Statistical interpretation of data--Test for parameter in Poisson distribution

The statistical terms used in this standard can be found in the national standard GB 3358-82 "Statistical Terms and Symbols". GB 4090-1983 Statistical processing and interpretation of data Test of Poisson distribution parameters GB4090-1983 standard download decompression password: www.bzxz.net
The statistical terms used in this standard can be found in the national standard GB 3358-82 "Statistical Terms and Symbols".

1 Introduction
National Standard of the People's Republic of China
Statistical interpretation of dataTest for parameter in Poisson distribution1.1 The statistical terms used in this standard can be found in the national standard GB3358-82 "Statistical Terms and Symbols". 1.2 The population discussed in this standard is a Poisson distributionX=xlal
x=0,1,2,
UDC 519.25
GB 4090--83
where ≥0 is the distribution parameter. This standard is based on independent random samples xI, x2, x, and specifies the method for testing given hypotheses related to the parameters. This standard can be used when there is sufficient reason to believe that the population follows a Poisson distribution. 1.3 H. represents the null hypothesis and H, represents the alternative hypothesis. H. >0 is a given value. This standard discusses three cases: Hot A = o,
Hi: a℃2=15 So reject Ho: =0.6=18
3 One-sided test Ho: ^o
3.1 Implementation steps
GB4090-83
a. Determine the critical value of the rejection region ℃, by u, sample size n and the given test significance level α (see 3.2 for the determination of c2). b. Calculate the value of T=≥,x,.
c. When Tc, reject Ha, when T, do not reject Ho3.2: Determination of the critical value of the rejection region c2
c, can be determined by input. , n and a. It is the smallest integer that satisfies the following formula: Tic2 / nad) -Z-(ndo)
The corresponding probability of the first and second type errors is shown in Appendix A (Supplement). C2 can be determined by the x' distribution table method or the normal approximation method.
3.2.1x distribution table method
C2 is the smallest integer that satisfies the following formula:
2n/0x? (2c2)Www.bzxZ.net
Formula: (2c2) is the a quantile of the x distribution with 2c degrees of freedom. 3.2.2 Normal approximation method
C2 is the smallest integer that satisfies the following formula:
Formula: ul-a is the 1-α quantile of the standard normal distribution. 2
As with the two-sided test, when it is difficult to use the distribution table method, the normal approximation method can be used. 4 One-sided test Ho: 1
4.1 Implementation steps
By. , sample size n and the given significance level α of the test, determine the critical value c of the rejection region; (the determination of c, see 4.2). a.
b. Calculate the value of T = yixi.
c. When Tc,, reject Ha, and when Tci, do not reject Ho4.2 Determination of the critical value of the rejection region c,
Ct is determined by, n and α, and it is the largest integer that satisfies the following formula. =(ndo)
The corresponding probability of the first type error and the probability of the second type error are shown in Appendix A (Supplement). ci can be determined by the 'distribution table method or the normal approximation method.
4.2.1×2 distribution table method
C, is the largest integer that satisfies the following formula:
2nAoxi-a (2ci+ 2)
Where: ×-α(2cl+2) is the 1-α quantile of the × distribution with 2c, +2 degrees of freedom. 4.2.2 Normal approximation method
Ci is the largest integer that satisfies the following formula:
Where: u- is the quantile of the standard normal distribution.
When it is difficult to use the x distribution table method, the normal approximation method can be used. 136
GB 4090—83
Appendix A
Two types of errors
(Supplement)
A.1 First type error
The probability of a first-type error is the probability of rejecting the null hypothesis when the null hypothesis is true. It can be expressed as a function of the critical value of the rejection region. It is denoted as a.
Ho: = Case: α'=P (≤c ln） + (Tlna)= (T≤c-Ina) + 1-P (T≤cz- 1 /nao)Ho：o Case: a' (Tc2lndo) = 1(Cz1lnao)(When <, there is always ())
Ho: Case: α'(c|n)
(When <, there is always (cn)α')
Due to the discreteness of the Poisson distribution, the critical value of the rejection region is an integer, and α is often not exactly equal to the given significance level α, but smaller than α.
A.2 Type 2 Error
The probability of a type 2 error is the probability of not rejecting the null hypothesis when the null hypothesis is wrong. It can be expressed as a function of the critical value of the rejection region and the specific input value in the alternative hypothesis considered, denoted as β. Ha：o Case: β\= p (ci<
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