upport thi hypothei than an alternate hypothei (H a ) i aepted a: H a : The mean are different, ie. the mean are not tatitially equal Thi tet i generally appliable when the number of tet (or oberation a i known in Statiti) i le than or equal to 30. Howeer, ine the approah ued in the t-tet i dependent upon whether or not the ariane (quare of the ample tandard deiation) are equal for the two et of data, it i neeary to tet the ariane of the tet reult before omparing the mean of the tet reult. F-tet for the Sample Variane The F-tet determine whether the differene in the ariability of the ontrator' QC tet and that of KDOT' erifiation tet i larger than might be expeted from hane if they ame from the ame population. In thi ae, a hypothei teting i done at a ertain leel of ignifiane. The null hypothei in the tet i: H o : There i no differene in the ample ariane, ie. the ariane are tatitially equal If the tet reult do not upport thi hypothei than an alternate hypothei i aepted a: H a : The ariane are different, i.e. the ariane are not tatitially equal The following tep need to be followed in doing an F-tet: i) Compute the ariane (the tandard deiation quared) for the QC tet,, and the KDOT erifiation tet, ii) Compute F tatiti a: F / or / Alway ue the larger of the two ariane in the numerator. iii) Chooe the leel of ignifiane,, for the tet. The reommended i 1%. Page 1/

2 i) Find the ritial F alue F rit, from the Table 1 uing the degree of freedom aoiated with eah et of tet reult. The degree of freedom for eah et of reult i the number of tet reult in the et, le one. If the number of QC tet i n and the number of erifiation tet i n, then the degree of freedom aoiated with i (n -1) and the degree of freedom aoiated with i (n -1 ). The alue in Table are tabulated to tet if there i a differene (either larger or maller) between two ariane etimate. Thi i known a a two-ided or two-tailed tet. Care mut be taken when uing other table of the F ditribution, ine they are uually baed on a one-tailed tet, ie., teting peifially whether one ariane i larger than another. When finding F rit be ure that the appropriate degree of freedom for the numerator and denominator are ued. ) Find the alue for F rit from Table 1. i) If F F rit, then the null hypothei i rejeted i.e. the two et of tet hae ignifiantly different ariabilitie. If F F rit then there i no reaon to beliee that the ariabilitie are ignifiantly different. t-tet for Sample Mean One the ariane hae been teted and been aumed to be either equal or not equal, the mean of the tet reult an be teted to determine whether they differ from one another or an be aumed equal. The deire i to determine whether it i reaonable to aume that the QC tet ame from the ame population a the erifiation tet. A mentioned before, a t-tet i ued to ompare the ample mean. Two approahe for the t-tet are neeary. If the ample ariane are aumed equal, then the t-tet i onduted baed on the two ample uing a pooled etimate for the ariane ( p ) and the pooled degree of freedom. If the ample ariane are found to be different in the F-tet, the t-tet i onduted uing the indiidual ample ariane, the indiidual ample ize, and the effetie degree of freedom (etimated from the ample ariane and ample ize). In either of the two ae diued earlier, the null hypothei ued i: H o : There i no differene in the ample mean, i.e. the mean are tatitially equal If the tet reult do not upport thi hypothei than an alternate hypothei i aepted a: H a : The mean are different, i.e. the mean are not tatitially equal Page /

3 DEGREES OF FREEDOM FOR DENOMINATOR Table 1 Critial Value, F rit for the F-tet for a Leel of Signifiane, =1% DEGREES OF FREEDOM FOR NUMERATOR NOTE : Thi i for a two-tailed tet with the null and alternate hypothee hown below: H o : = H a : Page 3/

4 DEGREES OF FREEDOM FOR DENOMINATOR Table 1 Critial Value, F rit, for the F-tet for a Leel of Signifiane, =1 % (ontd..) DEGREES OF FREEDOM FOR NUMERATOR NOTE : Thi i for a two-tailed tet with the null and alternate hypothee hown below: H o : H a : = Page 4/

5 Cae 1: Sample Variane Aumed to Be Equal a) To ondut the t-tet when the ample ariane are aumed equal, Equation 1 i ued to alulate the t alue from whih the deiion i reahed. t X n X p p n (1) where: X = mean of QC tet X = mean of erifiation tet p = pooled etimate for the ariane (deribed below) n = number of QC tet n = number of erifiation tet b) The pooled ariane, whih i the weighted aerage, uing the degree of freedom for eah ample a the weighting fator, i omputed from the ample ariane uing Equation. p n 1 n 1 n n () Where: p = pooled etimate for the ariane n = number of QC tet n = number of erifiation tet = ariane of the QC tet = ariane of the erifiation tet ) One the pooled ariane i etimated, the alue of t i omputed uing equation 1. d) To determine the ritial t alue againt whih to ompare the omputed t alue, it i neeary to elet the leel of ignifiane,. A alue of = 1 % i reommended. e) Determine the ritial t alue, t rit, from Table for the pooled degree of freedom. The pooled degree of freedom for the ae where the ample ariane are aumed equal i (n + n -). f) If t t rit, then deide that the two et of tet hae ignifiantly different mean. If t t rit, then deide that there i no reaon to beliee that the mean are ignifiantly different. Page 5/

6 Cae : Sample Variane Aumed to Be Not Equal a) To ondut the t-tet when the ample ariane are aumed not equal, Equation 3 i ued to alulate the t alue from whih the deiion i reahed. t X X n n (3) where: X = mean of QC tet X = mean of erifiation tet = ariane of the QC tet = ariane of the erifiation tet n = number of QC tet n = number of erifiation tet b) To determine the ritial t alue againt whih to ompare the omputed t alue, it i neeary to elet the leel of ignifiane,. A alue of = 1% i reommended. ) The effetie degree of freedom, f', for the ae where the ample ariane are aumed not equal i determined from Equation 4 (the alulated effetie degree of freedom i rounded down to a whole number). f ' n n n n n 1 n 1 (4) where all the ymbol are a deribed preiouly. d) Determine the ritial t alue, t rit, from Table for the effetie degree of freedom determined by Equation 4. e) If t t rit, then deide that the two et of tet hae ignifiantly different mean. If t t rit, then deide that there i no reaon to beliee that the mean are ignifiantly different. Page 6/

7 Table Critial t alue degree of = 0.01 = 0.05 = 0.10 freedom NOTE : Thi i for a two-tailed tet with the null and alternate hypothee hown below : H o : X X H a : X X Page 7/

8 Example Problem 1-Conrete A ontrator ha run 1 QC tet for ompreie trength and KDOT ha run 5 erifiation tet oer the ame period of time. The reult are hown below. I it likely that the tet ame from the ame population? Contrator QC Tet Reult KDOT VerifiationTet Reult (%) (%) X = X =37.30 A t-tet between the mean of thee two et of reult an be ued to tet whether the mean reult of the tet done by the ontrator and KDOT are tatitially different. If they are not different, then it i likely that they ame from the ame population. Howeer, firt the F-tet need to be done to determine whether or not to aume the ariane of the QC tet reult differ from the KDOT erifiation tet. Step 1. Compute the mean and tandard deiation for eah et of data: QC tet reult KDOT Verifiation tet reult X X =.736 = Step. Compute ariane,, for eah et of tet reult (ariane i quare of the tandard deiation): QC tet reult KDOT Verifiation tet reult Page 8/

9 Step 3. Compute F, uing the larget in the numerator. F Step 4. Determine F rit from Table 1 being ure to ue the orret degree of freedom for the numerator (n - 1 = 5-1 = 4) and the denominator (n - 1 = 1-1 = 0). From Table , at = 1 %, F rit = 5.17 Conluion: Step 5. Sine F < F rit (i.e., 1.34 < 5.17), there i no reaon to beliee that the two et of tet hae different ariabilitie. That i, they ould hae ome from the ame population. Sine we an aume that the ariane are equal, we an ue the pooled ariane to alulate the t-tet tatiti, and the pooled degree of freedom to determine the ritial t alue, t rit. Compute the pooled ariane, p, uing the ample ariane from aboe. p p 1 1 n n n n (4) 7.86 Step 6. Compute the t-tet tatiti, t. t X n X p p n t Step 7. Determine the ritial t alue, t rit, for the pooled degree of freedom degree of freedom = (n + n -) = ( ) = 4. Page 9/

10 From Table, for = 1 % and degree of freedom = 4, t rit =.80. Conluion: Sine.87 >.80, we aume that the ample mean are not equal. It i therefore probable that the two et of tet reult did not ome from the ame population (or lot). Example Problem - Cae -Aphalt A ontrator ha run 10 QC tet and KDOT ha run 5 erifiation tet oer the ame period of time for the aphalt paement denity (%G mm ). The reult are hown below. I it likely that the tet ame from the ame population or lot? Contrator QC Tet Reult KDOT Verifiation Tet Reult X X A t-tet between the mean of thee two et of reult an be ued to tet whether the mean reult of the %G mm done by the ontrator and KDOT are tatitially different. If they are not different, then it i likely that they ame from the ame population. Firt, we hae to determine whether the ariane of the QC tet differ from the erifiation tet uing F-tet. Step 1. Compute the mean and tandard deiation for eah et of data: QC tet reult KDOT Verifiation tet reult X X 936. = = 1.1 Step. Compute the ariane,, for eah et of tet (ariane i the quare of the tandard deiation): Page 10/

11 Step 3. Compute F, uing the larget in the numerator. F Step 4. Determine F rit from Table 1 (be ure to ue the orret degree of freedom for the numerator (n - 1 = 5-1 = 4) and the denominator (n - 1 = 10-1 = 9)). From Table 1, at = 1 %, F rit = 7.96 Conluion: Sine F F rit (i.e., ), there i reaon to beliee that the two et of tet hae different ariabilitie. Thu, it i likely that they ame from population with different ariane. Sine we CAN NOT aume that the ariane are equal, we annot ue the pooled ariane to alulate the t-tet tatiti, and the pooled degree of freedom to determine the ritial t alue, t rit. Step 5. Compute the t-tet tatiti, t. t X n X n Step t Determine the ritial t alue, t rit, for the approximate degree of freedom (the alulated effetie degree of freedom i rounded down to a whole number). f ' n n n n n 1 n 1 Page 11/

12 ' f From Table, for = 1 % and degree of freedom = 4 (rounded down to the nearet whole number) t rit = 4.60 Conluion: Sine t < t rit, (i.e., 1.3 < 4.60), there i no reaon to aume that the ample mean are not equal. It i, therefore, reaonable to aume that the et of tet reult ame from Aphalt Paing Exel Spreadheet The Air Void F & t portion of the EXCEL preadheet ompare the Contrator Quality Control (QC) reult and KDOT erifiation reult uing the following proe: In lot 1 and, the mean and tandard deiation of the QC reult are alulated and ompared to the mean of the erifiation reult. The omparion i onidered to be atifatory (Pa) if the mean of the erifiation reult for that lot i within the greater of: 1. the mean of the QC reult three tandard deiation of the QC reult for that lot. one perent of the mean of the QC reult for that lot Starting with lot 3, the F & t tet are ued to ompare the QC reult and erifiation reult. All of the QC reult and erifiation reult are ued in the omparion for lot 3, 4 and 5. Starting with lot 6, all of the QC reult and erifiation reult for the lat fie lot are ued in the omparion. For example, the tet reult from lot -6 are ued in the omparion for lot 6. The maximum peifi graity (G mm ) F & t portion of the EXCEL preadheet ompare the QC reult and erifiation reult uing the follow proe: In lot 1 and, the mean and tandard deiation of the QC reult are alulated and ompared to the mean of the erifiation reult. The omparion i onidered to be atifatory (Pa) if the mean of the erifiation reult for that lot i within the greater of : 1. the mean of the QC reult three tandard deiation of the QC reult for that lot. 0.0 of the mean of the QC reult for that lot Starting with lot 3, the F & t tet are ued to ompare the QC reult and erifiation reult. All of the QC reult and erifiation reult are ued in the omparion for lot 3, 4 and 5. Starting with lot 6, all of the QC reult and erifiation reult for the lat fie lot are ued in the omparion. For example, the tet reult from lot -6 are uing in the omparion for lot 6. Page 1/

13 If the reult of omparion of the G mm QC and erifiation reult for a lot are atifatory (Pa), the QC G mm reult hould be ued in the alulation of %G mm for both the QC and erifiation Denity reult. If the reult of the omparion of the G mm QC and erifiation reult for a lot are not atifatory (Fail), the erifiation G mm reult hould be ued in the alulation of %G mm for both the QC and erifiation Denity reult. The Denity F & t portion of the EXCEL preadheet ompare the QC reult and erifiation reult uing the follow proe: All of a lot QC reult and erifiation reult are ued in the omparion for that lot. Eah lot tand on it own for the Denity F & t omparion. For the Air Void, G mm and Denity F & t omparion, the reult are onidered atifatory (Pa) if the t-tet how that the Contrator QC reult and KDOT QA reult are from the ame population with a of 1%. Conrete Paing Exel Spreadheet The Compreie Strength and Thikne F & t portion of the EXCEL preadheet ompare the Contrator Quality Control (QC) reult and KDOT erifiation reult uing the following proe: In lot 1 and, the mean and tandard deiation of the QC reult are alulated and ompared to the mean of the erifiation reult. The omparion i onidered to be atifatory (Pa) if the mean of the erifiation reult for that lot i within the mean of the QC reult three tandard deiation of the QC reult for that lot. Starting with lot 3, the F & t tet are ued to ompare the QC reult and erifiation reult. All of the QC reult and erifiation reult are ued in the omparion for lot 3, 4 and 5. Starting with lot 6, all of the QC reult and erifiation reult for the lat fie lot are ued in the omparion. For example, the tet reult from lot -6 are ued in the omparion for lot 6. For the Compreie Strength and Thikne F & t omparion, the reult are onidered atifatory (Pa) if the t-tet how that the Contrator QC reult and KDOT QA reult are from the ame population with a of 1%. Page 13/

14 F & t Air Void Spreadheet for R1 HMA 1R R1 SEM 03/01/009 Lot: 1 10 Projet # KA Name of QC Teter Date: 3/5/009 4/9/009 Contrat # Certifiation # of QC Teter Mix Type SR-1.5A 64-8 HMA Oerlay Contrat Line #' 08 1 Metri/Englih: E Plaement ML Air Void Lot Date Contrator Quality Control Tet (%) KDOT Verifiation Tet (%) Number of Contrator Tet Number of KDOT Tet t Tet t(rit) Are Mean Ue Contrator The Same? Tet Reult? 1A 3/5/ B 3/31/ C 3/31/ D 4// E 1F 4 1 Pa Ye A 4// B 4/3/ C 4/3/ D 4/3/ E F 4 1 Pa Ye 3A 4/7/ B 4/7/ C 4/8/ D 4/8/ E 3F Pa Ye 4A 4/8/ B 4/9/ C 4/10/ D 4/10/ E 4F Pa Ye 5A 4/13/ B 4/13/ C 4/14/ D 4/14/ E 5F Pa Ye Comment Page 14/

15 F & t Denity Spreadheet for R1 HMA 1R R1 Lot: 1 Projet # KA Name of QC Teter Date: 3/5/009 4/30/009 Contrat # Certifiation # of QC Teter Mix Type SR-1.5A 64-8 HMA Oerlay Contrat Line # ' Lot Date Contrator Tet Reult (lb/ft3) Gmm Contrator Quality Control Tet (%Gmm) KDOT KDOT Tet Verifiation Reult (lb/ft3) Tet (%Gmm) Number of Contrator Tet Number of KDOT Tet t Tet t(rit) Are Mean Ue Contrator The Same? Tet Reult? Comment 1A A B B C1 3/5/ C D D E1 1E Fail No A A B B C /31/ C D D E1 E Pa Ye 3A A B B C /1/ C D D E E Pa Ye 4A A B B C // C D D E E Pa Ye 5A A B B C /3/ C D D E E Pa Ye Page 15/

16 F & t Gmm Spreadheet for R1 HMA 1R R1 Lot: 1 10 Projet # KA Name of QC Teter Date: 3/5/009 4/9/009 Contrat # Certifiation # of QC Teter Mix Type: SR-1.5A 64-8 HMA Oerlay Gmm Lot Date Contrator Quality Control Tet (Gmm) KDOT Verifiation Tet (Gmm) Number of Contrator Tet Number of KDOT Tet t Tet t(rit) Are Mean The Same? Ue Contrator Tet Reult? 1A 3/5/ Pa Ye 1B 3/31/ C 3/31/ Pa Ye 1D 4/0/ E 1F A 4/0/ Pa Ye B 4/03/ C 4/03/ D 4/03/ Pa Ye E F 3A 4/07/ B 4/07/ Pa Ye 3C 4/08/ D 4/08/ E 3F 4A 4/08/ Pa Ye 4B 4/09/ Pa Ye 4C 4/10/ D 4/10/ Pa Ye 4E 4F 5A 4/13/ B 4/13/ Pa Ye 5C 4/14/ D 4/14/ Pa Ye 5E 5F Comment Page 16/

17 F & t Compreie Strength Spreadheet for SS007 Setion 501 Lot: Projet # K Name of QC Teter Date: 5/16/08 Contrat # Certifiation # of QC Teter Junior Sample BR-549 Compreie Strength Comparion Lot Date Correted Contrator Compreie Strength (MPa) Correted KDOT Compreie Strength (MPa) Number of Contrator Tet Number of KDOT Tet T Tet T(rit) Are Mean The Same? Ue Contrator Tet Reult? 1A1 5/16/ A 1B B 1C C 1D D 1E E 5 1 Pa Ye A1 5/17/ A B B C C D D E1 E 4 1 Pa Ye 3A1 7// A 3B B 3C C 3D D 3E1 3E Pa Ye 4A1 7/3/ A 4B B 4C C 4D D 4E E Pa Ye 5A1 7/4/ A 5B B 5C C 5D D 5E1 5E Pa Ye Page 17/

18 F & t Thikne Spreadheet for SS007 Setion 501 Lot: Projet # K Name of QC Teter Date: 5/16/08 Contrat # Certifiation # of QC Teter Junior Sample BR-549 Conrete Thikne Comparion Lot Date Contrator Core Length (mm) KDOT Core Length (mm) Number of Contrator Tet Number of KDOT Tet T Tet T(rit) Are Mean The Same? Ue Contrator Tet Reult? 1A1 5/16/ A 1B1 61 1B 1C1 61 1C 1D1 41 1D 1E1 56 1E 5 1 Pa Ye A1 5/17/ A B1 64 B C1 64 C D1 75 D E1 E 4 1 Pa Ye 3A1 7// A 3B1 66 3B 3C1 60 3C 3D1 38 3D 3E1 3E Pa Ye 4A1 7/3/ A 4B1 77 4B 4C1 68 4C 4D1 60 4D 4E1 64 4E Pa Ye 5A1 7/4/ A 5B1 80 5B 5C1 67 5C 5D1 78 5D 5E1 5E Pa Ye Page 18/