88 2. FUNDAMENTAL RELIABILITY MATHEMATICS
900
800
700
600
500
400
300
200
100
0
1400
1200
1000
800
600
400
200
0
3
2.5
2
1.5
1
0.5
0
350
300
250
200
150
100
50
0
×10
4
4
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
4.5
0
0 500 1000 1500 2000 2500
0.5 1 1.5 2 32.5
5 5.5 6 6.5 7 87.5
(a) (b)
(c) (d)
Figure 2.27: Several histograms.
2.13.2 THE CHI-SQUARE TEST
A Chi-square (
2
) test can equally apply to any assumed distribution for a goodness-of-fit test. It
is a commonly used and versatile test to check whether a set of data fits an assumed distribution.
e test statistic W for the Chi-square (
2
) test is:
W D
2
D
J
X
j D1
o
j
e
j
2
e
j
; (2.129)
where
• W —e test statistic which represents the difference between the sample data and the
assumed distribution.
• W approximately follows a Chi-square (
2
) distribution and will be described by the Chi-
square (
2
) distribution with k D J m 1 degrees of freedom
k D J m 1: (2.130)
2.13. GOODNESS-OF-FIT TEST:
2
TEST 89
• k—e number of degrees of freedom of the Chi-square (
2
) distribution for the test
statistic W .
• J —e sample data X must be divided into J (J 6 ) non-overlapping intervals. e
lower limit for the first interval can be
1
, and the upper limit of the last interval can be
C1.
• m—e number of distribution parameters of the assumed distribution. For example, m
is equal to 2 for a uniform distribution, a normal distribution, a lognormal distribution, a
Weibull distribution, and 1 for an exponential distribution.
• o
j
—e observed frequency of the sample data in the j th non-overlapping interval
x
j L
; x
j U
. o
j
can be obtained through the histogram.
• e
j
—e expected frequency in the j th non-overlapping interval
x
j L
; x
j U
with the
assumed distribution. e
j
will be calculated from the assumed distribution by this formula:
e
j
D N P
x
j L
X x
j U
D N
F
x
j U
F
x
j L
:
(2.131)
• F
.
x
/
—e CDF of the assumed distribution whose distribution parameters are deter-
mined by the sample data.
• N —e total number of the sample data.
e hypothesis for the Chi-Square (
2
) goodness-of-fit test is:
H
0
: e random variable of sample data follows an assumed distribution whose distribution
parameters are determined by the sample data.
With a confidence level .1 ˛/, the critical value W
critical
can be determined from the
Chi-Square (
2
) distribution:
W
critical
D
2
1a
.k/; (2.132)
2
1a
.k/ can be determined by the Chi-Square (
2
) table, as shown in Table 2.11. In Microsoft
Excel,
2
1
a
.k/ can be calculated by the following function:
2
1a
.k/ D CHISQ:INV.1 a; k/: (2.133)
In MATLAB, the
2
1
a
.k/ can be calculated by the following command:
2
1a
.k/ D ch2inv.1 a; k/: (2.134)
Two possible results for the Chi-Square (
2
) goodness-of-fit test are as follows.
• If W W
critical
D
2
1a
.k/, the hypothesis H
0
should be rejected with a confidence level
.1 ˛/.
90 2. FUNDAMENTAL RELIABILITY MATHEMATICS
Table 2.11: e Chi-Square (
2
) table
2.13. GOODNESS-OF-FIT TEST:
2
TEST 91
• If W < W
critical
D
2
1a
.k/, the hypothesis H
0
should not be rejected with a confidence
level .1 ˛/.
˛ is a significant level, and (1 ˛) is a confidence level. For the goodness-of-fit test, ˛
should be at least 0.10 and typically is 0.05. e physical meaning of the confidence level (1 ˛)
is a probability of a judgment or statement. For example, when the hypothesis H
0
is rejected
with a confidence level
.
1 0:05
/
D 0:95, it means that this statement is true with a probability
of 95%.
e procedure for the Chi-Square (
2
) goodness-of-fit test can be summarized as follows.
• Step 1: Build the histogram of the sample data X . Based on the histogram or experience
about the random variable, it can be hypothesized as a specific type of distribution, that
is, the hypothesis H
0
.
• Step 2: Calculate the mean
x
and standard deviation
x
of the sample data. en use
them to determine the distribution parameters of the hypothesized distribution.
• Step 3: Divide the sample data into J (J 6 ) non-overlapping intervals.
• Step 4: o
j
is obtained from the histogram. Use Equation (2.131) to calculate e
j
in each
non-overlapping interval.
• Step 5: Use Equation (2.129) to calculate the test statistic
W
.
• Step 6: Select a significant level a, typically 0.05.
• Step 7: Determine the critical value of W
critical
D
2
1a
.k/ by Equations (2.133) or (2.134)
or the Chi-square Table 2.11.
• Step 8: If W < W
critical
D
2
1a
.k/, the hypothesis H
0
should not be rejected with a con-
fidence level .1 ˛/. Otherwise, the hypothesis H
0
should be rejected.
Example 2.46
One hundred sample data of a material’s yield strength is listed in Table 2.12. Make proper
assumption of the type of distribution for this set of data and then use the Chi-square (
2
)
goodness-of-fit test to check whether it follows a normal distribution.
Solution:
• Step 1: Histogram of the sample data X.
e histogram of this set of 100 sample data is shown in Figure 2.28. Per this histogram,
it might be a normal distribution. So, we can make this assumption:
H
0
: e material’s yield strength follows a normal distribution.
92 2. FUNDAMENTAL RELIABILITY MATHEMATICS
Table 2.12: Sample data of material’s yield strength
Yield Strength Data (ksi)
63.0, 56.4, 57.7, 56.4, 62.5, 55.9, 54.5, 62.4, 57.6, 61.5, 58.0, 61.0, 47.1, 62.0, 68.8, 64.8, 64.2, 58.5, 60.6, 60.9,
60.2, 55.6, 58.0, 58.8, 57.3, 63.8, 72.1, 53.2, 60.3, 52.6, 66.1, 66.6, 51.6, 69.2, 54.4, 60.8, 65.1, 60.4, 70.9, 73.2,
60.4, 50.4, 57.9, 55.5, 56.0, 54.2, 60.1, 70.0, 56.2, 58.3, 65.4, 65.6, 62.1, 64.7, 64.0, 57.8, 61.8, 58.2, 55.3, 62.8,
60.0, 58.8, 61.1, 64.5, 61.6, 60.7, 61.1, 60.0, 55.9, 63.6, 66.6, 57.1, 69.1, 58.8, 58.5, 55.3, 55.8, 55.0, 61.4, 67.5,
62.3, 63.9, 66.3, 47.5, 58.9, 61.4, 63.2, 53.3, 54.8, 63.6, 59.1,62.4, 55.0, 51.7, 63.4, 65.5, 67.7, 52.2, 53.1, 52.5
25
20
15
10
5
0
47.1 49.7 52.3 54.9 57.5 60.2 62.8 65.4 68.0 70.6 73.2
Bin
Frequency
Figure 2.28: Histogram of the sample data.
• Step 2: e mean
x
and the standard deviation
x
of the sample data.
Copy the sample data into a Microsoft Excel file. e mean and standard deviation based
on these sample data are:
x
D 60:1 .ksi/;
x
D 5:23 .ksi/:
So, the hypothesized normal distribution will have the distribution parameters: the mean
x
D 60:1 (ksi) and the standard deviation
x
D 5:23 (ksi).
• Step 3: Divide the sample data into ten non-overlapping intervals.
e ten non-overlapping intervals are shown in the first two columns of Table 2.13.
• Step 4: o
j
and e
j
in each non-overlapping interval.
e o
j
can be directly obtained from the histogram. e
j
can be calculated per Equa-
tion (2.131). For example, e
2
in the second non-overlapping interval (49.7, 52.36) is cal-
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