This standard specifies the estimation and test methods of binomial distribution parameters. This standard is integrated on the basis of GB/T 4087.1-1983 "Statistical processing and interpretation of data - point estimation of binomial distribution parameters", GB/T 4087.2-1983 "Statistical processing and interpretation of data - interval estimation of binomial distribution parameters" and GB/T 4088-1983 "Statistical processing and interpretation of data - test of binomial distribution parameters". This standard replaces GB/T 4087.1-1983, GB/T 4087.2-1983 and GB/T 4088-1983. Compared with GB/T4087.1-1983, GB/T4087.2-1983 and GB/T4088-1983, the changes in the technical content of this standard mainly include: --- The standard format has been modified according to the requirements of GB/T1.1-2000 "Guidelines for Standardization Part 1: Structure and Writing Rules of Standards"; --- Added p-value test. GB/T 4088-2008 Statistical processing and interpretation of data Estimation and test of binomial distribution parameters GB/T4088-2008 standard download decompression password: www.bzxz.net
This standard specifies the estimation and test methods of binomial distribution parameters.
This standard is integrated on the basis of GB/T 4087.1-1983 "Statistical processing and interpretation of data - point estimation of binomial distribution parameters", GB/T4087.2-1983 "Statistical processing and interpretation of data - interval estimation of binomial distribution parameters" and GB/T4088-1983 "Statistical processing and interpretation of data - test of binomial distribution parameters".
This standard replaces GB/T4087.1-1983, GB/T4087.2-1983 and GB/T4088-1983.
Compared with GB/T4087.1-1983, GB/T4087.2-1983 and GB/T4088-1983, the changes in technical contents of this standard mainly include:
--- The standard format has been modified according to the requirements of GB/T1.1-2000 "Guidelines for Standardization Work Part 1: Structure and Writing Rules of Standards";
--- Added the p value test.
Appendix A, Appendix B, Appendix C, Appendix D, Appendix E, Appendix F, Appendix G and Appendix H of this standard are all normative appendices.
This standard is proposed and managed by the National Technical Committee for Standardization of Statistical Methods (SAC/TC21). The
drafting units of this standard are: China National Institute of Standardization, Peking University, Guangzhou Product Quality Supervision and Inspection Institute, Hainan Product Quality Supervision and Inspection Institute.
The main drafters of this standard are Yu Zhenfan, Sun Shanze, Wu Yuluan, Deng Suixing, Ding Wenxing, Huang Yan, Cai Weihong, Hou Xiangchang and Fang Xiangzhong. The
previous versions of the standards replaced by this standard are:
---GB/T4087.1-1983;
---GB/T4087.2-1983;
---GB/T4088-1983.
The clauses in the following documents become the clauses of this standard through reference in this standard. For any dated referenced document, all subsequent amendments (excluding errata) or revisions are not applicable to this standard. However, the parties to the agreement based on this standard are encouraged to study whether the latest versions of these documents can be used. For any undated referenced document, the latest version shall apply to this standard.
GB/T4086.2 Numerical table of statistical distribution χ2 distribution
GB/T4086.4 Numerical table of statistical distribution F distribution
ISO3534-1:2006 Statistical vocabulary and symbols Part 1: General statistical terms and terms used in probability
ISO3534-2:2006 Statistical vocabulary and symbols Part 2: Applied statistics

Foreword III
Introduction IV
1 Scope 1
2 Normative references 1
3 Terms, definitions and symbols 1
4 Point estimation of binomial distribution parameters 2
4.1 Classical estimation method 2
4.1.1 Sample selection method 2
4.1.2 Estimator 2
4.1.3 Example 2
4.2 Sequential Sample Estimation Method 2
4.2.1 Sample Selection Method 2
4.2.2 Estimator 2
4.2.3 Example 2
5 Interval Estimation of Binomial Distribution Parameter 3
5.1 Two-Sided and One-Sided Confidence Intervals for
Proportion p 3 5.2 Method for Calculating Confidence Intervals 3
6 Tests of Binomial Distribution Parameter 7
6.1 Null Hypothesis and Alternative Hypothesis 7
6.2 Two-Sided Test H0:p=p0,H1:p≠p0 7
6.2.1 Implementation Steps 7
6.2.2 Determination of the critical values ​​c1 and c2 of the rejection region7
6.2.3 Example8
6.3 Upper limit one-sided test10
6.3.1 Implementation steps10
6.3.2 Determination of the critical value c2 of the rejection region10
6.4 Lower limit one-sided test10
6.4.1 Implementation steps10
6.4.2 Determination of the critical value c1 of the rejection region11
Appendix A (normative) Method for determining ˆ and c based on the absolute error limit of the point estimate12
Appendix B (normative) Several other estimation methods13
Appendix C (normative) Upper confidence limit table (ˆ=10(1)30) 14
Appendix D (normative) Graphical method 28
Appendix E (normative) Table of upper critical values ​​of the rejection region 32
Appendix F (normative) Two types of errors in significance tests 37
Appendix G (normative) Equivalent method for testing H0 42
Appendix H (normative) Estimation of sample size when first and second type errors are given 43

ICS 03.120.30
National Standard of the People's Republic of China
GB/T 4088--2008
Replaces GB/T 4087.1---1983, GB/T 4087.2—1983, GB/T 4088—1983Statistical interpretation of data-Estimation and hypothesis test of parameter in binomial distribution2008-07-16 Issued by
General Administration of Quality Supervision, Inspection and Quarantine of the People's Republic of China Standardization Administration of China
2009-01-01Implementation
1 Scope
2 Normative references
3 Terms, definitions and symbols.
4 Point estimation of a distribution parameter
4.1 Classical estimation method
4.1.1 Sample selection method
4.1.2 Estimator
4.2 Sequential sample estimation Method
4.2.1 Sample extraction method
4.2,2 Measurement
4.2.3 Example
5 Interval estimation of binomial distribution parameter
5.1 Two-sided confidence interval and one-sided confidence interval of ratio P5.2 Method of calculating confidence interval
Test of binomial distribution parameter
6.1 Null hypothesis and alternative hypothesis
6.2 Two-sided test Ho: p=po, H1: p≠po
6.2.1 Implementation steps． …
6.2.2 Determination of the critical values ​​c1 and C2 of the rejection region6.2.3 Example
6.3 Limited one-sided test
Implementation steps
Determination of the critical value c2 of the rejection region
6.4 Lower limit one-sided test-
6.4.1 Implementation steps
6. 4. 2 Determination of the critical value c1 of the rejection region Appendix A (Normative Appendix)
Appendix B (Normative Appendix)
Appendix C (Normative Appendix)
Appendix D (Normative Appendix)
Appendix E (Normative Appendix)
Appendix F (Normative Appendix)
Appendix G (Normative Appendix)
Appendix H (Normative Appendix)
Determine n and according to the absolute error limit of the point estimate. Methods of other estimation methods
........
Confidence upper limit table (n=10(1)30)
Graphical method
Rejection region upper critical value table
Two types of errors in significance test
Equivalent method of Hu test
Estimation of sample size n given the first and second type errors GB/T 4088—2008
CB/T 4088--2008
This standard is integrated based on GB/T4087.11983 "Statistical processing and interpretation of data - Point estimation of binomial distribution parameters",
CB/T4087.2:1983 "Statistical processing and interpretation of data - Area estimation of binomial distribution parameters" and GB/T40881983 "Statistical processing and interpretation of data - Test of binomial distribution parameters". This standard replaces GB/T4087.1---1983, GB/T4087.21983 and GB/T4088-1983. Compared with GB/T4087.1-1983, GB/T4087.2-1983 and GB/T4088-1983, the changes in technical contents of this standard mainly include:
The standard format has been modified according to the requirements of GB/T1.1:2000 "Guidelines for Standardization Work Part 1: Structure and Writing Rules of Standards";
Force value test has been added.
Appendix A, Appendix B, Appendix C, Appendix D. Appendix E, Appendix F, Appendix G and Appendix H of this standard are normative appendices. This standard is proposed and managed by the National Technical Committee for Standardization of Statistical Method Application (SAC/TC21). The drafting units of this standard are: China National Institute of Standardization, Beijing Institute of Standardization, Guangzhou Product Quality Supervision and Inspection Institute, Hainan Product Quality Supervision and Inspection Institute.
The main drafters of this standard are: Ding Zhenfan, Sun Shanze, Wu Yusong, Deng Suixing, Ding Wenxing, Huang Yan, Cai Weihong, Hou Xiangchang, Fang Xiangzhong. The previous versions of the standards replaced by this standard are: -GB/T 4087.1—1983;
CB/T4087.2—1983,
GB/T10881983.
GB/T 4088-2008
Scientific research, industrial and agricultural manufacturing, and management work are inseparable from data, and the organization, analysis, and interpretation of these data are inseparable from statistical methods. Statistics is a discipline that studies the organization, analysis, and correct interpretation of digital data. People obtain various digital data from different sources. These digital data are generally messy and must be organized and simplified before they can be used. The use of perfect statistical methods can make the data organized and arranged in an orderly manner. Using graphics or a few important parameters, the characteristics of a large number of data can be expressed, which can avoid incorrect interpretations and reduce the cost of obtaining satisfactory data to a minimum, thereby improving economic benefits. &Statistical Processing and Interpretation of Data" contains a number of national standards, which are: Determination of statistical tolerance interval (GB/T3359) Estimation and confidence interval of the mean (GB/T3360) Comparison of two means in the case of paired observations (GB/T13361) Estimation and test of a distribution parameter (GB/T4088) Estimation and test of Poisson distribution parameters (GB/T4089) Normality test (GB/T4882) Judgment and treatment of outliers in normal samples (GB/T4883) - Estimation and test methods for mean and variance of normal distribution (GB/T4889) Power of test for mean and variance of normal distribution (GB/T4890) Judgment and treatment of outliers in samples of type I extreme value distribution (GB/T6380) Parameter estimation of gamma distribution (Pearson type II distribution) (GB/T 8055) Judgment and treatment of outliers in exponential distribution samples (GB/T8056) 1 Scope
Statistical processing and interpretation of data
Estimation and test of binomial distribution parameters
This standard specifies the estimation and test methods of binomial distribution parameters. GB/T 4088-—2008
Assume that some individuals in the population have certain characteristics, and the probability is the ratio of individuals with this characteristic in the population. For example, the probability can be the ratio of defective products in a batch of products. Randomly and independently extract a certain number of individuals from the population as samples. This standard specifies the methods for point estimation, range estimation and test of population parameters based on such samples. For a finite population, let its size be N (N should be sufficiently large) and the sample size be n. When the extraction is with replacement, or when the extraction is without replacement, but the probability is <0.1, the n extractions can be considered independent. 2 Normative references
The clauses in the following documents become clauses of this standard through reference in this standard. For any dated referenced document, all subsequent amendments (excluding errata) or revisions are not applicable to this standard. However, parties that reach an agreement based on this standard are encouraged to study whether the latest versions of these documents can be used. For any undated referenced document, the latest version applies to this standard. GB/T4086.2 Distribution of numerical tables of statistical distributions GB/T4086.4 Distribution of numerical tables of statistical distributions F ISO3534-1:2006 Statistical vocabulary and symbols Part 1: General statistical terms and terms used for probability ISO3534-2:2006 Statistical vocabulary and symbols Part 2: Applied statistics 3 Terms, definitions and symbols
ISO3534-1:2006, ISO3534-2:The terms and definitions specified in GB/T 19000-2006 and GB/T 19000-2000 and the following terms, definitions and symbols apply to this standard. For ease of reference, some terms and symbols are directly quoted from the above standards. n Sample size
The ratio of individuals with specified characteristics in the population Lower confidence limit of interval estimate
Upper confidence limit of interval estimate of pu
1-a Confidence level of interval estimate
The number of individuals with specified characteristics in the sample Probability of type II error in significance test
β Probability of type II error in significance test Probability of rejecting the null hypothesis when it is true β Probability of not rejecting the null hypothesis when it is wrong Hc Null hypothesis of significance test
H1 Alternative hypothesis of significance test||t t||P(AI probability of event A
N the maximum number of individuals contained in a finite population
GB/T4088—2008
4 Point estimation of binomial distribution parameters
4.1 Classical estimation method
4.1.1 Sample extraction method
The sample size is specified in advance. The sample is randomly and independently extracted from the population. At this time, the number of individuals with certain characteristics in the sample is an observation value of a random variable X that obeys a binomial distribution. The probability of X taking the value: is P(X=ln,p)=(p(1 p)-+,r=0,l,2,,4. 1.2 Estimation The estimation of the quantity
is denoted by
where:
sample size. The determination method can be found in Appendix A formula (A,1); the number of individuals with specified characteristics in a sample. 4.1.3 ExampleWww.bzxZ.net
To estimate the defective rate of a batch of products (about 1000 pieces), 40 pieces are randomly selected as samples, of which 5 are defective. Test
0=year5
4.2 Sequential Sample Estimation Method
4.2. 1 The sampling method
does not specify the sample size in advance, but selects individuals from the population sequentially, randomly, and independently. When an individual is selected, check whether the individual has the specified characteristics immediately, and continue to accumulate the number of individuals with the specified characteristics. When the number of individuals with the specified characteristics reaches the predetermined number c (an integer greater than or equal to 2), stop sampling. At this time, the total number of individuals drawn cumulatively, n, is a random variable that obeys the negative binomial distribution, and the probability of the maximum value of n is: p(n-rlc,p)-(=1)pr(1-p),*,k=c,c+1,**4. 2. 2 Estimator
In the formula:
The number of individuals with specified characteristics that are specified in advance. The determination method can be found in Appendix A formula (A, 2); when individuals with specified characteristics are reached, the total number of individuals taken is accumulated. 4.2.3 Example
To estimate the defective rate of a batch of products (about 1000 pieces), sampling and inspection are carried out one by one in sequence. It is stipulated that positive sampling will be stopped when 5 defective products are found: when the 35th piece is drawn, the 5th defective product is found, then c-5
=C-1
-1 34
5Interval estimation of binomial distribution parameters
5.1 Two-sided confidence interval and one-sided confidence interval of ratio P GB/T 4088—2008
's two-sided confidence interval is (.u), where ≤r<, called the lower confidence limit, and called the upper confidence limit. The one-sided confidence interval of the force has the following two forms: a) a confidence interval with only the lower confidence limit (O≤<) (, 1. b) a confidence interval with only the upper confidence limit plus (0<≤1) [0.). These intervals must contain point estimates of the force. Which type of interval to choose depends on the nature of the specific problem and does not depend on the observed value. It is a function of the observed value. The confidence interval used in this standard satisfies the following conditions: the obtained confidence interval contains the true value. The frequency of this event is approximately equal to the confidence level 1-α. Specifically, the two-sided interval should satisfy Pi=a
and the one-sided interval should satisfy
P(p30, and 0.1<3
<0.9, the confidence limit of the confidence level is almost 1-α, which is given by the following approximate formulas (7) and (8): pl -p+ -u/p.(i=p.)/(n+2d)
pu -p* +u/p*(lp*)/(n+2d)
where:
p,=z+d-0.5
p*=z+d+o. 5
n→-2d
ratio, d is a constant. For different confidence levels, the values ​​of u and d are shown in Table 1. Table 1
Confidence level
Example: In an experiment: n=10, r=12, take 1-@=0.95, 4
·（8)
a) Find the one-sided confidence interval [0,pu),
1---α-0.95 Look up Table 1 and get u=1.645,d-1,n40,x-12
+*=±+0.5_121+0:5=0.321
40+2×1
/p*(1-p*)/(n+2d)-0.072
GB/T 4088—2008
pu=p*+uvp(1-p)/n+2d)-0.321+1.645×0.072=0.439The one-sided confidence interval is [0,0.439].
b) Find the one-sided confidence interval (PL, 1.
1a=0. 95, look up Table 1 to get u-1. 645, d=1, n=40, x12
p, -+do. 5
512-1-0.5
40+2×1
p(1=p.$/(n+2d)=0.071
P.=p, —uVp, (1-p, 57(n+2d)-0.298-1.645×0.071=0.181 The one-sided confidence interval is (0.181, 1.
c) Find the two-sided confidence interval (r.,).
1—=0. 95, look up Table 1 to get u=1. 960, - 1. 5n=40,r=12
p#+4-05±12+15-0. 5
40+2×1.5
p*=#+d+0.5_12+1.5+0.5
10+2×1.5
/p.(1-p.)/(n+2d)-0.070||t t||P,=p,—up(1—p.)/(n-2d)=0.302—1.960×0.070=0,165Vp*(1-p)/(n+2a)0.071
Prp\+wVp*(1 p*)/(n- 2d)
=0. 326 -1 1. 960× 0, 071
The two-sided confidence interval is (0.165, 0.466). 5 When n>30, and ≤0.1 or ≥0.9, the Poisson approximation can be used. This approximation requires the use of the x distribution table (see 5.2. 5
GB/T4086.2). At this time, the lower confidence limit is
2n--1+
n+aa'
In+xx
For a one-sided confidence interval, where
(2x）
When three is close to 0
When to close to 1
x-.[.2(n)+2]
GB/T4088—2008
For a two-sided confidence interval, where
The upper confidence limit is
For a one-sided confidence interval, where
For a two-sided Confidence interval, where
/2(2z)
xi-a/2[2(n—)+2]
2n—x+
++1—
when it is close to 0
when it is close to 1
xi-. (2+2)
x[2(n2)
-/2(2+2)
x2[2(n)
here,) represents the quantile of the distribution with degrees of freedom. For example, in an experiment; n=50, r=5, take 1…α=0.95, a) Find the one-sided confidence interval [0,).
n—50,x—5,
一0.1, use the formula close to zero.
.9(12)
x. (2+2) =
×21.026--10.513
2×10.513
5~2×50-5+10.513
2n—r+>
The one-sided confidence interval is [0.0.199).
b) Find the one-sided confidence interval (, 1]
n=50,2=5,
一0.1, using the formula close to zero,
xi-. (22)
.5(10)
×3.9401.970
2×1. 970
pL=2n-2+1f+\2x50-$+141. 970=0. 040 The one-sided confidence interval is (0.040,13
c) Find the two-sided confidence interval (pl, pu). 2=50, yuan-5, =0.1, use the formula close to zero. For.
x2(22）
xo.D23（10）
For, there is
X3.247=1.621
pl.2n*41+±-2x50-$+141,624-0. 0330.975（12)
-/2(22)~
X23. 337=11, 669
2×11.669
PU2n:4±-2×505+11.669
The two-sided confidence interval is (0.033,0.219). 6 Test of binomial distribution parameters
6.1 Null hypothesis and alternative hypothesis
GB/T 4088---2008
Use Ho to represent the null hypothesis and Hi to represent the alternative hypothesis. o is a given value, 0≤a≤1. This standard deals with three common situations: Hc:=po, H (two-sided test)
Ha:oH:>p (one-sided test)
Hc:pe.Hi:< (one-sided test)
Which type of test to choose depends on the needs of the specific problem. 6. 2 Two-sided test Ha:p=o,H1:ppo
6.2. 1 Implementation steps
a) Determine the critical values ​​1 and t2 of the rejection region based on po, sample size n and the given significance level of the test -u (see 6.2.2 for the determination of c1 and 2).
b) The number of individuals with this characteristic among the n individuals collected cumulatively. ) When ≤c or c, reject H
When 1
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