1. 6.3 Sampling Distributions

2. Sampling Distributions
Definition 6.9-P107
The probability distribution of a statistic is called a
sampling distribution.

3. 6.3.1 Chi-squared Distribution
Theorem
Let ,
the distribution of the
random variable ,
where
is given
by the density function
This is known as the chi-squared distribution with
degrees of freedom.
Denoted by

5. Properties: 1. If are independent ，and Xi~
then
2. If X and Y are independent and
respectively, then

6. 3. If X~ , then
E(X)=n, D(X)=2n.

7. Definition-α Given , the upper -point
of a PDF F is determined from the equation .

8. Notes:
represent the t-value above
which we find an area equal to
leaves an area of to the right.

9. Example 1
Assume
be a sample from
find
Example 6.3-P109 .

10. t-Distribution
Theorem 6.6
Let X ~N(0,1)，Y~
If X and Y are independent,
then the distribution of the random variable T,
where
is given by the density function
This is known as the t-distribution with
degrees of
denoted by
freedom.

11. Notes:
represent the t-value above which we find an
area equal to
leaves an area of to the right. 2. 3.

12. 4. 5. 6. 7.
Example 6.6-P112

13. 6.3.3 F-Distribution
Theorem 6.7
Then the distribution of the
random variable is given by the density
This is known as the F-distribution with
and
degrees of freedom.
Denoted by

14. Note: 1.
If n2>2
represent the t-value above which we find an
area equal to
leaves an area of to the right.

15. 6.3.5
Theorem 6.9-P117
be a sample from
Let
then
(1)
(2)

16. Let with sample mean and sample variance then Theorem 6.10-P118
be a sample from
with
sample mean
and sample variance
then

17. Theorem 6.11 Let with sample mean (1) (2) (3) then be a sample from
and sample variance
then
(1)
(2)
(3)
and
are independent.

18. Assume a sample of size n=21 from
Example
Assume a sample of size n=21 from
is given, find .

19. Theorem P118 Theorem P119

20. random samples, respectively, from
Theorem 6.13
Let
and
be independent
random samples, respectively, from
and
distributions,
Then . .

21. random samples, respectively, from
Theorem P120
Let
and
be independent
random samples, respectively, from
and the
distributions,
Then .

22. Example
Let
be a sample from
Find the distribution of .

23. Homework:
P123: 5,9,11