Chapter 7Chapter 7

Sample VariabilitySample Variability

Those who jump off a bridge in Paris are in Seine.

A backward poet writes inverse.

A man's home is his castle, in a manor of speaking.

SamplingSampling

The NeedThe Need Get information about a population without checking Get information about a population without checking

the entire populationthe entire population

AdvantagesAdvantages CostCost TimeTime Accuracy (can be achieved with low cost)Accuracy (can be achieved with low cost) Destruction is sometimes involved; checking all is not Destruction is sometimes involved; checking all is not

possible.possible.

[Insert Excel Simulation here][Insert Excel Simulation here]

Distribution of MeansDistribution of Means

Visual Mean of MeansVisual Mean of Means

Distribution of Sample MeansDistribution of Sample Means

Many different sample means are possibleMany different sample means are possible The sample means cluster closer to the The sample means cluster closer to the

population mean than the population population mean than the population values do.values do.

The larger the sample, the closer they The larger the sample, the closer they cluster around the population meancluster around the population mean

Therefore the likelihood of a single sample Therefore the likelihood of a single sample mean being close to the true mean is highmean being close to the true mean is high

Distribution of Sample MeansDistribution of Sample Means

When trying to use a sample to estimate a When trying to use a sample to estimate a population mean, we know we won’t get the population mean, we know we won’t get the exact valueexact value

We want some way of managing the error so as We want some way of managing the error so as to be as close as we need to beto be as close as we need to be

We can decide on a margin of error that we are We can decide on a margin of error that we are willing to accept (polls typically 2% - 4%).willing to accept (polls typically 2% - 4%).

We cannot eliminate the possibility of getting a We cannot eliminate the possibility of getting a value outside that range, but we can keep it value outside that range, but we can keep it small by adjusting the sample size.small by adjusting the sample size.

How Close Can We Get?How Close Can We Get?

The variance of the sample mean is the The variance of the sample mean is the population variance divided by n (sample size)population variance divided by n (sample size)

Thus larger n’s bring smaller variancesThus larger n’s bring smaller variances Let’s look at an example. In order to understand Let’s look at an example. In order to understand

the process, we will assume we actually know the process, we will assume we actually know the true mean and variance. Each of the the true mean and variance. Each of the following graphs is from a computer simulation following graphs is from a computer simulation of taking 100 samples from a normal population of taking 100 samples from a normal population with with μμ=15 and =15 and σσ=3, but with different =3, but with different sample sizessample sizes..

xx

μμ==15, 15, σσ==3, Sample Size 1 3, Sample Size 1 Number observed in [14,16]: 30Number observed in [14,16]: 30

252321191715131197

30

20

10

0

s=3

Per

cent

μ μ ==15, 15, σσ==3, Sample Size 4 3, Sample Size 4 Number observed in [14,16]: 52Number observed in [14,16]: 52

252321191715131197

50

40

30

20

10

0

s=1.5

Per

cent

μ μ ==15, 15, σσ==3, Sample Size 9 3, Sample Size 9 Number observed in [14,16]: 74Number observed in [14,16]: 74

252321191715131197

80

70

60

50

40

30

20

10

0

s=1

Per

cent

μ μ ==15, 15, σσ==3, Sample Size 16 3, Sample Size 16 Number observed in [14,16]: 81Number observed in [14,16]: 81

252321191715131197

80

70

60

50

40

30

20

10

0

s=3/4

Per

cent

μ μ ==15, 15, σσ==3, Sample Size 25 3, Sample Size 25 Number observed in [14,16]: 90Number observed in [14,16]: 90

252321191715131197

90

80

70

60

50

40

30

20

10

0

s=3/5

Per

cent

μ μ ==15, 15, σσ==3, Sample Size 36 3, Sample Size 36 Number observed in [14,16]: 97Number observed in [14,16]: 97

252321191715131197

100

90

80

70

60

50

40

30

20

10

0

s=1/2

Per

cent

Number in [14,16] vs Sample SizeNumber in [14,16] vs Sample Size

0

10

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25 30 35 40

So What?So What? In Real Life, we don’t know the true mean and In Real Life, we don’t know the true mean and

variance. We want to estimate them.variance. We want to estimate them. Furthermore, we will only take one sample, Furthermore, we will only take one sample,

which represents just one data point from the which represents just one data point from the distributions we have illustrated.distributions we have illustrated.

We will probably NEVER know where in the We will probably NEVER know where in the distribution that data point is coming from.distribution that data point is coming from.

Under these conditions, how can we provide an Under these conditions, how can we provide an estimate that is trustworthy? estimate that is trustworthy?

Clearly, the sample size directly affects the Clearly, the sample size directly affects the likelihood that the sample mean will be close to likelihood that the sample mean will be close to the true mean.the true mean.

Which one would you like to pick Which one would you like to pick from?from?

7 9 11 13 15 17 19 21 23 25

0

10

20

30

s=3

Pe

rce

nt

7 9 11 13 15 17 19 21 23 25

0

10

20

30

40

50

60

70

80

90

100

s=1/2

Pe

rce

nt

The situation: You have 100 balls in an urn (left). Each has an odd The situation: You have 100 balls in an urn (left). Each has an odd number on it, which may be from 7-25, but you don’t know how number on it, which may be from 7-25, but you don’t know how many of each there are. You will draw one ball and record its many of each there are. You will draw one ball and record its number. If this number matches the mean of the distribution, your number. If this number matches the mean of the distribution, your company will make lots of money and you will get a promotion. company will make lots of money and you will get a promotion. However, you have the opportunity, for a sizable fee, to trade in the However, you have the opportunity, for a sizable fee, to trade in the urn for the one on the right. If you do so, and are wrong, you will be urn for the one on the right. If you do so, and are wrong, you will be fired because of the excessive expense you incurred.fired because of the excessive expense you incurred.

Does the name Pavlov ring a bell?

Reading while sunbathing makes you well red.

When two egotists meet, it's an I for an I.

NotesNotes::

1.1. : the sample mean. : the sample mean.

2.2. : the standard deviation of the sample means. : the standard deviation of the sample means.

3.3. The theory involved with sampling distributions The theory involved with sampling distributions described in the remainder of this chapter requires described in the remainder of this chapter requires random samplingrandom sampling..

Random SampleRandom Sample: A sample obtained in such a way : A sample obtained in such a way that each possible sample of a fixed size that each possible sample of a fixed size nn has an equal has an equal probability of being selected.probability of being selected.

(Example: (Example: Every possible handfulEvery possible handful of size of size nn has the has the same probability of being selected.)same probability of being selected.)

x

xs

The Central Limit TheoremThe Central Limit Theorem

The most important idea in all of statistics.The most important idea in all of statistics. Describes the sampling distribution of the Describes the sampling distribution of the

sample mean.sample mean. Examples suggest: the sample mean (and Examples suggest: the sample mean (and

sample total) tend to be normally sample total) tend to be normally distributed.distributed.

Distribution of Sample MeansDistribution of Sample MeansIf all possible random samples of a particular size If all possible random samples of a particular size nn are are taken from any population with a mean taken from any population with a mean and a and a standard deviation standard deviation , the distribution of sample means , the distribution of sample means will:will:

1.1. have a mean equal to have a mean equal to ..

2.2. have a standard deviation equal to have a standard deviation equal to

Further, if the sampled population has a normal Further, if the sampled population has a normal distribution, then the sampling distribution of will also distribution, then the sampling distribution of will also be normal for samples of all sizes.be normal for samples of all sizes.

Central Limit TheoremCentral Limit Theorem

The distribution of sample means will come closer to The distribution of sample means will come closer to normal as the sample size increases.normal as the sample size increases.

x

x

x

.n

( )x

Graphical Illustration of the Central Limit Theorem:

Original Population

10 20 30 x 10 20 x

Distribution of : n = 2x

10 x

Distribution of : n = 10x

10 x

Distribution of : n = 30x

ExampleExample: Consider a normal population with : Consider a normal population with = 50 and = 50 and = 15. Suppose a sample of size 9 is selected at = 15. Suppose a sample of size 9 is selected at random. Find:random. Find:

1.1.

2.2.

SolutionSolution::

Since the original population is normal, the distribution Since the original population is normal, the distribution of the sample mean is also (exactly) normal.of the sample mean is also (exactly) normal.

P(45 60)x

P( 47.5)x

50

15 9 15 3 5

x

x n

P x Px

P z

( )

( )

. . .

45 6045 50

550

560 50

5

1 2

0 3413 0 4772 0 8185

5045 60 x0 1 2 z

0 3413. 0 4772.

P x Px

P z

( . ).

( . )

. . .

47 550

547 5 50

5

5

0 5000 01915 0 3085

5047 5. x0 .5 z

01915.0 3085.

ExampleExample: A recent report stated that the day-care cost : A recent report stated that the day-care cost per week in Boston is $109. Suppose this figure is per week in Boston is $109. Suppose this figure is taken as the mean cost per week and that the standard taken as the mean cost per week and that the standard deviation is known to be $20.deviation is known to be $20.

1.1. Find the probability that a sample of 50 day-care Find the probability that a sample of 50 day-care centers would show a mean cost of $105 or less per centers would show a mean cost of $105 or less per week.week.

2.2. Suppose the actual sample mean cost for the sample Suppose the actual sample mean cost for the sample of 50 day-care centers is $120. Is there any evidence of 50 day-care centers is $120. Is there any evidence to refute the claim of $109 presented in the report?to refute the claim of $109 presented in the report?

SolutionSolution::

The shape of the original distribution is unknown, but The shape of the original distribution is unknown, but the sample size, the sample size, nn, is large. The CLT applies., is large. The CLT applies.

The distribution of is approximately normal.The distribution of is approximately normal.x109 20 50 2.83

x xn

P x Px

P z

( ). .

( . )

. . .

105109

2 83105 109

2 83

141

0 5000 0 4207 0 0793

109105 x0 141. z

0 4207.0 0793.

To investigate the claim, we need to examine how To investigate the claim, we need to examine how likelylikely an observation is the sample mean of $120.an observation is the sample mean of $120.

Consider how far out in the tail of the distribution of the Consider how far out in the tail of the distribution of the sample mean is $120.sample mean is $120.

Compute the Compute the tail probabilitytail probability..

Since the tail probability is so small, this suggests the Since the tail probability is so small, this suggests the observation of $120 is very rare (if the mean cost is observation of $120 is very rare (if the mean cost is really $109).really $109).

There is evidence to suggest the claim of There is evidence to suggest the claim of = $109 is = $109 is wrong.wrong.

109 120 109P( 120) P

2.83 2.83

P( 3.89)

0.0001

xx

z

In democracy your vote counts. In feudalism your count votes.

She was engaged to a boyfriend with a wooden leg but broke it off.

A chicken crossing the road is poultry in motion.