Statistical interpretation of data; Power of tests relating to means and variances of normal distributions
Some standard content:
1 Introduction
National Standard of the People's Republic of China
Statistical interpretation of data -Power of tests relating to means and variances of normal distributions UDC 519.28
GB4890—85
1.1 This standard is a continuation of GB4889—85 "Statistical interpretation of data - Estimation and testing methods for the mean and variance of normal distributions".
1.2 Class I risk (denoted as α) is the probability of being rejected when the null hypothesis is correct. Class II risk (denoted as 8) is the probability of not being rejected when the null hypothesis is wrong. 1-β is the power of the test. 1.8 Class I risk and Class II risk are selected by the parties according to the possible consequences of each risk. Usually α=0.05 or 0.01.
1.4 The operating characteristic curve of the test represents the functional relationship between the type II risk β and the parameters of the alternative hypothesis. β also depends on the value selected for the type I risk, the sample size, and whether the test is one-sided or two-sided. 1.5 The operating characteristic curve of the test can solve the following problems Problem 1. Determine the value of type II risk β when the alternative hypothesis and sample size are known. Problem 2. Determine the sample size that should be selected when the alternative hypothesis and β value are known. To solve the above problems, two sets of curves are given in Figures 1 to 32. Figures 1, 4, 7, 10, 13, 15, 17, 19, 21, 23, 25, 27, 29 and 31 give the functional relationship between β and the alternative hypothesis for α=0.05 and 0.01 and different sample sizes, respectively. Figures 2, 3, 5, 6, 8, 9, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 and 32 show the functional relationship between the required sample size and the alternative hypothesis for α = 0.05 and 0.01 and different B values.
1.6 This standard is formulated with reference to the international standard ISO3494 "Statistical interpretation of data - Power of mean and variance tests" (1976 edition).
2 For comparison of the mean with a given value (with known variance), refer to Table 1 of GB4889-85 "Statistical processing and interpretation of data - Estimation and testing methods for the mean and variance of the normal distribution". 2.1 Symbols
n: sample size
: population mean,
uo: given value:
q: population standard deviation.
2.2 Assumptions of the test
For a two-sided test, the null hypothesis is μ=μo, and the alternative hypothesis is μμo. For a one-sided test:
Published by the National Bureau of Standards on January 29, 1985
Implemented on October 1, 1985
Or,
GB 4890--85
The null hypothesis is o, and the alternative hypothesis is o
The null hypothesis is μ>μ, and the alternative hypothesis is <μo. 2.3 Problem 1 Given n, determine β
For a two-sided test or a one-sided test, when a value is given, first calculate the value of the parameter as follows: A=Jnl u-μo i
Then use the following corresponding graphs: Figure!
Operating characteristic curve of two-sided test for mean comparison (Type I risk α=0.05)
Operating characteristic curve of two-sided test for mean comparison (Type I risk α=0.01),
Operating characteristic curve of one-sided test for mean comparison Figure 7
(Type I risk α=0.05)1
Figure 10 Operating characteristic curve of one-sided test for mean comparison (Type I risk α=0.01)
β is the curve with degrees of freedom =, and the abscissa is the ordinate of the entry point. 2.4 Problem 2 Given β, determine n
For a two-sided test or a one-sided test, when u is given, first calculate the value of parameter ^ as follows: Jμμ
Then use the corresponding graphs below: Figure 2 Required sample size for a two-sided test comparing the mean to a given value (Type I risk α = 0.05)
Required sample size for a two-sided test comparing the mean to a given value (Type I risk α = 0.01)
Figure 8 Required sample size for a one-sided test comparing the mean to a given value (Type I risk α = 0.05)
Figure 11 Required sample size for a one-sided test comparing the mean to a given value (Type I risk α = 0.01);
n is the imaginary straight line for the given β value, and the horizontal axis is the vertical axis of the entry point. 2.5 Example
A cotton mill claims that the average strength of each batch of cotton yarn it delivers is at least equal to μ. =2.30N. However, the user only agrees to do this: take a certain length of yarn from different cones, and conduct a one-sided test according to the method described in GB 4889-85, taking the Class I risk α=0.05. If the original hypothesis uμ. =2.30N is not rejected, then accept this batch of products, otherwise reject it. Experience has shown that the average strength of each batch of cotton yarn produced by the factory may change, but the dispersion of cotton yarn strength can be considered unchanged, and its standard deviation α=0.33N.
2.5.1 The user draws 10 cones from each batch for observation, and wants to know the probability βB that the original hypothesis μμp2.30N is still not rejected when the average strength is reduced to 2.10N.
When μ=2.10N
=niμμo l
10 (2.30 - 2.10)
Using Figure 7, we can find 1008=36 from the straight line of =α, so β=0.36 (or 36%). 2.5.2 The user believes that the above β value is too high and wants to select a sample of appropriate size to reduce the β value to 0.10 (or 10%)207
When μ=2.10Www.bzxZ.net
GB 4890---85
I μ-μo l
Using Figure 8, we can find n=22 from the dotted straight line of β=0.10. Comparison of mean and given value (variance unknown) Refer to Table 2 of GB4889--85
3.1 Symbols
n: sample size,
u: population mean,
uo: given value
α: (replaced by some approximate value) population standard deviation, : degrees of freedom.
3.2 Hypothesis of test
2.30-2.10
For a two-sided test, the null hypothesis is u=μo and the alternative hypothesis is uo.
For a one-sided test:
The null hypothesis is, and the alternative hypothesis is>μ
Or,
The null hypothesis is u>μo, and
The alternative hypothesis is u<μo
8.8Problem 1
Given n, determine B
For a two-sided test or a one-sided test, when the value of μ is given, first calculate the value of the parameter In as follows: =ynlu-uo
Then use the corresponding curve graphs below: Figure 1|| tt||Operating characteristic curve of two-sided test for mean comparison (Type I risk α=0.05)
Operating characteristic curve of two-sided test for mean comparison (Type I risk α=0.01),
Operating characteristic curve of one-sided test for mean comparison (Type I risk α=0.05)
Operating characteristic curve of one-sided test for mean comparison (Type I risk α=0.01)
β is the ordinate of the point on the curve with v=n-1 and the abscissa is α. 3.4 Problem 2 Given β, determine n
For a two-sided test or a one-sided test, when a value is given, first calculate the value of the parameter In as follows: =
Then use the corresponding graphs below: [μ-μo
Graph 2 of the required sample size for a two-sided test comparing the mean to a given value
(Type I risk α=0.05)
Graph 5 of the required sample size for a two-sided test comparing the mean to a given value
(Type I risk α=0.01)
Graph 8 of the required sample size for a one-sided test comparing the mean to a given value
(Type I risk α=0.05),
GB 4890--85
Figure 11 The curve of the sample size required for a one-sided test comparing the mean with a given value (type I risk α=0.01)
n is the curve for a given β value, and the abscissa is the ordinate of the point of entry. Example 3.5
Same as the example in 2.5, but the user does not know the exact value of the strong standard deviation, and only knows from experience that it is between the lower limit α, = 0.30N and the F limit αu=0.45N.
3.5.1 The user selects 10 tubes of yarn from each batch for observation. He wants to know the probability β that the null hypothesis μo=2.30N is still not rejected when the average strength is reduced to 2.10N.
When μ=2.10 and =0.30
When μ=2.10 and =0.45
Jniμ-μo
10 (2.30 2.10)
J10 (2.30 ~ 2.10)
Using Figure 7, we can find the value of 100β corresponding to 1 by the curve of 9 (by interpolation). The values are 40 and 64, so the lower limit value β is 0.40 (or 40%) and the upper limit value is =0.64 (or 64%). 3.5.2 The user believes that the above β value is too high and wants to select a sample of appropriate size so that even if α=0μ=0.45, the β value does not exceed 0.10 (or 10%).
When μ=2.10 and o=0.45
Jμ-μo l
Using Figure 8, from the curve of 8=0.10, n is about 45. Comparison of two means (known by the user)
Refer to Table 5 of GB4889-85
4.1 Symbols
Sample size
Standard deviation of the difference between two sample means
4.2 Hypothesis tested
2.30-2.10
For a two-sided test, the null hypothesis is μ,=μ2, and the alternative hypothesis is μ, rich μ2. For a one-sided test:
Or,
The null hypothesis is 2The alternative hypothesis is, μ2
The null hypothesis is, μ2, the alternative hypothesis is μ2. 4.3 Problem 1
4.3.1 Given n and n2, determine 8
Population 1
Population 2
For a two-sided test or a one-sided test, when the u value is given, first calculate the value of the parameter as follows: 209
Then use the corresponding curve graphs below: Figure 1
GB 4890--85
=lur-uz /
Operating characteristic curve of two-sided test for mean comparison (Type I risk α=0.05)
Operating characteristic curve of two-sided test for mean comparison (Type I risk α=0.01)
Operating characteristic curve of one-sided test for mean comparison (Type I risk α=0.05)
Operating characteristic curve of one-sided test for mean comparison Figure 10
(Type I risk α=0.01)
β is the vertical coordinate of the point with horizontal coordinate ^ on the curve with degrees of freedom = α. 4.8.2 Given nl+n2 (=2n), determine the minimum β. At this time,
The corresponding n and nz are
4.4 Problem 2
α=/2nlut=μ2 l
4.4.1 Given β, determine n, and n2 (general case) g
Using the curve with v=α in Figures 1, 4, 7 or 10 according to different situations, we can get the general solution to the problem: first determine the input and output, and then, any point (n1, n,) on the curve that fits the equation
is the solution to the problem. The most economical (the sum of n and n2 is the smallest) sample is suitable for ui-μ2)2
Therefore, take
ni=01 (01+α2)
n2=α2(α→+α2)(
4.4.2 Given β, determine n, and n2 (special case) in the special case of α, == and n=n=n, for different μμz values, the alternative hypothesis is determined by the parameter (0<) is more appropriate.
For two-sided test or one-sided test, when μ, and μ2 values are given, first calculate the value of parameter I as follows: =
Then use the following corresponding curve graphs: I μi - μ2 /
The curve of the sample size required for the two-sided test comparing two means 210
(Type I risk α=0.05)
GB 4890 --85
The curve of sample size required for a two-sided test comparing two means (type I risk α=0.01)
The curve of sample size required for a one-sided test comparing two means (type I risk α=0.05)
The curve of sample size required for a one-sided test comparing two means (type I risk α=0.01)
The vertical coordinate of the point on the imaginary straight line with a given β value is the horizontal coordinate. 4.5 Example
After changing the process, a cotton mill claimed that the average strength μ2 of the new process was the same as the average strength μ of the original process. The user prepared To accept the new process, in order to verify the manufacturer's statement, a certain length of yarn is taken from different bobbins, and a two-sided test is performed according to the method described in GB4889-85. The Class I risk α=0.05 is taken. If the hypothesis μ,=μ2 is not rejected, the new process is accepted. Experience shows that the discrete degree of strength of the cotton yarn produced by the factory can be considered unchanged, and its standard deviation α=0.33N. 4.5.1 The user takes a batch of cotton yarn for the new and old processes, and selects 10 bobbins from each batch for observation. The user wants to know the probability β that the original hypothesis μ,=μ2 is not rejected when the actual "1-21=0.30N.↓ μ1 - μ2 [ = 0. 30
0.33=0.1476
1μ-μ21
From the curve of V= in Figure 1, we can find that 100β=47, so B=0.47 (or 47%). 4.5.2 The user believes that the above 8 value is too high and wants to select a sample of appropriate size so that when μ, a μ2 [=0.30, the β value is reduced to 0.10 (or 10%).
A1μμ2 L
From the dotted straight line 8=0.10 in Figure 3, we can find that n=26. 0.33
5 Comparison of two means (the variances are unknown, but there is reason to assume that they are equal or approximately equal) Refer to Table 6 of GB4889-85
5.1 Symbol
Population 1
Sample size
Variance (approximate value)
Degrees of freedom
Standard deviation of the difference between two sample means
5.2 Hypothesis tested
Population 2
V= ni+nz- 2
(α_=2)
GB 4890-85
For a two-sided test, the null hypothesis is μ, =μ2, and the alternative hypothesis is μ,丰μ2. For a one-sided test:
The null hypothesis is μ, and the alternative hypothesis is,>μ2
Or,
The null hypothesis is μz, and the alternative hypothesis is μ20
5.3 Problem 1
5.3.1 Given n, and n2, determine β
For a two-sided test or a one-sided test, when the values of μ and 2 are given, first calculate the value of the parameter as follows: a= lu μ2 /
Then use the following corresponding curve graphs: Figure 1 Operating characteristic curve of two-sided test for mean comparison (Class I risk α=0.05)
Operating characteristic curve of two-sided test for mean comparison Figure 4
(Class I risk α=0.01)
Operating characteristic curve of one-sided test for mean comparison Figure 7
(Class I risk α=0.05):
Figure 10 Operating characteristic curve of one-sided test for mean comparison (Class I risk α=0.01)
β is the degrees of freedom v=n +n22 curve, the horizontal axis is the vertical axis of the point in. 5.3.2 Given n, +n2 (=2n), determine β At this time
[n/μ,μ2]
5.4 Problem 2 Given β, determine the common size n of n, and n2 For a two-sided test or a one-sided test, when the values of, and μ2 are given, first calculate the value of the parameter in as follows: μ-μ2
Then use the following corresponding curves
The curve of the sample size required for a two-sided test comparing two means 3
(Type I risk α=0.05)
The curve of the sample size required for a two-sided test comparing two means (Type I risk α=0. 01)
The curve of sample size required for one-sided test of two mean comparisons (type I risk α=0.05)
Figure 12 The curve of sample size required for one-sided test of two mean comparisons (type 1 risk α=0.01)
n is the ordinate of the point on the curve with a given β value and abscissa . Example 5.5
Same as the example in 4.5, but the user does not know the exact value of the standard deviation of the cotton yarn strength. It is known that the standard deviation of the strength of the two batches of cotton yarn may be the same, that is, 2
5.5.1 The user selects 10 bobbins from two batches of cotton yarn with different processes for observation. In order to know when the actual value of "μ, a μ2! = 0.When the null hypothesis μ=μ2 is 30N, the probability β that the null hypothesis μ=μ2 is not rejected. According to the results of observing the two samples, we get: 212
GB 4890—85
First batch: X = 2.176, Z (X1; - X)2 =1.2563, second batch: X2= 2.520, Z(Xz:- X)2=1.3897. Referring to Table 11 of GB488985, after testing, there is no reason to reject α=α. What is the common variance of the strength of the two batches of cotton yarn? The estimated value of is S2 =
1.2563+1.3897
10+10-2
If the confidence level is 1～α=0.95, the upper confidence limit of α2 is 2.6460
Xo.0s (18)
=0,1470
(refer to Table 10 of GB4889-85) Therefore, it is very unlikely that α exceeds αu=/0.2818=0.53. 100.30
[μ-μz
二1,27
In Figure 1, it can be seen by interpolation that =18, when ^=1.27, the corresponding value of 100β is about 80. Therefore, under the unfavorable situation of α=, the probability that the null hypothesis 4, =42 is not rejected is about 0.805.5.2 The user believes that the above B value is too high and wants to select a sample of appropriate size, so that when 1μ-μz1=0.30, the value of 8 does not exceed 0.20 (or 20%) even under the unfavorable situation of α=0u=0.53. [μi - μ2
From the curve of β=0.20 in Figure 3, n=49 is obtained. Comparison of variance or standard deviation with given value
Refer to Table 9 of GB4889-85
6.1 Symbols
n, sample size,
α\, population variance (α: standard deviation of the population)au
α: given value of variance (αo: given value of standard deviation). 6.2 Hypothesis tested
For a two-sided test, the null hypothesis is? =, and the alternative hypothesis is 2. For a one-sided test:
The null hypothesis is?, the alternative hypothesis is>
Or,
The null hypothesis is 2>, the alternative hypothesis is 2<.
When the value of α is given, first calculate the value of parameter In as follows: Then use the following corresponding curve graphs. 6.8 Problem 1 Given n, determine β
The curve graphs used for the various situations mentioned above are:qo
Figure 13 Operating characteristic curve of a two-sided test comparing variance with a given value (Type 1 risk α=0.05),
Operating characteristic curve of a two-sided test comparing variance with a given value (Type I risk α=0.01)
GB 4890--85
Operating characteristic curve of one-sided test for comparing variance with given value Figure 17
(null hypothesis is 2, class 1 risk α=0.05)Operating characteristic curve of one-sided test for comparing variance with given value Figure 19
(null hypothesis is 2, class risk α=0.01)Figure 21
Operating characteristic curve of one-sided test for comparing variance with given value (null hypothesis g2α, class risk α=0.05)Figure 23
Operating characteristic curve of one-sided test for comparing variance with given value (null hypothesis is 2≥, class 1 risk=0.01)β is on the curve for given n value, and the horizontal axis is the vertical axis of the entry point. 6.4 Question 2 Given B, determine n
. The graphs suitable for the various situations mentioned above are: Figure 14
The curve of the sample size required for a two-sided test comparing the variance with a given value (Type I risk α=0.05)
The curve of the sample size required for a two-sided test comparing the variance with a given value16
(Class risk α=0.01)
The curve of the sample size required for a one-sided test comparing the variance with a given value (the null hypothesis is 2, Type I risk α=0.05), the curve of the sample size required for a one-sided test comparing the variance with a given value20
(The null hypothesis is α2, Class risk α=0.01) The curve of the sample size required for a one-sided test comparing the variance with a given value22
(The null hypothesis is g2; Type 1 risk α=0.05)! Figure 24 The curve of the sample size required for the one-sided test of the variance compared with a given value (the null hypothesis is 2≥, and the type I risk α=0.01) n is the curve of the given β value, and the horizontal axis is the vertical axis of the point of entry. 6.5 Example
The standard deviation of the original cotton yarn strength of a cotton spinning mill was α=0.45 (α=0.2025). After the process was improved, it was claimed that the dispersion of the cotton yarn strength was reduced. The user was ready to purchase this product, but hoped that the risk of mistakenly accepting this product when the product quality was not actually improved would be small. For this reason, it was decided to take α2≥㎡=0.2025 as the null hypothesis and take the type I risk α=0.05 for a one-sided test. 6.5.1 The user selected 12 bobbins from a batch of cotton yarn produced by the new process for observation. To know the probability that the null hypothesis g>0.45 is not rejected when the standard deviation is actually reduced to g=0.30 8. a
From the curve of a=12 in Figure 21, the value of 1008 is about 51, so the value of B is about 0.51 (or 51%). 6.5.2 The user believes that the above β value is too high and wants to select a sample of appropriate size so that the β value is reduced to 0.10 (or 10%) when α is 0.30.
From the curve of b=0.10 in Figure 22, n=29. 7 Comparison of two variances or two standard deviations
Refer to Table 11 of GB 4889-85
This standard only gives the operating characteristic curve of the test for the case where the two samples are of the same size. 7.1 Symbols
g: variance of population 1 (α, standard deviation of population 1)214
GB 4890--85
α: variance of population 2 (α2 is standard deviation of population 2), ni=n: size of sample 1,
n2=n: size of sample 2.
7.2 Assumptions of the test
For a two-sided test, the null hypothesis is =, and the alternative hypothesis is. 92
When the alternative hypothesis is
When the alternative hypothesis is,
For a one-sided test:
°) is determined, or,
. The null hypothesis is, and the alternative hypothesis is, determined by the parameter =q2
b. The null hypothesis is, the alternative hypothesis is, and the parameter = 7.3 Problem 1 Given n, determine β
The curve graphs suitable for the above-mentioned various situations are: Figure 25 Operating characteristic curve of the two-sided test for comparison of two variances (Type I risk α=0.05),
Operating characteristic curve of the two-sided test for comparison of two variances Figure 27
(Type I risk α=0.01)
Operating characteristic curve of the one-sided test for comparison of two variances (Type I risk α=0.05)
Operating characteristic curve of the one-sided test for comparison of two variances Figure 31
(Type I risk α=0.01)
β is on the curve for a given n value, and the horizontal axis is the vertical axis of the entry point. 7.4 Problem 2 Given β, determine n
The graphs used for the above different situations are: Figure 26 Curve of sample size required for two-sided test of two variance comparisons (Type I risk α=0.05),
Curve of sample size required for two-sided test of two variance comparisons Figure 28
(Type I risk α=0.01)
Curve of sample size required for one-sided test of two variance comparisons (Type I risk α=0.05)
Curve of sample size required for one-sided test of two variance comparisons (Type I risk α=0.01)
n is the vertical coordinate of the point on the curve with the horizontal axis being the vertical coordinate of the point of entry for the given β value. 7.5 Example
Decision.
A cotton mill provides two batches of cotton yarn to a user. The manufacturer claims that the dispersion of strength of batch No. 1 is lower, so the price is slightly higher. If this is true, the user is prepared to use batch No. 1. The user hopes that the risk of misjudging α2 when the actual value is α,≥α2 will be small. For this reason, it is decided to take α≥ as the null hypothesis, take the Class I risk α=0.05, and conduct a one-sided test. 7.5.1 The user selects 20 bobbins from each batch for observation. When the actual value is 0.1, the probability of 8.
2 is 100. The probability of not rejecting the null hypothesis is
GB489085
In Figure 29, it can be seen by interpolation that when n=20 and
=1.5, the corresponding value of
is about 48, so β is about 0.48 (or 48%). When
=1.5, the value of
is reduced to 0.10 (or 10%). From Figure 30, B=0.10, we can find n=55. 8
Graph
GB 4890—85
a. Test μ= μo
If o is known, use the curve of =e,. ! If g is unknown. Use the curve of =n－1, b. Test μr-u2
If a, and are known, use the curve of v=, Avh lμHo l.
μ-μ
If o, =α2=a and unknown, use the curve of =n, +n2-2 Iμi- μ2
Figure 1 Operating characteristic curve of two-sided test for mean comparison (Type I risk α=0.05) 100(1 - 8)
Tip: This standard content only shows part of the intercepted content of the complete standard. If you need the complete standard, please go to the top to download the complete standard document for free.