E1C04 09/14/2010
14:7:43 Page 136
For the normal distribution, the x
2 statistic (1, 3, 4) in Figure 4.7 is
x
2
¼ ns
2
x =s
2
ð4:26Þ
with degrees of freedom n ¼ N À 1.
Precision Interval in a Sample Variance
A precision interval for the sample variance can be formulated by the probability statement
P x
2
1Àa=2
x
2
x
2
a=2
¼ 1 À a
ð4:27Þ
with a probability of P(x
2 ) ¼ 1 À a. The term a is called the level of significance. Combining
Equations 4.26 and 4.27 gives
P ns
2
x =x
2
a=2
s
2
ns
2
x =x
2
1Àa=2
¼ 1 À a
ð4:28Þ
For example, the 95% precision interval by which s
2
x estimates s
2 , is given by
ns
2
x =x
2
0:025
s
2
ns
2
x =x
2
0:975
ð95%Þ
ð 4:29Þ
Note that this interval is bounded by the 2.5% and 97.5% levels of significance (for 95% coverage).
The x
2 distribution estimates the discrepancy expected due to random chance. Values for x
2
a are
tabulated in Table 4.5 as a function of the degrees of freedom. The P(x
2 ) value equals the area under
Table 4.5 Values for x
2
a
n
x
2
0:99
x
2
0:975
x
2
0:95
x
2
0:90
x
2
0:50
x
2
0:05
x
2
0:025
x
2
0:01
1
0.000
0.000
0.000
0.016
0.455
3.84
5.02
6.63
2
0.020
0.051
0.103
0.211
1.39
5.99
7.38
9.21
3
0.115
0.216
0.352
0.584
2.37
7.81
9.35
11.3
4
0.297
0.484
0.711
1.06
3.36
9.49
11.1
13.3
5
0.554
0.831
1.15
1.61
4.35
11.1
12.8
15.1
6
0.872
1.24
1.64
2.20
5.35
12.6
14.4
16.8
7
1.24
1.69
2.17
2.83
6.35
14.1
16.0
18.5
8
1.65
2.18
2.73
3.49
7.34
15.5
17.5
20.1
9
2.09
2.70
3.33
4.17
8.34
16.9
19.0
21.7
10
2.56
3.25
3.94
4.78
9.34
18.3
20.5
23.2
11
3.05
3.82
4.57
5.58
10.3
19.7
21.9
24.7
12
3.57
4.40
5.23
6.30
11.3
21.0
23.3
26.2
13
4.11
5.01
5.89
7.04
12.3
22.4
24.7
27.7
14
4.66
5.63
6.57
7.79
13.3
23.7
26.1
29.1
15
5.23
6.26
7.26
8.55
14.3
25.0
27.5
30.6
16
5.81
6.91
7.96
9.31
15.3
26.3
28.8
32.0
17
6.41
7.56
8.67
10.1
16.3
27.6
30.2
33.4
18
7.01
8.23
9.39
10.9
17.3
28.9
31.5
34.8
19
7.63
8.91
10.1
11.7
18.3
30.1
32.9
36.2
20
8.26
9.59
10.9
12.4
19.3
31.4
34.2
37.6
30
15.0
16.8
18.5
20.6
29.3
43.8
47.0
50.9
60
37.5
40.5
43.2
46.5
59.3
79.1
83.3
88.4
136 Chapter 4 Probability and Statistics
14:7:43 Page 136
For the normal distribution, the x
2 statistic (1, 3, 4) in Figure 4.7 is
x
2
¼ ns
2
x =s
2
ð4:26Þ
with degrees of freedom n ¼ N À 1.
Precision Interval in a Sample Variance
A precision interval for the sample variance can be formulated by the probability statement
P x
2
1Àa=2
x
2
x
2
a=2
¼ 1 À a
ð4:27Þ
with a probability of P(x
2 ) ¼ 1 À a. The term a is called the level of significance. Combining
Equations 4.26 and 4.27 gives
P ns
2
x =x
2
a=2
s
2
ns
2
x =x
2
1Àa=2
¼ 1 À a
ð4:28Þ
For example, the 95% precision interval by which s
2
x estimates s
2 , is given by
ns
2
x =x
2
0:025
s
2
ns
2
x =x
2
0:975
ð95%Þ
ð 4:29Þ
Note that this interval is bounded by the 2.5% and 97.5% levels of significance (for 95% coverage).
The x
2 distribution estimates the discrepancy expected due to random chance. Values for x
2
a are
tabulated in Table 4.5 as a function of the degrees of freedom. The P(x
2 ) value equals the area under
Table 4.5 Values for x
2
a
n
x
2
0:99
x
2
0:975
x
2
0:95
x
2
0:90
x
2
0:50
x
2
0:05
x
2
0:025
x
2
0:01
1
0.000
0.000
0.000
0.016
0.455
3.84
5.02
6.63
2
0.020
0.051
0.103
0.211
1.39
5.99
7.38
9.21
3
0.115
0.216
0.352
0.584
2.37
7.81
9.35
11.3
4
0.297
0.484
0.711
1.06
3.36
9.49
11.1
13.3
5
0.554
0.831
1.15
1.61
4.35
11.1
12.8
15.1
6
0.872
1.24
1.64
2.20
5.35
12.6
14.4
16.8
7
1.24
1.69
2.17
2.83
6.35
14.1
16.0
18.5
8
1.65
2.18
2.73
3.49
7.34
15.5
17.5
20.1
9
2.09
2.70
3.33
4.17
8.34
16.9
19.0
21.7
10
2.56
3.25
3.94
4.78
9.34
18.3
20.5
23.2
11
3.05
3.82
4.57
5.58
10.3
19.7
21.9
24.7
12
3.57
4.40
5.23
6.30
11.3
21.0
23.3
26.2
13
4.11
5.01
5.89
7.04
12.3
22.4
24.7
27.7
14
4.66
5.63
6.57
7.79
13.3
23.7
26.1
29.1
15
5.23
6.26
7.26
8.55
14.3
25.0
27.5
30.6
16
5.81
6.91
7.96
9.31
15.3
26.3
28.8
32.0
17
6.41
7.56
8.67
10.1
16.3
27.6
30.2
33.4
18
7.01
8.23
9.39
10.9
17.3
28.9
31.5
34.8
19
7.63
8.91
10.1
11.7
18.3
30.1
32.9
36.2
20
8.26
9.59
10.9
12.4
19.3
31.4
34.2
37.6
30
15.0
16.8
18.5
20.6
29.3
43.8
47.0
50.9
60
37.5
40.5
43.2
46.5
59.3
79.1
83.3
88.4
136 Chapter 4 Probability and Statistics