psyc200_0805

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Distribution of Sample Means

The distribution of the means of an infinitenumber of samples of a certain size selectedfrom a population

Before, we talked about distributions of scores, but nowwe are talking about the distribution of all possiblesample means of a specific size from a population

SAMPLING ERROR:

The amount of error, or the difference, between asample statistic and the corresponding populationparameter

If we start with a population, and take a number ofsamples from that population, each of the samplemeans will be different from the population mean invarying degrees

For samples of size ntaken from a population

with mean, !, and a standard deviation, !,

As nincreases, the sampling distribution approaches

the normal distribution

Holds for all population distributions in other words, as long as your sample size is large (30

or above usually) the sampling distribution will benormal even if the population (parent distribution) fromwhich it was sampled is not

CENTRAL LIMIT THEOREM

X

=

"X

=

"

n

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Central Limit Theorem

=100

Population

Sampling Distribution

Though the population

distribution is skewed thesampling distribution will still be

normal as long as you have alarge enough n.

"=20

"

X

=

20

n

X

=100

STANDARD ERROR OF THE MEAN: The

standard deviation of the sample means

Formula:

Like the standard deviation, the standarderror of the mean gives an average distancefrom all of the sample means to the

population mean The tells us how much error, onaverage, you would expect between a samplemean and a population mean

n

x

!

! =

x!


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As n(sample size) gets larger, the distribution ofsample means will approximate a normal distribution

-- when n>30, the shape of the sampling distributionis almost perfectly normal regardless of the shape of

the original distribution

The mean of the sampling distribution of the mean is

represented by

or

The mean of the sample means ( ) will equal thepopulation mean (!)

x

x


X

=

X

LAW OF LARGE NUMBERS:

The larger the sample, the more accurately thesample represents the population

Therefore, the larger the sample size, thesmaller the standard error


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From the previous class example.

POPULATION

20

50

30

40

p(20) =1

4

p(30) =1

4

p(40) =1

4

p(50) =1

4

Sampling Distribution of the Mean

0

0.05

0.1

0.15

0.2

0.25

0.3

20 30 40 50

Sample Means

p(

)

Rectangular

Distribution

LAW OF LARGE NUMBERS:

For samples of size 2.Sampling Distribution of the Mean

0

0.05

0.1

0.15

0.2

0.25

0.3

20 25 30 35 40 45 50

Sample Means

p(

)

Sampling Distribution of the Mean

0

2

4

6

8

10

12

14

20 20.3 2 6.7 30 33.3 3 6.7 40 43.3 4 6.7 50

Sample Means

p(

)

For samples of size 3.

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EXAMPLE:

What is the standard error for a sample ofn=25, !=15?

The average distance of a sample mean fromthe population mean is 3 points

35

15

25

15====

n

x

!

!


As n increases, the distribution approachesthe normal

Knowing or assuming the population meanand standard deviation, we can use the tableof the standard normal distribution todetermine the probability of obtainingsample means within any interval or beyondany point

Thus, we can test simple hypotheses about

sample means (called the z-test) Calculate z-score for sample mean, zobt Compare to critical z-score, zcrit

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The primary use of the distribution of sample

means is to find the probability associatedwith getting a specific sample value

Because the distribution of sample means is

normal, we can use the z-score table to findprobabilities associated with given samples

This is called a z-test

Formula:

wherex

x

z

!

"=

n

x

!

! =

Distribution of Sample Means / z-tests

METHOD 1:

Step 1: Convert sample mean to zobtStep 2: Compare to zcrit,

which is either z "for a 1-tailed test, orz"/2for a 2-tailed test

Step 3: Decide whether or not to reject H0METHOD 2:

Step 1: Convert sample mean to z

Step 2: Determine the probability from the z-table

Step 3: Compare to "

Step 4: Decide whether or not to reject H0


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EXAMPLE: A population of heights has a !=68and !=4. What is the probability of selecting

a sample of size n=25 that has a mean of 70or greater?


Summary of Power:

Power is the sensitivity of an experiment todetect a real effect if there is one.

Power is the probability the experiment will

result in rejecting Ho if the IV has a realeffect.

Power + #= 1

Power increases with N. Power increases with ".

Power increases with effect size.

Power & z-tests

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Calculating Power

1. Choose 1-tail or 2-tail test

2. Set alpha value

3. Select statistical test

4. Determine critical values to reject H05. Calculate the probability of obtaining

those values under a specified H1 That is the power of the test under that

version of H1

Power & z-test : N

1. Choose 1-tail or 2-tail test1-tail

2. Set alpha value

0.05

3. Select statistical testz-test

From previous example:A population of heights has a !=68 and !=4. What is the probabilityof selecting a sample of size n=25 that has a mean of 70 or greater?

Calculate the power of this experiment.

What if we had a sample or size n=100?

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Power & z-test : N

4. Determine critical values to rejectH0"=0.05!zcrit=1.645

zcrit

=

Xcrit

"null

#X

Xcrit

= null

+"X

(zcrit

)!

80.5

4

25

4====

n

x

!

!

"X

=

4

100= 0.4

Xcrit = 68+ 0.40(1.645)

Xcrit

= 68.66

Xcrit = 68+ 0.80(1.645)

Xcrit

= 69.32

n=25

n=25

n=100

n=100

Power & z-test : N

5. Calculate the probability of obtainingthose values under a specified H1 That is the power of the test under that

version of H1

z =X

crit"

reall

#X

z =69.32" 70

0.80= "0.855

p= 1-0.1949 = 0.8023

Power = 0.8023

- - 2 4 6 8

z= -0.855

-4 -2 0 2 4 6 8

z = 68.66" 70

0.40= "3.355

p= 1-0.0005 = 0.9995

Power = 0.9995

n=25

n=25

n=100

n=100

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Power & z-test :

1. Choose 1-tail or 2-tail test1-tail

2. Set alpha value

0.05 & 0.013. Select statistical test

z-test

From previous example:A population of heights has a !=68 and !=4. What is the probabilityof selecting a sample of size n=25 that has a mean of 70 or greater?

Calculate the power of this experiment for "=0.05 and "=0.01

Power & z-test :

4. Determine critical values to rejectH0"=0.05!zcrit=1.645

"=0.01!zcrit=2.335

zcrit

=

Xcrit

"null

#X

Xcrit

= null

+"X

(zcrit

)!

Xcrit

= 68+ 0.80(1.645)

Xcrit

= 69.32

"=0.05

Xcrit

= 68+ 0.80(2.335)

Xcrit

= 69.87"=0.01

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Power & z-test :

5. Calculate the probability of obtainingthose values under a specified H1 That is the power of the test under that

version of H1

z =X

crit"

reall

#X

z =69.32" 70

0.80= "0.855

p= 1-0.1949 = 0.8023

Power = 0.8023

- - 2 4 6 8

z= -0.855

-4 -2 0 2 4 6 8

p= 1-0.4325 = 0.5675

Power = 0.5675

"=0.05

"=0.05

"=0.01

z =69.87 " 70

0.80= "0.165

"=0.01

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