1

Chapter Seven

Introduction to Sampling Distributions

Section 1

Sampling Distribution

2

Key Points

• Review such commonly used terms as random sample, relative frequency, parameter, statistic, and sampling distributions

• From raw data, construct a relative frequency distribution for x-bar values and compare the result to a theoretical sampling sitribution

3

Review of Statistical Terms

• Population

• Sample

• Parameter

• Statistic

4

Population

the set of all measurements (either existing or conceptual) under

consideration

5

Sample

a subset of measurements from a population

6

Statistic

a numerical descriptive measure of a sample

7

Parameter

a numerical descriptive measure of a population

8

We use a statistic to make inferences about a population

parameter.

9

Principal types of inferences

• Estimate the value of a population parameter

• Formulate a decision about the value of a population parameter

• Make a prediction about the value of a statistical variable.

10

Sampling Distribution

a probability distribution for the sample statistic we are using

11

Example of a Sampling Distribution

Select samples with two elements each (in sequence with

replacement) from the set

{1, 2, 3, 4, 5, 6}.

12

Constructing a Sampling Distribution of the Mean for

Samples of Size n = 2List all samples and compute the mean of each sample.

sample: mean: sample: mean

{1,1} 1.0 {1,6} 3.5{1,2} 1.5 {2,1} 1.5{1,3} 2.0 {2,2} 4{1,4} 2.5 … ...{1,5} 3.0

There are 36 different samples.

13

Sampling Distribution of the Mean

p

1.0 1/361.5 2/362.0 3/362.5 4/363.0 5/363.5 6/364.0 5/364.5 4/36 5.0 3/365.5 2/36 6.0 1/36

x

14

Sampling Distribution Histogram

| | | | | | | | | | | |

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

36

6

36

1

15

Pinedale Related Review Problems

Pinedale, Wisconsin, is a rural community with a children’s fishing pond. All fish caught under 6 inches in length must be released. 100 children are selected and the lengths of 5 fish they caught are recorded along with the mean length of the 5 fish for each child.

16

17

What type of measurement is the mean of the average recorded fish lengths for the 100 children?

Pinedale Related Review Problems

-The mean is a statistic because it is from a sample of fish lengths

18

How can the relative frequencies of the individual fish length means be used to help us estimate probabilities occurring in the sampling distributions of fish lengths?

Pinedale Related Review Problems

-Relative frequencies are measures or estimates of the likelihood of a certain statistic falling within the class bounds.

19