Page 447 - DMTH404_STATISTICS

Unit 32: Hypothesis Concerning Standard Deviation

32.5 Summary Notes

2
1 2 n 1 æ 2 å X ö
2 1 2 1i
s =
 We can write 1 å (X - X 1 ) = S = çå X - ÷ and
1i
1
1i
n - 1 n - 1 n - 1 n
1 1 1 è 1 ø
2
1 2 n 2 1 æ 2 å X ö
2i
2
s = å (X - X ) = 2 S = çå X -
2 2i 2 2 2i ÷ .
n - 1 n - 1 n - 1 è n 2 ø
2
2
2
s 1 2
 In the variance ratio F = 2 , we take, by convention the largest of the two sample
s
2
2
variance as s . Thus, this test is always a one tailed test with critical region at the right
1
hand tail of the F - curve.
s 2
 The 100(1 - a)% confidence limits for the variance ratio 1 2 , are given by
s 2
s é 2 1 s 2 s 2 1 ù
-
P ê 1 ×  1  1 × ú = 1 a .
2 2 2
s ê ë 2 F a /2 s 2 s 2 F 1 a /2 ú û
-
32.6 Keywords
s 2 /s 2
F - distribution: If H : s = s against s > s . Refer to § 20.6, the statistic F = 1 1 , would
0 1 2 1 2 2 2
s /s
2 2
s 2
become 1 2 under H , follows F - distribution with n (= n - 1) and n (= n - 1) degrees of
2
2
0
1
1
s
2
freedom.
32.7 Self Assessment
Fill in the blanks:
1. These tests can be divided into two broad categories depending upon whether the .................
of the sample is large or small.

s 2 /s 2 s 2
2. If H : s = s against s > s . Refer to § 20.6, the statistic F = 1 1 , would become 1
0 1 2 1 2 2 2 2
s /s s
2 2 2
under H , follows ................. with n (= n - 1) and n (= n - 1) degrees of freedom.
0 1 1 2 2
s 1 2
3. In the ................. F = 2 , we take, by convention the largest of the two sample variance as
s
2
2
s . Thus, this test is always a one tailed test with critical region at the right hand tail of the
1
F - curve.

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