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1、J IS JIS Z 9041-31999 JAPANESE I N D U STR I AL STANDARD Translated and Published by Japanese Standards Association (IS0 1 1453 : 196) Statistical Interpretation of data - Part 3: Tests and confidence intervals relating to proportions ICs 03.120.30 Descriptors : cusum charts, data, research work, es
2、timation, verification, statistical Reference number : JIS 2 9041-3 : 1999 (E) quality control, statistical distribution, sampling equipment 30 S Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/
3、2007 10:03:31 MDTNo reproduction or networking permitted without license from IHS -,-,- Z 9041-3 : 1999 (IS0 11453 : 1996) Foreword This translation has been made based on the original Japanese Industrial Standard established by the Minister of International Trade and Industry through deliberations
4、at the Japanese Industrial Standards Committee in accordance with the Industrial Standardization Law: To conform with the International Standard, IS0 11453:1996 has been basically employed. JIS 2 9041:1999 consists of the following 4 parts under the title “Statistical interpretation of data”. Part 1
5、: Statistical presentation of data Part 2 : Techniques of estimation and test relating to means and variances Part 3: Tests and confidence intervals relating to proportions Part 4 : Power of tests relating to means and variances Date of Establishment: 1999-05-20 Date of Public Notice in Official Gaz
6、ette: 1999-05-20 Investigated by: Japanese Industrial Standards Committee Divisional Council on Basic Items TIS Z 9041-3 : 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 an
7、y doubts arising as to the contents, the original JIS is to be the final authority. c L 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, wi
8、thout permission in writing from the publisher. 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 10:03:31 MDTNo reproduction or networking permitted without license from IHS
9、 -,-,- STD-JIS Z 7041-3-ENGL 1779 4733b08 05b7735 7Li5 2 9041-3 : 1999 (IS0 11453 : 1996) Contents Page Introduction . 7.1 7.2 7.3 8 8.1 8.2 8.3 9 9 . 1 9.2 Scope Normative references Definitions Symbols . Point estimator of the proportion p . Confidence limits for the proportion p Significance test
10、s on proportions p General Comparison of a proportion with a given value po Comparison of two proportions . Forms . A forms: Confidence interval for the proportion p B forms: Comparison of the proportion p with a given value po . C forms: Comparison of two proportions Tables and nomographs Interpola
11、tion in Table 4 of the quantiles of the F-distribution Example . Annex A (normative) Computation of the operating characteristic of the test according to the B forms . Examples of completed forms Annex B (informative) Annex C (informative) Bibliography 1 1 1 2 2 2 3 3 3 3 4 5 6 10 16 23 23 23 37 40
12、53 . Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 10:03:31 MDTNo reproduction or networking permitted without license from IHS -,-,- JAPANESE INDUSTRIAL STANDARD JIS Z 9041-3 : 1999 (IS0
13、 11453 : 1996) Statistical interpretation of data Part 3: Tests and confidence intervals relating to proportions Introduction This Japanese Industrial Standard has been prepared based on IS0 11453, Statistical interpretation of data - Tests and confidence intervals relating to proportions issued in
14、1996 as the first edition together with Technical Corrigendum 1: 1999 without changing the technical contents. Portions underlined with dots in this Standard show the matters not specified in the original International Standard. 1 Scope This Standard describes specific statistical methods for addres
15、sing the following questions. a) Given a population of items from which a sample of n items has been drawn, x of the sample items are found to show a specified characteristic. What proportion of the population has that characteristic ? (See A forms, 8 . 1 . ) b) Is the proportion estimated in a) dif
16、ferent from a nominal (specified) value ? (See B forms, 8.2.) c) Given two distinct populations, are the proportions with the characteristic in the two populations different ? (See C forms, 8 . 3 . ) d) In b) and c) how many items must be sampled in the population(s) to be suffi- ciently sure that t
17、he result of the test is correct ? (See 7 . 2 . 3 and 7 . 2 . 4 ) It is essential that the drawing of samples does not have any appreciable effect on the population. If the sample drawn at random is less than 10 ? 6 of the population this is usually satisfactory, but if the sample is greater than th
18、is, reliable results can be obtained only by replacing each item sampled before drawing the next item at random from the population. 2 Normative references The following standards contain provisions which, through reference in this Standard, constitute provisions of this Standard. If the indication
19、of the year of coming into effect is given to these referred standards, only the edition of indicated year constitutes the provision of this Standard but the revision and amendment made thereafter are not applied. The normative references without the indication of the year of coming into effect appl
20、y limiting only to the most recent edition (including the amendment). Statistics - Vocabulary and symbols - Part 1: Probability and gen- eral statistical terms JIS Z 8101-1 JIS Z 8101-2 Statistics - Vocabulary and symbols - Part 2: Statistical quality control terms Copyright Japanese Standards Assoc
21、iation Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 10:03:31 MDTNo reproduction or networking permitted without license from IHS -,-,- 2 Z 9041-3 1999 (IS0 11453 : 1996) 3 Definitions For the purposes of this Standard, the defi
22、nitions given in JIS Z 8101-1 and the following definition apply. 3.1 target item One in which the specified characteristic is found. 4 Symbols significance level chosen achieved significance level confidence level chosen probability of the error of the second kind sample size; sample size of sample
23、 1; sample size of sample 2 number of target items in the sample (random variable) value of X proportion of target items in the population upper limit of the one-sided confidence interval for p lower limit of the one-sided confidence interval for p upper limit of-the two-sided confidence interval fo
24、r p lower limit of the two-sided confidence interval for p value from Table 2 for the determination of confidence limits for n S 30 critical value of the test of the null hypothesis Ho: p h po critical value of the test of the null hypothesis Ho: p 5 po lower critical value of the test of the null h
25、ypothesis Ho: p = po upper critical value of the test of the null hypothesis Ho: p = po given value for p value ofp for which the probability of not rejecting the null hypothesis (pa is to be determined probability of not rejecting the null hypothesis number of degrees of freedom of the F-distributi
26、on test statistics q-quantile of the F-distribution with v, and v2 degrees of freedom test statistics q-quantile of the standard normal distribution auxiliary values 5 Point estimator of the proportion p The estimator of p from a sample of n items including x target items is j = x l n This estimator
27、 is unbiased if the sample is drawn at random, whatever the sample size and population size may be, even if the sample forms an appreciable part of the population. Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not
28、for Resale, 03/12/2007 10:03:31 MDTNo reproduction or networking permitted without license from IHS -,-,- 3 Z 9041-3 : 1999 (IS0 11453 : 1996) 6 Confidence limits for the proportionp The calculation of a confidence inter- val for p is described in forms A-1 to A-3. The confidence limits will depend
29、on sample size (n), the number of target items in the sample (x), and the desired confidence level (i - a). It is not possible in gen- eral to achieve the desired confidence level exactly because the probability distribu- tion governing the outcome x is discrete. Thus, the procedure yields the neare
30、st confidence level greater than or equal to (i - a). The procedure used in this International Standard for determining the two-sided confidence limits for the desired confidence level (i - a) uses the limits for the upper and lower one-sided limits each for the desired confidence level (i - a/2). I
31、t thereby guarantees that the error probability is less than or equal to a/2 on each side of the interval. 7 Significance tests on proportions p 7.1 General For practical applications, forms B-1 to B-3 and C-1 to C-3 present the null hypotheses concerning proportions and schemes for carrying out tes
32、ts. At the beginning of the procedures, the appropriate null hypothesis and the sample size n (the sample sizes v1 and VZ) are to be determined and the significance level is to be chosen. Because the underlying sampling distributions are discrete, the procedures are designed to achieve the nearest s
33、ignificance level less than or equal to the de- sired (nominal) level. The alternative hypotheses are not indicated in the forms be- cause in each application the alternative hypothesis is implicitly assumed to be complementary to the null hypothesis. Example: At the beginning of the B forms (proced
34、ure for the comparison of a pro- portion with a given value), one of the following three null hypotheses Ho (with the complementary alternative hypothesis HI) is to be chosen: a) b) c) one-sided test with Ho: p 2 po one-sided test with HO: p S po one-sided test with Ho: p = po H1: p po HI: p + po Th
35、e result of each test is either to reject or not to reject the null hypothesis. Rejecting the null hypothesis means adopting the alternative hypothesis. Not re- jecting the null hypothesis does not necessarily mean accepting the null hypothesis (see 7.2.2). 7.2 Comparison of a proportion with a give
36、n value po 7.2.1 Test procedure The test procedures for the null hypotheses Ho: P 2 Po Ho: P 5 Po Ho: P = Po (where po is the given value) are described in forms B-1 to B-3. They are especially easy to apply if the critical value(s1 for the specified values of n, p and a is (are) known. The critical
37、 value(s1 may have been determined determine the critical val- ues is the one given in the same forms. Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 10:03:31 MDTNo reproduction or network
38、ing permitted without license from IHS -,-,- STD*JIS Z 7041-3-ENGL 1999 4933b08 05b7739 390 = Case 4 Z 9041-3 : 1999 (IS0 11453 : 1996) Given value Straight line 1 Straight line 2 from po to from p to 7 . 2 . 2 Operating characteristics The computation of the operating characteris- tics (including t
39、he probability of the error of the first kind, the achieved significance level and the probability of the error of the second kind) is described in Annex A. For this purpose the critical value(s) need(s) to be known (see 7 . 2 . 1 ) and the alterna- tive hypothesis, p = p1, for which the probability
40、 of the error of the second kind is to be computed has to be chosen. 7 . 2 . 3 Determination of sample size n If the sample size is not already specified (for example for economic or technical reasons), its minimum value shall be deter- mined such that for a given null hypothesis Ho (see 7 . 2 . 1 )
41、 , the achieved value of the significance level a is not greater than its chosen or given value. In addition, the achieved value of the type II error, probability b, shall be approximately equal to its chosen or given value if p equals a particular chosen or given value p. For this purpose, po and p
42、are to be marked on thep-scale and a, (1 - a), a/2, (1 - a/2) on the p-scale and the straight lines 1 and 2 as defined through the procedure shown in Table 1 are drawn in the Larson nomograph (Fig. 2) Table 1 Procedure for determining the sample size from the Larson nomograph (Fig. 2) The point of i
43、ntersection of the two lines leads to the values Cl,o (Cu,o) on the x-scale. If x is not an integer, round up or down to the next integer. 7 . 3 Comparison of two proportions 7 . 3 . 1 Test procedure The test procedures for the null hypothesis Ho: Pi 2 P2 Ho: Pi 5 P 2 Ho: Pi = P 2 (where pl is the p
44、roportion of target items in population 1 and p2 is the proportion of target items in population 2) are described in forms C-1 to C-3. These procedures are also suitable for testing the independence of two attributes (dichotomous charac- teristics) of items in a population. 7 . 3 . 2 Operating chara
45、cteristics It is assumed: a) that for a one-sided test of Ho: pi S p2, the power (1 - p) must be determined for a given pair of proportions pi and p2, with pi p2: b) that the test is carried out with two samples of the same size, i.e. nl = n2 = n. Copyright Japanese Standards Association Provided by
46、 IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 10:03:31 MDTNo reproduction or networking permitted without license from IHS -,-,- STD*JIS Z 70VL-3-ENGL 1777 = V733b08 05b77LiO O02 5 Z 9041-3 : 1999 (IS0 11453 : 1996) The significance level
47、is a. Then a very accurate approximate value of the power can be obtained by the arc sine transformation (proposed by Walters i) as follows: 1 - p = q ? (2 - U1-J where 4 is the distribution function of the standard normal distribution, ul-= is the (1 - = 0 0.90 0.95 0.99 0.677 0.960 1.659 Computati
48、on: pi,; = i - (a/2)“ - C a s e O Pu.; = 9 P1.i P Pu.; Copyright Japanese Standards Association Provided by IHS under license with JSALicensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 03/12/2007 10:03:31 MDTNo reproduction or networking permitted without license from IHS -,-,- 1 0 Z 9041-3 : 1999 (IS0 11453 : 1996) 8.2 B forms: Comparison of the proportionp with a given valuepo 8.2.1 Form B-1 : Comparison of the proportion p with a given value po and with one-sided test with HO : p 1 . po Characteristic: Determ