1、BRITISH STANDARD BS2846-6: 1976 ISO 3301:1975 Guide to Statistical interpretation of data Part 6: Comparison of two means in the case of paired observations UDC 31:519.24:519.233.8BS2846-6:1976 This British Standard, having been prepared under the directionof The Advisory Committee on Statistical Me
2、thodswas published under theauthority of the Executive Board on 30June1976 BSI 03-1999 The following BSI references relate to the work on this standard: Committee reference OC/8/2 and draft for approval ISO/DIS3301 ISBN 0 580 09294 1 Cooperating organizations The Advisory Committee on Statistical Me
3、thods, under whose supervision this British Standard was prepared, consists of representatives from the following Government department and scientific and industrial organizations. Biometrika* British National Committee for Surface Active Agents British Paper and Board Industry Federation (PIF) Econ
4、omist Intelligence Unit Electronic Components Board Institute of Petroleum Institute of Quality Assurance* Institute of Statisticians* Institution of Production Engineers Ministry of Defence* National Coal Board* Post Office Royal Statistical Society* Society of Analytical Chemists University of Ess
5、ex The Government department and scientific and industrial organizations marked with an asterisk in the above list were directly represented on the committee entrusted with the preparation of this British Standard. Amendments issued since publication Amd. No. Date of issue CommentsBS2846-6:1976 BSI
6、03-1999 i Contents Page Cooperating organizations Inside front cover Foreword ii 0 Introduction 1 1 Scope 1 2 Definition 1 3 Field of application 1 4 Conditions for application 1 5 Formal presentation of calculations 2 6 Errors of the second kind 5 Figure 1 Power of Students one-sample test (one-sid
7、ed), =0.01 5 Figure 2 Power of Students one-sample test (one-sided), =0.05 6 Table 1 Values of the ratio t 1 /3n for = n 1 3 Table 2 Shaft-wear after a given working time in0.00001in 4BS2846-6:1976 ii BSI 03-1999 Foreword The correct interpretation and presentation of test results have been assuming
8、 increasing importance in the analysis of data obtained from manufacturing processes based on sample determinations and prototype evaluations in industry, commerce and educational institutions. It was for this reason that Subcommittee2of Technical Committee69, “Applications of Statistical Methods”,
9、of the International Organization for Standardization (ISO), was charged with the task of preparing a guide to statistical methods for the interpretation of test results. As international agreement is reached on the statistical tests relevant to specific situations it is proposed to publish them as
10、parts of a revised BS28461957: “The reduction and presentation of experimental results” as follows. Statistical interpretation of data Part 1: Routine analysis of quantitative data; Part2: Estimation of the mean-confidence interval ISO2602; Part3: Determination of a statistical tolerance interval IS
11、O3207; Part 4: Techniques of estimation and tests relating to means and variances ISO2854 1) ; Part 5: Efficiency of tests relating to means and variances ISO3494 1) ; Part 6: Comparison of two means in the case of paired observations ISO3301. A situation often encountered is one in which a decision
12、 in favour of one of two alternatives, whether they be manufacturing processes, machines or medical drugs etc., needs to be taken. There exists a number of criteria for making such judgements depending upon whether it is the “average effect” or the number of “individual items being processed that ar
13、e better than some tolerance” that is of prime importance. This Part of the standard specifies a technique for testing for (significant) differences between two mean values (e.g.two process averages) and consequently is concerned with problems of the former kind. The latter problem is dealt with in
14、Part3, and also BS6002 “Sampling procedures and charts for inspection by variables for percent defective”. 1)The technique of this Part provides a procedure for deciding between two alternatives on the basis of differences between their average performances. This Part of this British Standard is ide
15、ntical with ISO3301 “Statistical interpretation of data Comparison of two means in the case of paired observations”. For the purposes of this British Standard the text of ISO3301given in this publication should be modified as follows. Terminology. The words “British Standard” should replace “Interna
16、tional Standard” wherever they appear. The decimal point should replace the decimal comma, wherever it appears. Cross-reference. The references to other International Standards should be replaced by references to British Standards as follows. 1) In course of preparation. Reference to ISO Standard Ap
17、propriate British Standard ISO2854 “Statistical interpretation of data Techniques of estimation and tests relating to the means and variances” BS2846 “Guide to statistical interpretation of data” Part4 “Techniques of estimation and tests relating to means and variances” a a In course of preparation.
18、BS2846-6:1976 BSI 03-1999 iii The following British Standards also provide practical guidance on the application of statistical methods. BS 600, Application of statistical methods to industrial standardization and quality control. BS 1313, Fraction-defective charts for quality control. BS 2564, Cont
19、rol chart technique when manufacturing to a specification, with special reference to articles machined to dimensional tolerances. BS 6000, Guide to the use of BS6001. Sampling procedures and tables for inspection by attributes. BS6001, Sampling procedures and tables for inspection by attributes. BS6
20、002 2) , Sampling procedures and charts for inspection by variables for percent defective. BS 2) , Determination of repeatability and reproducibility for a standard test method. A British Standard does not purport to include all the necessary provisions of a contract. Users of British Standards are
21、responsible for their correct application. Compliance with a British Standard does not of itself confer immunity from legal obligations. Summary of pages This document comprises a front cover, an inside front cover, pagesitoiv, pages1to6 and a back cover. This standard has been updated (see copyrigh
22、t date) and may have had amendments incorporated. This will be indicated in the amendment table on theinside front cover. 2) In course of preparation.iv blankBS2846-6:1976 BSI 03-1999 1 0 Introduction The method specified in this International Standard, known as the method of paired observations, is
23、 a special case of the method described in Table A of ISO2854, Statistical interpretation of data Techniques of estimation and tests relating to means and variances. 3) This special case is mentioned in Section 2 of ISO2854immediately after the numerical illustration of Table A, and a complete examp
24、le of applications of the method of paired comparisons has been given in Annex A of that International Standard. The importance and wide applicability of the method justify a separate International Standard being devoted to it. 1 Scope This International Standard specifies a method for comparing the
25、 mean of a population of differences between paired observations with zero or any other preassigned value. 2 Definition paired observations two observations x iand y iof a certain property or characteristic are said to be paired if they are made: on the same element i from a population of elements b
26、ut under different conditions (for example, comparison of results of two methods of analysis on the same product); on two distinct elements considered similar in all respects except for the systematic difference which is the subject of the test (for example, comparison of the yield from adjacent plo
27、ts sown with two distinct varieties of seed). however, it should be noted that in the second case the efficiency of the test depends on the validity of the hypothesis that there is no other systematic difference between the individuals in the same pair other than the systematic difference under test
28、 3 Field of application The method may be applied to establish a difference between two treatments. In this case, the observations x iare carried out after the first treatment and y iafter the second treatment. The two series of results of the observations are not independent because each result x i
29、of the first series (first treatment) is associated with a result y iof the second series (second treatment). The term “treatment” should be understood in a wide sense. The two treatments to be compared may, for instance, be two test methods, two measuring instruments or two laboratories, in order t
30、o detect a possible systematic error. Two treatments carried out successively on the same experimental material might interact and the value obtained might depend on the order. Good experimental design should enable such biases to be eliminated. Alternatively, only one treatment may be applied and i
31、ts effect may be compared to the absence of treatment; the purpose of this comparison is then to establish the effect of that treatment. 4 Conditions for application The method can be applied validly if the following two conditions are satisfied: the series of differences d i = x i y ican be conside
32、red as a series of independent random items; the distribution of the differences d i = x i y ibetween the paired observations is supposed to be normal or approximately normal. If the distribution of these differences deviates from the normal, the technique described remains valid, provided the sampl
33、e size is sufficiently large; the greater the deviation from normality, the larger the sample size required. Even in extreme cases, however, a sample size of 100will be sufficient for most practical applications. 3) At present at the stage of draft.BS2846-6:1976 2 BSI 03-1999 5 Formal presentation o
34、f calculations Statistical data Calculations Sample size: n = Sum of the observed values: Cx i = Cy i = Sum of the differences: Cd i= Sum of the squares of the differences: Given value: d o = Degrees of freedom: = n1= Chosen significance level: = Results Two-sided case: The hypothesis that the popul
35、ation mean of the difference is equal to d o(null hypothesis) is rejected if: One-sided cases: a) The hypothesis that the population mean of the differences is at least equal to d o(null hypothesis) is rejected if: b) The hypothesis that the population mean of the differences is at most equal to d o
36、(null hypothesis) is rejected if: NOTEt 1 ! (v) is the fractile of order1! of Students variate t with v degrees of freedom. The values of t 1 ! ()/ are given in Table 1. d 1 n -Cx i Cy i () = 1 n -Cd i = = s 2 d 1 n 1 -Cd 2 i 1 n -Cd i () 2 = = Cd 2 i= B * d s 2 d= = A 1 t 1 ! ()n B* d= = A 2 t 1 !
37、2 ()n B* d= = dd 0 A 2 dd 0 A 1 nBS2846-6:1976 BSI 03-1999 3 Table 1 Values of the ratio t 1 ()/ for = n 1 = n 1 Two-sided case One-sided case 1 8,985 45,013 4,465 22,501 2 2,434 5,730 1,686 4,021 3 1,591 2,920 1,177 2,270 4 1,242 2,059 0,953 1,676 5 1,049 1,646 0,823 1,374 6 0,925 1,401 0,734 1,188
38、 7 0,836 1,237 0,670 1,060 8 0,769 1,118 0,620 0,966 9 0,715 1,028 0,580 0,892 10 0,672 0,956 0,546 0,833 11 0,635 0,897 0,518 0,785 12 0,604 0,847 0,494 0,744 13 0,577 0,805 0,473 0,708 14 0,554 0,769 0,455 0,678 15 0,533 0,737 0,438 0,651 16 0,514 0,708 0,423 0,626 17 0,497 0,683 0,410 0,605 18 0,
39、482 0,660 0,398 0,586 19 0,468 0,640 0,387 0,568 20 0,455 0,621 0,376 0,552 21 0,443 0,604 0,367 0,537 22 0,432 0,588 0,358 0,523 23 0,422 0,573 0,350 0,510 24 0,413 0,559 0,342 0,498 25 0,404 0,547 0,335 0,487 26 0,396 0,535 0,328 0,477 27 0,388 0,524 0,322 0,467 28 0,380 0,513 0,316 0,458 29 0,373
40、 0,503 0,310 0,449 30 0,367 0,494 0,305 0,441 40 0,316 0,422 0,263 0,378 50 0,281 0,375 0,235 0,337 60 0,256 0,341 0,214 0,306 70 0,237 0,314 0,198 0,283 80 0,221 0,293 0,185 0,264 90 0,208 0,276 0,174 0,248 100 0,197 0,261 0,165 0,235 200 0,139 0,183 0,117 0,165 500 0,088 0,116 0,074 0,104 0 0 0 0
41、n t 0975 , n - t 0995 , n - t 095 , n - t 099 , n -BS2846-6:1976 4 BSI 03-1999 Example: The data tabled below were collected during an investigation designed to determine whether the average rate of shaft-wear caused by various bearing metals in an internal combustion engine differed between metals.
42、 Table 2 Shaft-wear after a given working time in0.00001 in Shaft i Wear with Difference d i = x i y i copper-lead x i white metal y i 1 3.5 1.5 2.0 2 2.0 1.3 0.7 3 4.7 4.5 0.2 4 2.8 2.5 0.3 5 6.5 4.5 2.0 6 2.2 1.7 0.5 7 2.5 1.8 0.7 8 5.8 3.3 2.5 9 4.2 2.3 1.9 Total 34.2 23.4 10.8 Statistical data C
43、alculations Sample size: n=9 Sum of the observed values: Cx i= 34.2 Cy i= 23.4 Sum of the differences: Cd i =10.8 Sum of the squares of the differences: Given value: d o =0 Degrees of freedom: v=8 Chosen significance level: =0.01 Result Comparison of the population mean with the given value0: Two-si
44、ded case: The hypothesis of the equality of the rate of shaft-wear by the two metals is rejected at the1% level. d 1 9 -34.223.4 ()1.2 = s 2 d 1 8 -19.22 10.8 2 9 -0.782 5 = = B * d 0.782 50.884 6 = Cd 2 i 19.22 = t 0.995/91.118 = A 2 1.1180.884 6 0.99 = dd 0 1.20.99 =BS2846-6:1976 BSI 03-1999 5 6 E
45、rrors of the second kind The probability of rejecting the null hypothesis when it is true is at most equal to the significance level . Rejecting the null hypothesis when it is true is called an error of the first kind, and the choice of therefore limits the risk of such an error. On the other hand,
46、it is possible to commit an error of the second kind, that is, accepting the null hypothesis when it is false. The probability 1 of rejecting the null hypothesis when it is false is called the power of the test; the probability of an error of the second kind is therefore . For a given sample n and e
47、rror of the first kind, these probabilities depend not only on the true mean D of the observed differences d i = X i Y ifor which one can postulate different alternative hypotheses but also on the standard deviation B dof these differences. This standard deviation is in general unknown and if n is s
48、mall the sample will provide only a poor estimator. The result is that it is impossible to set an upper limit to the probability of an error of the second kind. Nevertheless, in the following graphs the relation is shown between the power of the test, 1, and the actual population mean divided by the
49、 corresponding standard deviation, D/B dfor one-sided tests of the hypothesis H o : D k 0, and for various values of n and for the significance levels0,05and0,01respectively. From these graphs the following conclusions may be drawn: 1) The power of the test is uniquely determined by the true mean of the differences, measured in units of their standard
