1、BS 2846 : Part 5 : 1977 IS0 3494-1976 Confirmed October 1985 UDC 31 :/519.222 : 1519.724.6 Guide to Statistical interpretation of data Part 5. Power of tests relating to means and variances IS0 title: Statistical interpretation of data -Power of tests relating to means and variances Guide pour linte
2、rprtation statistique des donnes Partie 5. Efficacit des tests portant sur des moyennes et des variances Hinweise fr die statistische Auswertung von Daten Teil 5. Schre von Prfungen mit Bezug auf Durchschnittswerte und Variante British Standards Institution - BSI BS*E!84b: PART85 77 E Lb24bb OLL5257
3、 O 9 Clause BS 2846 : Part 5 : 1977 to table Comparison tests Variancek) I 1 150y854 1 Contents Page Foreword. .Inside front cover Cooperating organizations. Back cover Clause Guide Section one: Comparison tests Generalremarks . 1 Historicalnote 2 MEANS Reference Reference to table of IS0 2854 Compa
4、rison tests 6 1 Comparison of two variances G 2 3 4 Comparison of a mean with a given value Comparison of a mean with a given value Comparison of two means Comparison of two means known unknown known unknown . 3 4 6 8 Comparison of a variance with a given 10 . 11 Section two: Sets of curves Referenc
5、es of the sets of curves 12 Sets 13 Foreword This Part of BS 2846 is identical with IS0 3494 - 1976 Statistical interpretation of data - Power of tests relating to means and variances. The correct interpretation and presentation of test results have been assuming increasing importance in the analysi
6、s 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 Subcommittee 2 of Technical Committee 69, Application of statistical methods, of the International Organization for
7、 Standardization (ISO), was charged with the task of preparing a guide to statistical methods for the interpretation of test results. As international agreement was reached on the statistical tests relevant to specific situations, it was decided to publish them as Parts of a revised BS 2846 as follo
8、ws: Statistical interpretation of data Part 1 Routine analysis of quantitative data Part 2 Estimation of the mean-confidence interval IS0 26021 Part 3 Determination of a statistical tolerance interval IS0 32071 Part 4 Techniques of estimation and tests relating to means and variances IS0 28541 Part
9、5 Power of tests relating to means and variances IS0 34941 Part 6 Comparison of two means in the case of paired observations IS0 3301 1 Part 5 of this standard is complementary to Part 4 which provides a series of statistical techniques applicable when the characteristic of interest is measured on a
10、 continuous scale and its distribution is Normal. In presenting these techniques Part 4 was only concerned with the error of the first kind, .e. the error of rejecting the null hypothesis when true, or more particularly the maximum value of the probability of committing such an error that could be t
11、olerated, the significance level of the test. continued on inside back cover British Standard Guide to Statistical interpretation of data Part 5. Power of tests relating to means and variances SECTION ONE : COMPARISON TESTS GENERAL REMARKS 1) This International Standard follows on from IS0 2854, Sta
12、tistical interpretation of data - Techniques of estimation and tests relating to means and variances, The conditions of application of this International Standard are as stated in the “General remarks in IS0 2854. It will be recalled that the tests used are valid if the distribution of the observed
13、variable is assumed to be normal in each population (see comments on paragraph 3 of the “General remarks“ in IS0 2854). IS0 2854 is concerned only with the type I risk (or significance level). This International Standard puts forward notions of the type II risk and of power of the test. 2) It will a
14、lso be recalled that the type I risk is the probability rejecting the null hypothesis (tested hypothesis) if this hypothesjs is true (case 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, acceptan
15、ce of the hypothesis, yet non-rejection does not mean that the hypothesis is true. Accordingly, the type II risk, designated by 0, is the 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 -0) is the “pow
16、er“ of the test (see !Historical note“ following these general remarks). 3) Whereas the value of the type I risk is chosen by the consumers according to the consequences that could arise from that risk (either of the values a = 0,05 or (Y = 0,Ol is commonly employed), the type II risk is dependent o
17、n the true hypothesis (the null hypothesis Ho being false), .e. the alternative hypothesis to the null hypothesis. In the comparison of a population mean with a given value mo, for example, a specific alternative corresponds to a value of the population mean of m #mo being a deviation m -mo #O. As a
18、 general rule, in tests of comparison of means and variances, the alternatives are defined by the values that might be assumed by a parameter. 4) The operating characteristic curve of a test is the curve which shows the value 0 of the type II risk as a function of the parameter defining the alternat
19、ive. p is also dependent on the value chosen for the type 1 risk, on size(s) of sample(s) and on the nature of the test (two-sided or one-sided). In the tests of comparison ot means, p also ciepenas on tne standard deviation of the population(s). Where this is unknown, the risk 0 cannot be known exa
20、ctly. 5) The operating characteristic curves allow the following problems to be solved. a) problem 1 : For a given alternative and given size of sample, determine the probability 0 of not rejecting the null hypothesis (type I I risk). b) problem 2 : For a given alternative and a given value of 0 det
21、ermine 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 order to facilitate practical applications : - sets 1.1 to 14.1, giving the risk 0 as a function of the alternative, for a = 0,05 or a = 0,Ol a
22、nd 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 (Y = 0,05 or (Y = 0,OI and for different values of the risk 0. 6) Attention is drawn to the practical significance of interpreting statistics by mea
23、ns of tests of hypotheses and curves. When testing a hypothesis such .as m =mo (or ml = m2), it is generally desired to know whether it can be concluded with little risk of mistake, that m does not differ too greatly from mo (or ml does not differ too greatly from m2). Moreover, the choice of the va
24、lue 01 = 0,05 or a = 0,Ol 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 would be with values close to mo (or value of the difference D =ml -m2 close to O), possibly using both values of the type I risk a
25、 = 0,05 and a = 0,Ol and, in these circumstances, to evaluate by means of the operating characteristic curves the risk 0 associated with different alternatives. 7) The sets of curves which are given in section two of this International Standard are described and discussed in six clauses which corres
26、pond to the tables in IS0 2854. The detailed correspondence between the different sets, the problems which they allow to be solved, the clauses of this International Standard and the tables of IS0 2854, .appear at the top of the group of sets. 1 IS1 BSSZ84b: PART*5 77 W L24689 0315259 4 m BS 2846 :
27、Part 5 : 1977 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 of the second kind - which they calle
28、d “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 deviation from the null hypothesis, which the
29、y designated by the symbol 0. 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 X of the population differs (when such is truly the case) by at leas
30、t a given quantity from the specified value Ao, 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 0, might b
31、e taken into consideration at the same time as the “producers risk a. The symbol 0 was adopted for the type II risk in IS0 3534, Statistics - Vocabulary and symbols, and it has therefore been adopted with the same significance in this International Standard. However, as this symbol is used, and will
32、 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. 2 BSI BS*214b: PART*S 77 3b246b 03152b0 O BS 2846 : Part 5 : 1977 - - 1 COMPARISON OF A MEAN WITH A GIVEN VALUE (VARIANCE
33、KNOWN) See table A of IS0 2854. m -mo b) A=- (one-sided test m mo 1 .I Notations n = sample size m = population mean mo = given value u = standard deviation for the population 1.2 Tested hypotheses For a two-sided test, the null hypothesis is m =mo, the alternative hypothesis corresponding to m #mo.
34、 For a one-sided test, the null hypothesis is a) either m mo; b) or m mo, the alternative hypothesis corresponding tom mo -6m -mo) c) A= (one-sided test m 2 mo) alter- -u natives m mo) alternatives U m mo = 2,30, the value chosen for the type I risk being a = 0,05 (a is therefore here the “producers
35、 risk“). The consumer knows from experience that the mean of the different batches may vary, but the dispersion of the breaking loads within any one batch is practically constant with a standard deviation u = 0,33. 1.5.1 The consumer envisages selecting n = 10 bobbins per batch, and wishes to know t
36、he probability that he will not reject the hypothesis m 2,30 (hence to accept the batch) where in fact the mean breaking load would be m = 2,lO. The set to be consulted is set 3.1. The value of the parameter X form = 2,lO is = 192 -6im-mo) - fi0 (2,30-2,101 A= - U 0,33 The straight line u =Q) gives
37、for 1000 the value 36 : 0 = 0,36 or 36 %. 1.5.2 This value being considered by the consumer as much too high, he decides to select a sample of sufficient size for the risk 0 to be reduced to 0,lO (or 10%) if m = 2,lO. The set to be consulted is set 3.2. The value of the parameter A for m = 2,l O is
38、m-mo 2.30-2,lO U 0,33 = 0.61 A=-= The value of n, read on the straight lines (broken) 0 = 010 is n = 22. 3 ElSI BS*284b: PART*5 77 Lbi94bb9 OLL5261 2 = BS 2846 : Part 5 : 1977 2 COMPARISON OF A MEAN WITH A GIVEN VALUE (VARIANCE UNKNOWN) See table A of IS0 2854. IMPORTANT NOTE The type II risk 0 depe
39、nds on the true value u of the standard deviation for the population, which is unknown. Hence, 0 can only be known approximately, and this provided that an order of magnitude of u is available. In the absence of any valid previous information, one will take for u the estimation s obtained from the s
40、ample. It is strongly recommended that the influence on the values read from the operating characteristic curve of an error made for the standard deviation u should be considered. The inaccuracy can be very great where u has been estimated from a sample of small size : allowance for this situation c
41、an be made by placing s within the confidence limits for u calculated by the method in table F of IS0 2854. 2.1 Notations n = sample size m = population mean mo = given value u = standard deviation for the population (which will be replaced by an approximate value) v =n-I 2.2 Tested hypotheses For a
42、 two-sided test, the null hypothesis is m =mo, the alternative hypotheses corresponding to m #mo. For a one-sided test, the null hypothesis is a) either m mo; b) or m 2 mo, the alternative hypothesescorresponding tom mg (one-sided test m mo) alter- fi (m -mo) c) X=- U natives m mo U m -mo c) A=- (on
43、e-sided test m mo) alternatives U m mp; b) or ml m2, the alternative hypotheses corresponding to ml m2 m2 -m1 cl h= (one-sided test ml 2 m2) alternatives Ud ml m2) alternatives ml m2; b) or ml m2, the alternative hypotheses corresponding to ml m2) alternatives Ud ml m2 (one-sided test ml 2 m2) alter
44、natives m2 -m1 c) x=- ml u$ (u uo); b) or u2 au; (u-,), the alternative hypotheses corresponding to 02 u2 For a one-sided test, the null hypothesis is a) either u: u$ (ul u2) and defined by the parameter X = u1 /u2 (1 u$ (u1 u2), the alternative hypotheses corresponding to u: up. He decides to carry
45、 out the one-sided test u: u2 (u1 2 up) as described in IS0 2854, taking for the type I risk the value (Y = 0,05 (a is therefore here the “consumers risk“). 6.5.1 The consumer envisages selecting n = 20 bobbins from each batch, and wishes to know the probability that he will not reject the hypothesi
46、s ul up (hence not to find that batch number one has a smaller dispersion than batch number two) while in fact one would have u1 =-u2. The set to be consulted is set 13.1, with for the parameter A the value 2 3 For n = 20, one finds (by interpolation) that the corresponding value of 1 O0 0 is close
47、to 48 : 0 = 048 or 48 %. 6.5.2 The consumer recognizes that he runs a high risk of not revealing an effectively interesting improvement. He therefore decides to select from each batch a sufficiently large sample for the value of 0 to be reduced to 010 (or 10 %), when u1 /u2 = 2/3. The set to be cons
48、ulted is set 13.2. For 0 = 0,l and X = 15 one finds n = 55. 11 BSI BS*284b: PART*5 77 m 3b24bb9 0335269 7 BS 2846 : Part 5 : 1977 SECTION TWO : SETS OF CURVES REFERENCES OF THE SETS OF CURVES Reference of the set of curves Reference of the table in Is0 2854 Clause of Section one Problem 22) Test Pro
49、blem 11) 0.05 0,Ol 0,05 0,Ol Comparison of means m = mo; u known m = mo; o unknown m1 = m2; u1,u2 known ml = m2; ul = 02 unknown m 4 mo, m mo; u known m 4 mo, m mo; u unknown ml 4 m2, ml m2; ut, 02 known ml u. Comparison of two variances or of two standard deviations u2 = u2 12 0,05 0,Ol 0,05 0.01 11.1 12.1 13.1 14.1 11.2 12.2 13.2 14.2 6 G 1) Sample size(c) given, determine p. 2) p given, determine the sample cize(s) to be taken. Scale Sets of curves Abscissae Ordinate 1.1,2.1,3.1,4.1,7.1,8.1, 11.1, 12.1,13.1,14.1 Linear Nor
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