6.3 Sampling Distributions

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

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

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

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

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

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

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

6.3.2 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.

Notes: 1. represent the t-value above which we find an area equal to leaves an area of to the right. 2. 3.

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

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

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

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

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

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.

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

Theorem 6.12-P118 Theorem 6.14-P119

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

random samples, respectively, from Theorem 6.16-P120 Let and be independent random samples, respectively, from and the distributions, Then .

Example Let be a sample from Find the distribution of .

Homework: P123: 5,9,11