Page 446 - DMTH404_STATISTICS

Statistics

Notes The critical value of t at 5% level of significance and 17 d.f. is 2.11. Since this value is greater than
the calculated, there is no evidence against H . Thus, we conclude that the two samples may be
0
regarded to have drawn from a population with same means and same standard deviations.
32.4 Test of Hypothesis Concerning Equality of Standard Deviations
(Large Samples)

It can be shown that when sample sizes are large, i.e., n , n > 30, the sampling distribution of the
1 2
2 2
s 1 s 2
statistic S - S is approximately normal with mean s - s and standard error + .
1 2 1 2
2n 1 2n 2
S - S ) (s- - s )
( 1 2 1 2
Therefore z = ~N (0,1 )
2 2
s s
1 + 2
2n 2n
1 2
S - S 2
1
or z = under H : s = s = s.
0
1
2
1 1
s +
2n 1 2n 2
Very often s is not known and is estimated on the basis of sample. The pooled estimate of s is
2
n S + n S 2
S = 1 1 2 2 . Thus, the test statistic becomes
n + n 2
1
S - S 2 S - S 2 2n n
1 2
1
1
z cal = = ´ .
1 1 S n + n 2
1
S +
2n 2n
1 2
Example 7: The standard deviation of a random sample of the heights of 500 individuals
from country A was found to be 2.58 inches and that of 600 individuals from country B was found
to be 2.35 inches. Do the data indicate that the standard deviation of heights in country A is
greater than that in country B?
Solution.
We have to test H : s = s against H : s > s .
0 1 2 a 1 2
It is given that S = 2.58, n = 500, S = 2.35 and n = 600.
1 1 2 2
2
´
500 2.58 + 600 2.35 2
´
The pooled estimate of s is S = = 2.46
1100
2.58 2.35 600000
-
The test statistic is z = ´ = 2.17
cal
2.46 1100
Since this value is greater than 1.645, H is rejected at 5% level of significance. Thus, the sample
0
evidence indicates that the standard deviation of heights in country A is greater.

438 LOVELY PROFESSIONAL UNIVERSITY