This standard specifies the method for comparing the population mean of the differences between paired observations with zero or other pre-specified values. GB 3361-1982 Statistical processing and interpretation of data Comparison of two means in the case of paired observations GB3361-1982 standard download decompression password: www.bzxz.net
This standard specifies the method for comparing the population mean of the differences between paired observations with zero or other pre-specified values.

National Standard of the People's Republic of China
Statistical Processing and Interpretation of Data
Comparison of Two Means in the Case of Paired Observations
UDC 519.28
GB 336182bzxZ.net
This standard specifies the method for comparing the population mean of the difference between paired observations with zero or other pre-specified values. For two observations X and Y with certain characteristics, they are called paired observations if they are obtained in the following situations: a. The same individual is taken from the same population, but the observation conditions are the same (for example: comparison of the results of two different analytical methods of the same product).
b, two different individuals, except for the systematic difference involved in the test, are similar in all aspects (for example: the yield of two adjacent plots of land sown with two different varieties of seeds). It must be noted that in this case, the power of the test depends on whether the following assumption is correct: there are no other systematic differences between the paired individuals except the systematic difference being tested. This standard is based on the international standard ISO3301 "Statistical interpretation of data - comparison of two means in the case of paired observations" (first edition in 1975).
1 Scope of application
This method It can be used to determine the difference between two treatments. In this case, it is the ith observation of the first treatment and it is the ith observation of the second treatment. These two series of test results are not independent. The term "treatment" should be understood in a broad sense. For example: the two treatments compared can be two test methods, an instrument or a laboratory, so as to discover possible systematic errors between the two treatments. Two treatments carried out successively with the same test materials may have an influence on each other, and the values ​​obtained may be related to each other. A good experimental design should be able to eliminate this bias. In addition, it can also be used when there is a treatment and its effect can be compared with the effect when there is no treatment. Comparison, the purpose of this comparison is to make sure that the treatment is correct. Conditions for application
If the following two conditions are met, this method can be effectively applied: a, difference d, X-series spring case independent random Li series bd: the distribution of is normal distribution or approximately normal distribution, if the distribution of deviates from normality, then when the sample size is sufficiently large, the method described is still valid, the greater the deviation from normality, the larger the sample size required. However, even in special cases, the sample size is 100, which is sufficient for most practical applications. The National Bureau of Standards issued on December 30, 1982
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The problem studied
Test service…·
Statistical items
The sample is too small
The sum of observations:
EXi=ZY,=
The sum of differences:
The sum of squares of differences,
Given value:
Degrees of freedom
Significance level,
Two-sided expansion,
GB3861-82
a-(X-)
Ai- [ti-aivi'? -
d,= [ti-a?2(v)ivn.
If —α-d,IA.
then reject the hypothesis that the population mean of the difference is equal to d,. Side case:
a. If dde-A
then reject the hypothesis that the population mean D is at least equal to d, b. If d>d,+A,
then reject the hypothesis that the population mean of the difference is at most equal to d. Note: (. (v) is the (1-a) quantile of the variable with degree v. The values ​​of tz()' are given in the table.
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ratio ti-a()/n
double-sided case
single-sided case
0, 823
.0.410
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GB 8861--82
Example: The data in the following table were collected to determine whether the average magic loss rate of the rotating shaft is different when different metal bearings are used in internal combustion engines.
: Wear of the shaft after a given time
Technical characteristics
Statistical item H
Sum of the small
observations:
I = 86h.68
, = 54. 36
Sum of differences
Ed, = 27 1.32
Sum of squares of differences
165,10
Z d - 12 399.975 2
Given value:
Significant result:
Result:
Overall mean DGiven value comparison
Two-sided color:
White metal
Unit: 0.001 mm
d,=X,-Y
1868.68 591.361- 30.48
L12 399.9752
= 501. 837 7
0 = , 501.837 7 = 22.468 6
A4, = 1.118 ×22.4686
- 25. 139 8
Under the extreme water level of Sheng, reject the hypothesis that the wear rates of the two metal bearings are equal.
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4 Type II error
GB 3361--82
When the null hypothesis is correct, the probability of rejecting this hypothesis is at most equal to the significance level. When the null hypothesis is correct, rejecting this hypothesis is called making a type I error. Therefore, α limits the risk of making this type of error. On the other hand, it is possible to make a type II error, that is, accept this hypothesis when the null hypothesis is incorrect. When the null hypothesis is incorrect, the probability of rejecting it is 1-β, which is called the test efficiency. Therefore, the probability of making a type III error is. When the sample size II and the probability of making a Type I error are known, the probability of making a Type II error depends not only on the population mean D of the differences d, X, Y (different alternative hypotheses can be assumed for D), but also on the standard deviation of these differences. This standard deviation is generally unknown, and when II is small, the sample provides only a single estimate. As a result, it is impossible to determine an upper limit on the probability of making a Type II error.
The following figure shows the relationship between the power of the test 1 and the population mean divided by the standard deviation D/ for different values ​​of n and significance levels of 0.05 and 0.01, respectively, for a one-sided test of hypothesis H. [-8 1.00
Figure 1 Power of a single sample test (one-sided) = .01 From Figures 1 and 2, we can conclude that the power of the test is uniquely determined by the ratio of the population mean to the standard deviation of the differences, the significance level α, and the sample size 1. at
b. The power function is a strictly (increasing) function of the population mean of the difference. Standard photography network various practical standard cylinder industry information free download 1.
GB3861
Figure 2 Power of single sample 1 test (one-sided) α=0.05If D>0, α is 0 and 1, then the power function is still a strictly (increasing) function of sample size n and significance level α. c, for significance level 0.05 and sample size n = 50, when the true mean of the difference exceeds half of 1 standard deviation of the difference (i.e. a > 0.5
, the power is at least 0.95. When n = 20 and D/αd ≥ 0.78, the power is also at least 0.95. Additional notes:
This standard was proposed by the Standardization Institute of the Ministry of Electronics Industry. This standard was jointly drafted by the Standardization Institute of the Ministry of Electronics Industry, the Institute of Systems Science of the Academy of Sciences and Harbin Institute of Technology. Plant
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