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1、JIS JAPANESE INDUSTRIAL STANDARD Translated and Published by Japanese Standards Association JIS Z 9041-4:1999 (IS0 3494 : 1974) Statistical interpretation of data - Part 4: Power of tests relating to means and variances ICs 03.120.30 Descriptors : cusum charts, data, research work, average, estimati
2、on, variance, sampling Reference number : JIS Z 90414 : 1999 (E) equipment, statistical quality control 28 S Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 21:33:43 MDTNo reproduction or n
3、etworking permitted without license from IHS -,-,- Z 9041-4 : 1999 (IS0 3494 : 1976) Foreword This translation has been made based on the original Japanese Industrial Standard established by the Minister of International Trade and Industry through deliberations at the Japanese Industrial Standards C
4、ommittee in accordance with the Industrial Standardization Law: To conform with the International Standard, IS0 3494:1976 has been especially employed. JIS Z 90411999 consists of the following 4 parts under the title “Statistical interpretation of data”. Part 1 : Part 2 : Part 3: Part 4 : Statistica
5、l presentation of data Techniques of estimation and test relating to means and variances Tests and confidence intervals relating to proportions Power of tests relating to means and variances Date of Establishment: 1999-05-20 Date of Public Notice in Official Gazette: 1999-05-20 Investigated by: Japa
6、nese Industrial Standards Committee Divisional Council on Basic Items JIS Z 9041-4: 1999, First English edition published in 2001-02 Translated and published by: Japanese Standards Association 4-1-24, Akasaka, Minato-ku, Tokyo, 107-8440 JAPAN In the event of any doubts arising as to the contents, th
7、e original JIS is to be the final authority. O JSA2001 All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm, without permission in writing from the publis
8、her. Printed in Japan Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 21:33:43 MDTNo reproduction or networking permitted without license from IHS -,-,- Z 9041-4 : 1999 (IS0 3494 1976) Comp
9、arison of two variances Contents G . Section One: Comparison tests - General remarks - Historical note . Clause 4 Means Comparison tests Comparison of a mean with a given value Comparison of a mean with a given value Comparison of two means Comparison of two means table of known unknown known unknow
10、n A C C Clause 5 6 Variances table of JIS Z 9041-2 . . . . Comparison of a variance with a given value EI Section Two: Sets of curves - References of the sets of curves . - Sets Page 1 2 4 6 8 11 14 16 18 18 Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS
11、Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 21:33:43 MDTNo reproduction or networking permitted without license from IHS -,-,- JAPANESE INDUSTRIAL STANDARD JIS Z 9041-4 : 1999 (IS0 3494 : 1976) Statistical interpretation of data - Part 4: Power of tests relating to means and v
12、ariances Introduction This Japanese Industrial Standard has been prepared based on IS0 3494 Statistical interpretation of data - Power of tests relating to means and variances issued in 1976 as the first edition without changing the technical contents. The portion underlined with dots in this Standa
13、rd is not stated in the original International Standard. Section One: Comparison tests General remarks 1) This Standard follows on from “JIS Z 9041-2 Statistical interpretation of datu - Techniques of estimation and tests relating to means and variances“. The conditions of application of this Standa
14、rd are as stated in the “General remarks“ in JIS Z 9041-2 It will be recalled that the tests used are valid if the distribution of the observed variable is assumed to be normal in each population (see comments on paragraph 3 of the “General remarks“ in JIS 2 9041-2). JIS Z 9041-2 is concerned only w
15、ith the type I risk (or significance level). This International Standard puts forward notions of the type II risk and of power of the test. JIS Z 9041-2 will also be recalled that the type I risk is the probability of rejecting the null hypothesis (tested hypothesis) if this hypothesis is true (case
16、 of two-sided tests), or the maximum value of this probability (case of one-sided tests). The non-rejection of the null hypothesis produces, in practice, acceptance of the hypothesis, yet non-rejection does not mean that the hypothesis is true. Accordingly, the type II risk, designated by p, is the
17、probability of not rejecting the null hypothesis when it is false. The complement of the probability of committing the error of the second kind (1 - p) is the power of the test (see “Historical note“ following these general remarks). Whereas the value of the type I risk is chosen by the consumers ac
18、cording to the consequences that could arise from that risk (either of the values a = 0.05 or a = 0.01 is commonly employed), the type II risk is dependent on the true hypothesis (the null hypothesis H o being false) i.e. the alternative hypothesis to the null hypothesis. In the comparison of a popu
19、lation mean with a given value PO, for example, a specific alternative corresponds to a value of the population mean of p 2 p o being a deviation p - po+ O. As a general rule, in tests of comparison of means and variances, the alternatives are defined by the values that might be assumed by a paramet
20、er. The operating characteristic curve of a test is the curve which shows the value /3 of the type II risk as a function of the parameter defining the alternative. /3 is also dependent on the value chosen for the type I risk, on size(s) of sample(s) and on the nature of the test (two-sided or one-si
21、ded). In the tests of comparison of means, p also depends on the standard deviation of the population(s). Where this is unknown, the risk /3 cannot be known exactly. 2) 3) 4) Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing,
22、Bernie Not for Resale, 03/12/2007 21:33:43 MDTNo reproduction or networking permitted without license from IHS -,-,- 2 Z 9041-4 : 1999 (IS0 3494 1976) 5) The operating characteristic curves allow the following problems to be solved. a) problem 1 For a given alternative and given size of sample, dete
23、rmine the probability p of not rejecting the null hypothesis (type II risk). b) problem 2: For a given alternative and a given value of determine the size of sample to be selected. Although a single series of curve sets allows both problems to be solved, two series of sets will be presented, in orde
24、r to facilitate practical applications: - sets 1.1 to 14.1, giving the risk p as a function of the alternative, for a = 0.05 or a = 0.01 and for different values of the size(s) of sample. - sets 1.2 to 14.2, giving the size(s) of sample to be selected as a function of the alternative, for a = 0.05 o
25、r a = 0.01 and for different values of the risk p. 6) Attention is drawn to the practical significance of interpreting statistics by means of tests of hypotheses and curves. When testing a hypothesis such as p = (or p1= p2), it is generally desired to know whether it can be concluded with little ris
26、k of mistake, that p does not differ too greatly from po (or pl does not differ too greatly from p2). Moreover, the choice of the value a = 0.05 or a = 0.01 for the type I risk associated with the test has a degree of arbitrariness. Therefore, it may be useful to examine what the result of the test
27、would be with values close to (or value of the difference D = pl - p2 close to O), possibly using both values of the type I risk a = 0.05 and a = 0.01 and, in these circumstances to evaluate by means of the operating characteristic curves the risk associated with different alt ernatives. The sets of
28、 curves which are given in section two of this Standard are described and discussed in six clauses which correspond to the table in JIS Z 9041-2. The detailed correspondence between the different sets, the problems which they allow to be solved, the clauses of this Standard and the tables of JIS Z 9
29、041- 2, appear at the top of the group of sets. 7) Historical note The concepts “type I risk“ and “type II risk“ were introduced by J. Neyman and E. S. Pearson in an article which appeared in 1928. Subsequently, these authors considered that the complement of the probability of committing the error
30、of the second kind - which they called “power“ of the test, in its aptitude to reveal as significant a specified alternative to the null hypothesis (tested hypothesis) - was in general an easier concept for the users to understand. It is this “power“, or the probability of revealing a given deviatio
31、n from the null hypothesis, which they designated by the symbol B. Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 21:33:43 MDTNo reproduction or networking permitted without license from I
32、HS -,-,- 3 Z 9041-4 1999 (IS0 3494 1976) It is moreover not necessary to introduce the term power. One can more simply speak of the probability that a statistical test applied to a sample, at a significance level a, reveals that a parameter A. of the population differs (when such is truly the case)
33、by at least a given quantity from the specified value ;io, or, in relation to it, in a ratio at least equal to a given number. The change in notation was probably introduced in the United States by users of industrial applications of statistics, in order that the “consumers risk“, when designated by
34、 p, might be taken into consideration at the same time as the “producers risk d. The symbol p was adopted for the type II risk in JIS Z 8101 Statistics - Vocabulary and symbols, and it has therefore been adopted wit the same significance in this International Standard. However, as this symbol is use
35、d, and will continue no doubt to be used, with both meanings in statistical literature it is advisable to find out in each case of use the meaning which is effectively attributed to it. Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001,
36、User=Wing, Bernie Not for Resale, 03/12/2007 21:33:43 MDTNo reproduction or networking permitted without license from IHS -,-,- 4 Z 9041-4 : 1999 (IS0 3494 : 1976) 1 Comparison of a mean with a given value (variance known) See Table A of JIS Z 9041-2 1.1 Notations n = sample size p = population mean
37、 po = given value o = standard deviation for the population 1.2 Tested hypotheses For a two-sided test, the null hypothesis is p = PO, the alternative hypothesis corresponding to p f PO. For a one-sided test, the null hypothesis is a either p i po, the alternative hypothesis corresponding to p po; b
38、) or p 2 PO, the alternative hypothesis corresponding to p po o -4- c) A = (p - (one-sided test p 2 PO alternatives y po o CI a = - (one-sided test p 2 po) alternatives p PO; b) or p 2 po, the alternative hypotheses corresponding to p c) A = - .In p - Po) (one-sided test p 1 p u 0 1 alternatives p e
39、 fi o o o According to the case, the set to be consulted is 1.1 (two-sided test) type I risk a = 0.05 2.1 (two-sided test) type I risk a = 0.01 3.1 (one-sided test) type I risk a = 0.05 4.1 (one-sided test) type I risk a = 0.01 is the ordinate of the point on the abscissa A on the curve v = n - 1 of
40、 the suitable set. Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 21:33:43 MDTNo reproduction or networking permitted without license from IHS -,-,- 7 1999 (IS0 3494 : 1976) Z 9041-4 2.4 P
41、roblem 2: /3 being given, determine the size n For the different values of p, the alternative is defined by the parameter A ( O po c) A = - - (one-sided test p 2 po) alternatives p p2; or pl B p2, the alternative hypotheses corresponding to p1 p2 OD c) A. = - - (one-sided test p 2 pd alternatives p1
42、 p2; or pl I 1-12? the alternative hypotheses corresponding to p1 e p2. 4.3 Problem 1: ni and n2 being given, determine the risk p parameter A. (O 1-12 c) A. = P2 - I-11 (one-sided test 1-11 2 1-12 alternatives 1-11 e 1-12 OD GD OD Copyright Japanese Standards Association Provided by IHS under licen
43、se with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 21:33:43 MDTNo reproduction or networking permitted without license from IHS -,-,- 12 2 9041-4 : 1999 (IS0 3494 : 1976) According to the case, the set to be consulted is 1.1 (two-sided test) type I risk a = 0.
44、05 2 . 1 (two-sided test) type I risk a = 0.01 3.1 (one-sided test) type I risk a = 0.05 4 . 1 (one-sided test) type I risk a = 0.01 p is the ordinate of the point on the abscissa A on the curve i l = ni + n 2 - 2 of the suitable set. When only the total size of the two samples is fixed, interest to
45、 take ni + 722 = n (p minimum). One then has: 4.4 parameter A ( O p2 (one-sided test p 2 p2) alternatives p1 o o 2 (o 00); or 4 2 o o 2 (o 2 a o ) , the alternative hypotheses corresponding to 8 0 2 For a one-sided test, the null hypothesis is a) b) or o12 2 either o12 5 oz2 (01 5 021, the alternati
46、ve hypotheses corresponding to oi2 oz2 ( o 1 0 2 ) and defined by the parameter A = 0 1 / 0 2 (i 1 Onlinate Normal Logarithmic Normal Logarithmic Normal Normal 18 Z 9041-4 : 1999 (IS0 3494 : 1976) Section Two: Sets of curves References of the sets of curves of curves a Clause of Section one Referenc
47、e of the table in JIS Z 9041-2 Test Problem 22) Problem 1” 0.01 2.1 A A C e A A C c 1.2 2.2 3.2 4.2 Zomparison of variance, or of a standard leviation, with a given value =d 5 E 5.2 6.2 7.2 8.2 9.2 10.2 omparison of two variances or of two rtandard deviations .-+- , . . - if ul and u 2 are unknown,
48、but equal, use the curve v=n1+n2-2, with Q I c - 8 c 0.0 1 0.05 o. 1 0.2 0.5 1 2 5 x Set 3.1 One-sided tests of comparison of means (type I risk a = 0.05) Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 21:33:43 MDTNo reproduction or networking permitted without license from IHS -,-,- 24 Z 9041-4 : 1999 (IS0 3494 : 1976) a Test p 5 PO or p 2 PO - if o is known, use the straigh