1. Chapter 4
Probability and Sampling
Distributions
1

2. Random Variable

Definition: A random variable is a variable
whose value is a numerical outcome of a
random phenomenon.
 The
statistic calculated from a randomly chosen
sample is an example of a random variable.

We don’t know the exact outcome beforehand.
A
statistic from a random sample will take
different values if we take more samples from the
same population.
2

3. Section 4.4
The Sampling Distribution of a
Sample Mean
3

4. Introduction
A statistic from a random sample will take
different values if we take more samples
from the same population
 The values of a statistic do no vary
haphazardly from sample to sample but
have a regular pattern in many samples

 We

already saw the sampling distribution
We’re going to discuss an important
sampling distribution. The sampling
distribution of the sample mean,4x-bar( )

5. Example


Suppose that we are interested in the workout
times of ISU students at the Recreation center.
Let’s assume that μ is the average workout time of
all ISU students
 To
estimate μ lets take a simple random sample of 100
students at ISU


We will record each students work out time (x)
Then we find the average workout time for the 100 students



x
The population mean μ is the parameter of interest.
The sample mean, x , is the statistic (which is a random variable).
Use x to estimate μ (This seems like a sensible thing to do).
5

6. Example

A SRS should be a fairly good representation of
the population so the x-bar should be somewhere
near the µ.
from a SRS is an unbiased estimate of µ due to
the randomization
 x-bar

We don’t expect x-bar to be exactly equal to µ
 There

is variability in x-bar from sample to sample
If we take another simple random sample (SRS) of
100 students, then the x-bar will probably be different.
 Why,
then, can I use the results of one sample to
estimate µ?
6

7. Statistical Estimation

If x-bar is rarely exactly right and varies from
sample to sample, why is x-bar a reasonable
estimate of the population mean µ?
 Answer:
if we keep on taking larger and larger
samples, the statistic x-bar is guaranteed to get
closer and closer to the parameter µ

We have the comfort of knowing that if we can
afford to keep on measuring more subjects,
eventually we will estimate the mean amount of
workout time for ISU students very accurately
7

8. The Law of Large Numbers

Law of Large Numbers (LLN):
 Draw
independent observations at random from
any population with finite mean µ
 As the number of observations drawn increases,
the mean x-bar of the observed values gets closer
and closer to the mean µ of the population
 If n is the sample size as n gets large x → µ

The Law of Large Numbers holds for any
population, not just for special classes such
as Normal distributions
8

9. Example

Suppose we have a bowl with 21 small pieces of paper
inside. Each paper is labeled with a number 0-20. We
will draw several random samples out of the bowl of
size n and record the sample means, x-bar for each
sample.



What is the population?
Since we know the values for each individual in the
population (i.e. for each paper in the bowl), we can
actually calculate the value of µ, the true population
mean. µ = 10
Draw a random sample of size n = 1.
Calculate x-bar for this sample.
9

10. Example

Draw a second random sample of size n = 5. Calculate x for this
sample.
x

Draw a third random sample of size n = 10. Calculate
sample.

Draw a fourth random sample of size n = 15. Calculate
sample.

Draw a fifth random sample of size n = 20. Calculate
sample.

What can we conclude about the value of
increases?
for this
x
for this
x for this
x as the sample size
THIS IS CALLED THE LAW OF LARGE NUMBERS.
10

11. Another Example
5.710
5.705
5.700
5.695
Mean of first n observations

Example: Suppose we know that the average height of all high
school students in Iowa is 5.70 feet.
We get SRS’s from the population and calculate the height.
mean of first n observations (feet)

0
5000
10000
number of observations
15000
11
20000

12. Example 4.21 From Book


Sulfur compounds such as dimethyl sulfide (DMS)
are sometimes present in wine
DMS causes “off-odors” in wine, so winemakers
want to know the odor threshold
 What
is the lowest concentration of DMS that the
human nose can detect

Different people have different thresholds, so we
start by asking about the mean threshold µ in the
population of all adults
µ
is a parameter that describes this population
12

13. Example 4.21 From Text


To estimate µ, we present tasters with both
natural wine and the same wine spiked with DMS
at different concentrations to find the lowest
concentration at which they can identify the
spiked wine
The odor thresholds for 10 randomly chosen
subjects (in micrograms/liter):
 28

40 28 33 20 31 29 27 17 21
The mean threshold for these subjects is 27.4
 x-bar
is a statistic calculated from this sample
 A statistic, such as the mean of a random sample of
13
10 adults, is a random variable.

14. Example
Suppose µ = 25 is the true value of the
parameter we seek to estimate
 The first subject had threshold 28 so the
line starts there
 The second point is the mean of the first
28 + 40
two subjects:
x=
= 34
2


This process continues many many times,
and our line begins to settle around µ = 25
14

15. Example 4.21From Book
The law of large
numbers in action: as
we take more
observations, the
sample mean x
always approaches the
mean of the population
µ = 25
15

16. The Law of Large Numbers

The law of large numbers is the foundation of
business enterprises such as casinos and
insurance companies
 The
winnings (or losses) of a gambler on a few plays are
uncertain -- that’s why gambling is exciting(?)

But, the “house” plays tens of thousands of times
 So
the house, unlike individual gamblers, can count on
the long-run regularity described by the Law of Large
Numbers
 The average winnings of the house on tens of thousands
of plays will be very close to the mean of the distribution
of winnings
 Hence, the LLN guarantees the house a profit!
16

17. Thinking about the
Law of Large Numbers
The Law of Large Numbers says broadly that
the average results of many independent
observations are stable and predictable
 A grocery store deciding how many gallons of
milk to stock and a fast-food restaurant
deciding how many beef patties to prepare
can predict demand even though their
customers make independent decisions

 The
Law of Large Numbers says that the many
individual decisions will produce a stable result
17

18. The “Law of Small Numbers”
or “Averages”

The Law of Large Numbers describes the
regular behavior of chance phenomena in the
long run

Many people believe in an incorrect “law of
small numbers”
 We
falsely expect even short sequences of
random events to show the kind of average
behaviors that in fact appears only in the long run
18

19. The “Law of Small Numbers”
or “Averages”

Example: Pretend you have an average free throw
success rate of 70%. One day on the free throw
line, you miss 8 shots in a row. Should you hit the
next shot by the mythical “law of averages.”

No. The law of large numbers tells us that the long run
average will be close to 70%. Missing 8 shots in a row
simply means you are having a bad day. 8 shots is hardly
the “long run”. Furthermore, the law of large numbers says
nothing about the next event. It only tells us what will
happen if we keep track of the long run average.
19

20. The Hot Hand Debate




In some sports If player makes several consecutive
good plays, like a few good golf shots in a row, often
they claim to have the “hot hand”, which generally
implies that their next shot is likely to a good one.
There have been studies that suggests that runs of golf
shots good or bad are no more frequent in golf than
would be expected if each shot were independent of the
player’s previous shots
Players perform consistently, not in streaks
Our perception of hot or cold streaks simply shows that
we don’t perceive random behavior very well!
20

21. The Gambling Hot Hand


Gamblers often follow the hot-hand theory,
betting that a “lucky” run will continue
At other times, however, they draw the opposite
conclusion when confronted with a run of
outcomes
 If
a coin gives 10 straight heads, some gamblers feel
that it must now produce some extra tails to get back
into the average of half heads and half tails
 Not true! If the next 10,000 tosses give about 50%
tails, those 10 straight heads will be swamped by the
later thousands of heads and tails.
 No short run compensation is needed to get back to
21
the average in the long run.

22. Need for Law of Large Numbers

Our inability to accurately distinguish
random behavior from systematic
influences points out the need for
statistical inference to supplement
exploratory analysis of data

Probability calculations can help verify
that what we see in the data is more than
a random pattern
22

23. How Large is a Large
Number?

The Law of Large Numbers says that the actual
mean outcome of many trials gets close to the
distribution mean µ as more trials are made

It doesn’t say how many trials are needed to
guarantee a mean outcome close to µ
 That

depends on the variability of the random outcomes
The more variable the outcomes, the more trials
are needed to ensure that the mean outcome xbar is close to the distribution µ
23

24. More Laws of Large Numbers

The Law of Large Numbers is one of the central
facts about probability
 LLN
explains why gambling, casinos, and insurance
companies make money
 LLN assures us that statistical estimation will be accurate
if we can afford enough observations

The basic Law of Large Numbers applies to
independent observations that all have the same
distribution
 Mathematicians
general settings
have extended the law to many more
24

25. What if Observations are not
Independent




You are in charge of a process that
manufactures video screens for computer
monitors
Your equipment measures the tension on the
metal mesh that lies behind each screen and is
critical to its image quality
You want to estimate the mean tension µ for the
process by the average x-bar of the
measurements
The tension measurements are not independent
25

26. AYK 4.82

Use the Law of Large Numbers applet on
the text book website
26

27. Sampling Distributions

The Law of Large Numbers assures us that if
we measure enough subjects, the statistic xbar will eventually get very close to the
unknown parameter µ
27

28. Sampling Distributions

What if we don’t have a large sample?
 Take
a large number of samples of the same
size from the same population
 Calculate
 Make

the sample mean for each sample
a histogram of the sample means
the histogram of values of the statistic
approximates the sampling distribution that we
would see if we kept on sampling forever…
28

29. 
The idea of a sampling distribution is
the foundation of statistical inference
 The
laws of probability can tell us about
sampling distributions without the need to
actually choose or simulate a large number
of samples
29

30. Mean and Standard Deviation of a
Sample Mean

Suppose that x-bar is the mean of a SRS of size
n drawn from a large population with mean µ and
standard deviation σ

The mean of the sampling distribution of x-bar is
σ
µ and its standard deviation is n
 Notice:
averages are less variable than individual
observations!
30

31. Mean and Standard Deviation of a
Sample Mean

The mean of the statistic x-bar is always the same
as the mean µ of the population
sampling distribution of x-bar is centered at µ
 in repeated sampling, x-bar will sometimes fall above
the true value of the parameter µ and sometimes below,
but there is no systematic tendency to overestimate or
underestimate the parameter
 because the mean of x-bar is equal to µ, we say that the
statistic x-bar is an unbiased estimator of the
parameter µ
 the
31

32. Mean and Standard Deviation of a
Sample Mean

An unbiased estimator is “correct on the
average” in many samples
 how
close the estimator falls to the parameter in most
samples is determined by the spread of the sampling
distribution
 if individual observations have standard deviation σ,
then sample means x-bar from samples of size n
σ
have standard deviation
n

Again, notice that averages are less variable
than individual observations
32

33. Mean and Standard Deviation of a
Sample Mean

Not only is the standard deviation of the
distribution of x-bar smaller than the standard
deviation of individual observations, but it gets
smaller as we take larger samples
 The
results of large samples are less variable than
the results of small samples

Remember, we divided by the square root of n
33

34. Mean and Standard Deviation of a
Sample Mean

If n is large, the standard deviation of x-bar is
small and almost all samples will give values of xbar that lie very close to the true parameter µ
 The
sample mean from a large sample can be trusted
to estimate the population mean accurately

Notice, that the standard deviation of the sample
distribution gets smaller only at the rate n
 To
cut the standard deviation of x-bar in half, we must
take four times as many observations, not just twice as
many (square root of 4 is 2)
34

35. Example

Suppose we take samples of size 15
from a distribution with mean 25 and
standard deviation 7
 the
distribution of x-bar is:
 the
mean of x-bar is:

25
 the

7 

 25,


15 
standard deviation of x-bar is:
1.80739
35

36. What About Shape?

We have described the center and spread
of the sampling distribution of a sample
mean x-bar, but not its shape

The shape of the distribution of x-bar
depends on the shape of the population
distribution
36

37. Sampling Distribution of a
Sample Mean

If a population has the N(µ, σ) distribution,
then the sample mean x-bar of n
independent observations has the
σ

N µ
,
 distribution
n 

37

38. Example

Adults differ in the smallest amount of
dimethyl sulfide they can detect in wine

Extensive studies have found that the
DMS odor threshold of adults follows
roughly a Normal distribution with mean µ
= 25 micrograms per liter and standard
deviation σ = 7 micrograms per liter
38

39. Example

Because the population distribution is
Normal, the sampling distribution of xbar is also Normal

If n = 10, what is the distribution of xbar?
7 

N  25,

10 

39

40. What if the Population Distribution
is not Normal?

As the sample size increases, the distribution
of x-bar changes shape
 The
distribution looks less like that of the
population and more like a Normal distribution

When the sample is large enough, the
distribution of x-bar is very close to Normal
 This
result is true no matter what shape of the
population distribution as long as the population
has a finite standard deviation σ
40

41. Central Limit Theorem

Draw a SRS of size n from any population
with mean µ and finite standard deviation σ

When n is large, the sampling distribution of
the sample mean x-bar is approximately
Normal:
x-bar is approximately
σ

N µ
,

n 

41

42. Central Limit Theorem

More general versions of the central limit
theorem say that the distribution of a sum or
average of many small random quantities is
close to Normal

The central limit theorem suggests why the
Normal distributions are common models for
observed data
42

43. How Large a Sample is Needed?

Sample Size depends on whether the
population distribution is close to Normal
 We
require more observations if the shape of
the population distribution is far from Normal
43

44. Example



The time X that a technician requires to perform
preventive maintenance on an air-conditioning unit
is governed by the Exponential distribution (figure
4.17 (a)) with mean time µ = 1 hour and standard
deviation σ = 1 hour
Your company operates 70 of these units
The distribution of the mean time your company
spends on preventative maintenance is:
1 

N 1,
 = N (1,0.12 )
70 

44

45. Example

What is the probability that P ( x > 0.83)
your company’s units


average maintenance time


 x − µ > 0.83 − 1 
exceeds 50 minutes?
=P
 σ 
0.12 
50/60 = 0.83 hour



 So we want to know P(x-bar >
 n 

0.83)
= P ( z > −1.42 )


Use Normal distribution
calculations we learned in
Chapter 2!
= 1 − P ( z < −1.42 )
= 1 − 0.0778 = 0.9222
45

46. 4.86 ACT scores

The scores of students on the ACT college
entrance examination in a recent year had
the Normal distribution with mean µ = 18.6
and standard deviation σ = 5.9
46

47. 4.86 ACT scores

What is the probability that a single
student randomly chosen from all those
taking the test scores 21 or higher?
P( x ≥ 21)
 x − µ 21 − 18.6 
= P
≥

5.9 
 σ
= P ( z ≥ 0.4068) = 1 − P ( z < 0.41)
= 1 − 0.6591 = 0.3409
47

48. 4.86 ACT scores

About 34% of students (from this
population) scored a 21 or higher on the
ACT

The probability that a single student
randomly chosen from this population
would have a score of 21 or higher is 0.34
48

49. 4.86 ACT scores

Now take a SRS of 50 students who took
the test. What are the mean and standard
deviation of the sample mean score x-bar
of these 50 students?
 Mean
= 18.6 [same as µ]
 Standard Deviation = 0.8344 [sigma/sqrt(50)]
49

50. 4.86 ACT scores

What is the probability that the mean
score x-bar of these students is 21 or
higher?
P ( x ≥ 21)




x − µ 21 − 18.6 
= P
≥
 σ 
0.834 



 n 

= P ( z ≥ 2.8778) = 1 − P ( z < 2.88)
= 1 − 0.9980 = 0.002
50

51. 4.86 ACT scores

About 0.2 % of all random samples of size
50 (from this population) would have a
mean score x-bar of 21 or higher.

The probability of having a mean score xbar of 21 or higher from a sample of 50
students (from this population) is 0.002.
51

52. Section 4.4 Summary

When we want information about the
population mean µ for some variable, we
often take a SRS and use the sample
mean x-bar to estimate the unknown
parameter µ.
52

53. Section 4.4 Summary

The Law of Large Numbers states that
the actually observed mean outcome xbar must approach the mean µ of the
population as the number of observations
increases.
53

54. Section 4.4 Summary

The sampling distribution of x-bar
describes how the statistic x-bar varies in
all possible samples of the same size from
the same population.
54

55. Section 4.4 Summary

The mean of the sampling distribution is
µ, so that x-bar is an unbiased estimator
of µ.
55

56. Section 4.4 Summary

The standard deviation of the sampling
distribution of x-bar is sigma over the
square root of n for a SRS of size n if the
population has standard deviation sigma.
That is, averages are less variable than
individual observations.
56

57. Section 4.4 Summary

If the population has a Normal distribution,
so does x-bar.
57

58. Section 4.4 Summary

The Central Limit Theorem states that for large
n the sampling distribution of x-bar is
approximately Normal for any population with
finite standard deviation sigma. That is,
averages are more Normal than individual
observations. We can use the fact that x-bar
has a known Normal distribution to calculate
approximate probabilities for events involving xbar.
58