Page 388 - DMTH404_STATISTICS

Statistics

Notes For this calculation, the numerator degrees of freedom v are 12 – 1 or 11; and the denominator
1
degrees of freedom v are 7 – 1 or 6.
2
When you are trying to find the cumulative probability associated with an f statistic, you need
to know v and v . This point is illustrated in the next example.
1 2

Example 2: Find the cumulative probability associated with each of the f statistics from
Example 1, above.
Solution: To solve this problem, we need to find the degrees of freedom for each sample. Then,
we will use the F Distribution Calculator to find the probabilities.
 The degrees of freedom for the sample of women is equal to n – 1 = 7 – 1 = 6.

 The degrees of freedom for the sample of men is equal to n – 1 = 12 – 1 = 11.
Therefore, when the women’s data appear in the numerator, the numerator degrees of freedom
v is equal to 6; and the denominator degrees of freedom v is equal to 11. And, based on the
1 2
computations shown in the previous example, the f statistic is equal to 1.68. We plug these
values into the F Distribution Calculator and find that the cumulative probability is 0.78.
On the other hand, when the men’s data appear in the numerator, the numerator degrees of
freedom v is equal to 11; and the denominator degrees of freedom v is equal to 6. And, based
1 2
on the computations shown in the previous example, the f statistic is equal to 0.595. We plug
these values into the F Distribution Calculator and find that the cumulative probability is 0.22.

Figure 28.3

28.2 Summary

 Let there be two independent random samples of sizes n and n from two normal
1 2
1 2
2
2
2
populations with variances s and s respectively. Further, let s = å (X - X )
1 2 1 1i 1
n - 1
1
1 2
2
and s = å (X - X ) be the variances of the first sample and the second samples
2 2i 2
n - 1
2
respectively. Then F - statistic is defined as the ratio of two c 2 - variates. Thus, we can
write
c n 2 1 1- ( 1 1 s 2 s 1 2
n -
) 1
( 1
n - 1 s 2 / n - ) 1 s 2
F = 1 = 1 = 1
c 2 n 2 1- (n - 1 s 2 ( / n - ) 1 s 2 2
2
) 2
n - 1 s 2 2 s 2 2
2 2
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