Chapter 6 Introduction to Sampling Distributions

[Pages:14]Chapter 6

Student Lecture Notes

6-1

Chapter 6 Introduction to Sampling Distributions

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Chapter Goals

To use information from the sample to make inference about the population

Define the concept of sampling error Determine the mean and standard deviation for the

sampling distribution of the sample mean _

Describe the Central Limit Theorem and its importance

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Fundamentals of Business Statistics ? Murali Shanker

Chapter 6

Student Lecture Notes

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Sampling Error

Sample Statistics are used to estimate Population Parameters

ex: X is an estimate of the population mean,

Problems:

Different samples provide different estimates of the population parameter

Sample results have potential variability, thus sampling error exits

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Statistical Sampling

Parameters are numerical descriptive measures of populations.

Statistics are numerical descriptive measures of samples

Estimators are sample statistics that are used to estimate the unknown population parameter.

Question: How close is our sample statistic to the true, but unknown, population parameter?

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Chapter 6

Student Lecture Notes

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Notations

Parameter

2

p

Statistic

^, X, Mode, Median ^ 2 , s2

p^

,M^Xp ed

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Calculating Sampling Error

Sampling Error:

The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population

Example: (forStahemmpelainng) Error = x -

where:

x = sample mean = population mean

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Fundamentals of Business Statistics ? Murali Shanker

Chapter 6

Student Lecture Notes

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Example

If the population mean is = 98.6 degrees and a sample of n = 5 temperatures yields a

sample mean of x = 99.2 degrees, then the

sampling error is

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Sampling Distribution

A sampling distribution is a distribution of the possible values of a statistic for a given sample size n selected from a population

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Chapter 6

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Sampling Distributions

Objective: To find out how the sample mean X

varies from sample to sample. In other words, we want to find out the sampling distribution of the sample mean.

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Sampling Distribution Example

Assume there is a population ...

Population size N=4 Random variable, X,

is age of individuals

A BC D

Values of X: 18, 20, 22, 24 (years)

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Chapter 6

Student Lecture Notes

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Developing a Sampling Distribution

(continued)

Summary Measures for the Population Distribution:

=

x i

P(x)

N

.3

= 18 + 20 + 22 + 24 = 21 .2

4

.1

2 = (xi - )2 = 5 N

0

18 20

22 24 x

A B CD

Uniform Distribution

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Developing a Sampling Distribution

(continued)

Now consider all possible samples of size

1st n=22nd Observation Obs 18 20 22 24

18 18,18 18,20 18,22 18,24

16 Sample Means

20 20,18 20,20 20,22 20,24 22 22,18 22,20 22,22 22,24 24 24,18 24,20 24,22 24,24

16 possible samples (sampling with replacement)

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1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

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Chapter 6

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Developing a Sampling Distribution (continued)

Sampling Distribution of All Sample

16 SMamepalnesMeans

1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24

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Sample Means Distribution

P(x) .3

.2

.1

0

_

18 19 20 21 22 23 24 x

(no longer uniform) 13

Developing a Sampling Distribution (continued)

Summary Measures of this Sampling Distribution:

=

x i

=

18 + 19 + 21+ L + 24

= 21

x

N

16

2 =

(xi

-

x

)2

x

N

= (18 - 21)2 + (19 - 21)2 +L + (24 - 21)2 = 2.5 16

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Fundamentals of Business Statistics ? Murali Shanker

Chapter 6

Student Lecture Notes

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Expected Values

E(X)=

_

P(x) .3

.2

2 (X ) =

.1

0

_

18 19 20 21 22 23 24 X

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Comparing the Population with its Sampling Distribution

Population: N = 4

= 21

2 = 5

P(x) = 2.236

.3

.2

Sample Means Distribution: n = 2

= 21 x

2 = 2.5

_

P(x)

x

x = 1.58

.3

.2

.1

0

18 20 22

A B C

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.1

_ 24 x

0

18 19 20 21 22 23 24

x

D

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