sampling distributions - mrs. hamilton ap...

Sampling

Distributions

Chapter 9

First, a word from our textbook

A statistic is a numerical value computed from

a sample. EX. Mean, median, mode, etc.

A parameter is a numerical value determined

by the entire population and is assumed that

the value is fixed,unchanging and unknown.

Introduction to Statistics and

Sampling Variability

Consider a small population consisting of the

board of directors of a day care center.

Board member and number of children: Jay Carol Allison Teresa Anselmo Bob Roxy Vishal

5 2 1 0 2 2 1 3

Find the average number of children for the entire

group of eight:

m = 2 children

Discovery question ONE:

How is the parameter of the population

related to a sampling distribution based on

the population?

Introduction to Statistics and

Sampling Variability

Board member and number of children: Jay Carol Allison Teresa Anselmo Bob Roxy Vishal

5 2 1 0 2 2 1 3

List all possible samples of size 2. Calculate the

average number of children represented by the

group.

Samples:

3.5x

Jay Carol

5 2

Jay Allison

5 1 3x

Answer question ONE:

the average of all possible values for a

sampling distribution will equal the population

parameter

xm m

Variability of a statistic

What is the relationship between the population parameter and each sample statistic?

The observed value of a statistic will vary from sample to sample. This fact is called sampling variability.

Sampling distributions

If we calculated using only the first 3

columns of values, would we get the same

results? Explain.

How did the spread change from the

population to the sampling distribution?

Definition

In summary, a sampling distribution is the

distribution of all possible values for a given

sample size for a fixed population.

Sampling distribution applet

Discovery question TWO:

For a normal population, how will the shape

and spread of a sampling distribution change

as we increase the sample size?

Population distribution: m = 16 s = 5

Discovery question TWO:

16X

m 16X

m

16X

m 16X

m

3.535X

s 1.581X

s

1.25X

s 1X

s

Answer question TWO:

For a normal population, the shape of the

sampling distribution remains mound shaped

and symmetrical (taller/thinner)for all sample

sizes. We can conclude the sampling

distribution remains approximately normal.

The standard deviation for the sampling

distribution is equal to the population

standard deviation divided by the square root

of the sample size.

Xn

ss

Sample means

parameter statistic

mean m x

standard deviation s s

Formulas:

xm m Xn

ss

sampling

distribution

xm

Xs

Example ONE

The average sales price of a single-family

house in the United States is $243,756.

Assume that the sales prices are normally

distributed with a standard deviation of

$44,000.

Draw the normal distribution. Within what

range would the middle 68% of the houses

fall?

$243,756

$243,756m

$44,000s

$287,756 $199,756

Draw the sampling distribution for a sample

size of 4 houses. Within what range would the

middle 68% of the samples of size 4 houses

fall?

$243,756

$243,756xm

$44,000xs

$265,756 $221,756

$44,000

4xs

$22,000xs

Draw the sampling distribution for a sample

size of 16 houses. Within what range would

the middle 68% of the samples of size 16

houses fall?

$243,756

$243,756xm

$254,756 $232,756

$44,000

16xs

$11,000xs

Draw the sampling distribution for a sample

size of 25 houses. Within what range would

the middle 68% of the samples of size 25

houses fall?

$243,756

$243,756xm

$252,556 $234,956

$44,000

25xs

$8,800xs

Example TWO

Suppose the mean room and board

expense per year at a certain four-year

college is $7,850. You randomly select 9

dorms offering room and board near the

college. Assume that the room and board

expenses are normally distributed with a

standard deviation of $1125.

Draw the population distribution.

$7,850

$7,850m

1125s

$8,975 $6,725 $10,100 $11,225 $5,600 $4,475

$7,850m

$1125s

$8,180

( 8180)P x

What is the probability that a randomly

dorm has room and board of less than

$8,180?

$7,850 $8,975 $6,725 $10,100 $11,225 $5,600 $4,475

What is the probability that a randomly dorm

has room and board of less than $8,180?

$7,850m $1125s

( 8180)P x

Given normal distribution

xz

m

s

8180 7850

1125

0.29

.6141

Draw the sampling distribution for a sample

size of 9 dorms.

$7,850

$7,850xm

1125$375

9xs

$8,225 $7,475 $8,600 $8,975 $7,100 $6,725

What is the probability that the mean room

and board of the nine dorms is less than

$8,180?

$7,850

$7,850xm

1125$375

9xs

$8,225 $7,475 $8,600 $8,975 $7,100 $6,725

$8,180

( 8180)P x

What is the probability that the mean room

and board of the nine dorms is less than

$8,180?

$7,850xm 1125

$3759

xs

( 8180)P x

Given normal distribution

xz

n

m

s

8180 7850

375

0.88

.8106

What is the probability that the mean cost of a

sample of four dorms is more than $7,250?

$7,850xm 1125

$562.504

xs

( 7250)P x

Given normal distribution

xz

n

m

s

7250 7850

562.5

1.067

1 .1423 .8577

Central Limit Theorem

Take a random sample of size n from any

population with mean m and standard

deviation s. When n is large, the sampling

distribution of the sample mean is close to

the normal distribution.

How large a sample size is needed depends

on the shape of the population distribution.

Uniform distribution

Sample size 1

Uniform distribution

Sample size 2

Uniform distribution

Sample size 3

Uniform distribution

Sample size 4

Uniform distribution

Sample size 8

Uniform distribution

Sample size 16

Uniform distribution

Sample size 32

Triangle distribution

Sample size 1

Triangle distribution

Sample size 2

Triangle distribution

Sample size 3

Triangle distribution

Sample size 4

Triangle distribution

Sample size 8

Triangle distribution

Sample size 16

Triangle distribution

Sample size 32

Inverse distribution

Sample size 1

Inverse distribution

Sample size 2

Inverse distribution

Sample size 3

Inverse distribution

Sample size 4

Inverse distribution

Sample size 8

Inverse distribution

Sample size 16

Inverse distribution

Sample size 32

Parabolic distribution

Sample size 1

Parabolic distribution

Sample size 2

Parabolic distribution

Sample size 3

Parabolic distribution

Sample size 4

Parabolic distribution

Sample size 8

Parabolic distribution

Sample size 16

Parabolic distribution

Sample size 32

Loose ends

An unbiased statistic falls sometimes above

and sometimes below the actual mean, it

shows no tendency to over or underestimate.

As long as the population is much larger than

the sample (rule of thumb, 10 times larger),

the spread of the sampling distribution is

approximately the same for any size

population.

Loose ends

As the sampling standard deviation continually decreases, what conclusion can we make regarding each individual sample mean with respect to the population mean m?

As the sample size increases, the mean of the observed sample gets closer and closer to m. (law of large numbers)