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

x

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.

If we added Audra (5 kids) to the board, would we get the same results? Explain.

If we created a distribution based on a sample size of 4 comment on the mean, spread and shape of 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 16X

16X 16X

3.535X 1.581X

1.25X 1X

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.

X n

Sample means

parameter statistic

mean x

standard deviation s

Formulas:

x X n

samplingdistribution

x

X

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

$243,756

$44,000

$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

$243,756x

$44,000x

$265,756$221,756

$44,000

4x

$22,000x

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

$243,756x

$254,756$232,756

$44,000

16x

$11,000x

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

$243,756x

$252,556$234,956

$44,000

25x

$8,800x

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

$7,850

1125

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

$7,850

$1125

$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?

$7,850 $1125

( 8180)P x

Given normal distribution

xz

8180 7850

1125

0.29

.6141

Draw the sampling distribution for a sample

size of 9 dorms.

$7,850

$7,850x 1125

$3759

x

$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

$7,850x 1125

$3759

x

$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?

$7,850x 1125

$3759

x

( 8180)P x

Given normal distribution

xz

n

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?

$7,850x 1125

$562.504

x

( 7250)P x

Given normal distribution

xz

n

7250 7850

562.5

1.067

1 .1423.8577