Download Stat 281 Chapter 7

Chapter 7
Sample 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.
Sampling

The Need


Advantages





Get information about a population without checking
the entire population
Cost
Time
Accuracy (can be achieved with low cost)
Destruction is sometimes involved; checking all is not
possible.
[Insert Excel Simulation here]
Distribution of Means
Visual Mean of Means
Distribution of Sample Means
 Many
different sample means are possible
 The sample means cluster closer to the
population mean than the population
values do.
 The larger the sample, the closer they
cluster around the population mean
 Therefore the likelihood of a single sample
mean being close to the true mean is high
Distribution of Sample Means

When trying to use a sample to estimate a
population mean, we know we won’t get the
exact value
 We want some way of managing the error so as
to be as close as we need to be
 We can decide on a margin of error that we are
willing to accept (polls typically 2% - 4%).
 We cannot eliminate the possibility of getting a
value outside that range, but we can keep it
small by adjusting the sample size.
x
How Close Can We Get?

The variance of the sample mean is the
population variance divided by n (sample size)
 Thus larger n’s bring smaller variances
 Let’s look at an example. In order to understand
the process, we will assume we actually know
the true mean and variance. Each of the
following graphs is from a computer simulation
of taking 100 samples from a normal population
with μ=15 and σ=3, but with different sample
sizes.
μ=15, σ=3, Sample Size 1
Number observed in [14,16]: 30
30
Percent
20
10
0
7
9
11
13
15
17
s=3
19
21
23
25
μ =15, σ=3, Sample Size 4
Number observed in [14,16]: 52
50
Percent
40
30
20
10
0
7
9
11
13
15
17
s=1.5
19
21
23
25
μ =15, σ=3, Sample Size 9
Number observed in [14,16]: 74
80
70
Percent
60
50
40
30
20
10
0
7
9
11
13
15
17
s=1
19
21
23
25
μ =15, σ=3, Sample Size 16
Number observed in [14,16]: 81
80
70
Percent
60
50
40
30
20
10
0
7
9
11
13
15
17
s=3/4
19
21
23
25
μ =15, σ=3, Sample Size 25
Number observed in [14,16]: 90
90
80
70
Percent
60
50
40
30
20
10
0
7
9
11
13
15
17
s=3/5
19
21
23
25
μ =15, σ=3, Sample Size 36
Number observed in [14,16]: 97
100
90
80
Percent
70
60
50
40
30
20
10
0
7
9
11
13
15
17
s=1/2
19
21
23
25
Number in [14,16] vs Sample Size
100
90
80
70
60
50
40
30
20
10
0
0
5
10
15
20
25
30
35
40
So What?





In Real Life, we don’t know the true mean and
variance. We want to estimate them.
Furthermore, we will only take one sample,
which represents just one data point from the
distributions we have illustrated.
We will probably NEVER know where in the
distribution that data point is coming from.
Under these conditions, how can we provide an
estimate that is trustworthy?
Clearly, the sample size directly affects the
likelihood that the sample mean will be close to
the true mean.
Which one would you like to pick
from?
100
30
90
80
70
Percent
Percent
20
10
60
50
40
30
20
10
0
0
7
9
11
13
15
17
s=3

19
21
23
25
7
9
11
13
15
17
19
21
23
25
s=1/2
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
many of each there are. You will draw one ball and record its
number. If this number matches the mean of the distribution, your
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
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.
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.
Notes:
1. x : the sample mean.
2. sx : the standard deviation of the sample means.
3. The theory involved with sampling distributions
described in the remainder of this chapter requires
random sampling.
Random Sample: A sample obtained in such a way that
each possible sample of a fixed size n has an equal
probability of being selected.
(Example: Every possible handful of size n has the
same probability of being selected.)
The Central Limit Theorem
 The
most important idea in all of statistics.
 Describes the sampling distribution of the
sample mean.
 Examples suggest: the sample mean (and
sample total) tend to be normally
distributed.
Distribution of Sample Means
If all possible random samples of a particular size n are
taken from any population with a mean m and a
standard deviation s, the distribution of sample means ( x )
will:
1. have a mean m x equal to m.
2. have a standard deviation s x equal to s n .
Further, if the sampled population has a normal
distribution, then the sampling distribution of x will also
be normal for samples of all sizes.
Central Limit Theorem
The distribution of sample means will come closer to
normal as the sample size increases.
Graphical Illustration of the Central Limit Theorem:
Distribution
of x : n = 2
Original Population
10
20
30
x
10
Distribution
of x : n = 30
Distribution
of x : n = 10
10
x
20 x
10
x
Example: Consider a normal population with m = 50 and
s = 15. Suppose a sample of size 9 is selected at
random. Find:
1. P(45  x  60)
2. P( x  47.5)
Solution:
Since the original population is normal, the distribution
of the sample mean is also (exactly) normal.
m x  m  50
sx s
n  15
9  15 3  5
0.4772
0.3413
45
1
50
0
60
2
 45  50 x  50 60  50
P(45  x  60)  P



 5
5
5 
 P( 1  z  2)
 0.3413  0.4772  0.8185
x
z
0.3085
01915
.
47.5 50
.5 0
 x  50 47.5  50
P( x  47.5)  P


 5
5 
 P( z  .5)
 0.5000  01915
.
 0.3085
x
z
Example: A recent report stated that the day-care cost
per week in Boston is $109. Suppose this figure is
taken as the mean cost per week and that the standard
deviation is known to be $20.
1. Find the probability that a sample of 50 day-care
centers would show a mean cost of $105 or less per
week.
2. Suppose the actual sample mean cost for the sample
of 50 day-care centers is $120. Is there any evidence
to refute the claim of $109 presented in the report?
Solution:
The shape of the original distribution is unknown, but
the sample size, n, is large. The CLT applies.
The distribution of x is approximately normal.
m x  m  109
sx s
n  20
50  2.83
0.4207
0.0793
105
141
.
109
0
 x  109 105  109 
P( x  105)  P


 2.83
2.83 
 P( z  141
. )
 0.5000  0.4207  0.0793
x
z

To investigate the claim, we need to examine how
likely an observation is the sample mean of $120.
 Consider how far out in the tail of the distribution of
the sample mean is $120.
 Compute the tail probability.
 x  109 120  109 
P( x  120)  P 


2.83 
 2.83
 P( z  3.89)
 0.0001


Since the tail probability is so small, this suggests the
observation of $120 is very rare (if the mean cost is
really $109).
There is evidence to suggest the claim of m = $109 is
wrong.
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.