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MER301: Engineering
Reliability
LECTURE 8:
Chapter 4:
Statistical Inference, Point Estimation
and Hypothesis Testing
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
1
Summary
 Statistical Inference
 Estimates of Population Parameters
 Point Estimation
 Standard error
 Hypothesis Testing
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
2
Statistical Inference
 Consists of methods used to
 make decisions
 draw conclusions about populations
 Methods utilize information from
Samples drawn from Populations
 Two major areas
 parameter estimation
 hypothesis testing
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
3
Comment on Engineering Use
 Statistical Inference has long been
used to establish manufacturing
process capability and to monitor
manufacturing lines as part of
quality programs
 More recently, it has become part
of the Statistical Design Methods
toolkit to take account of variation
in the design phase
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
4
Relationship Between
Population and Sample
 Parameters of
populations are
what we want to
2
know  , 
 Estimates from
sample data are
2
what we get
 Need to estimate
parameters from
the sample data
 ,
x, s
L Berkley Davis
Copyright 2009
2
X,S
x, s
MER301: Engineering Reliability
Lecture 8
2
2
5
Relationship Between
Population and Sample
 Parameters of
populations are
what we want to
2
know  , 
 Estimates from
sample data are
2
what we get
 Need to estimate
parameters from
the sample data
Point Estimate
 , 2
x, s
L Ber kle y Davis
Cop yrig ht 20 09
X, S2
x, s2
MER30 1: Engi ne erin g R eliability
Le ct ur e 8
5
 An ESTIMATOR ̂is the statistic (Random Variable)
used to generate the estimate of a parameter
1 n
1 n
2
2
S 
(
X

X
)
X   Xi
 i
n  1 i 1
n i 1

 An ESTIMATE ˆ is a number derived from Sample
Data
“The Objective of Point Estimation is to Select a
Single Number ˆ , Based on Sample Data, that is the
Most Plausible Value for
…..”

L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
6
Point Estimators for a Population
 Let X be the Random Variable of a
Population

 Population Mean:
 Population Variance:
2
 Estimators for Population Mean and
Variance for a Random Sample of Size n
i n
X 
L Berkley Davis
Copyright 2009
 Xi
i 1
n
n
S2 
MER301: Engineering Reliability
Lecture 8
2
(
X

X
)
 i
i 1
n 1
7
Parameters, Statistics,
and Point Estimates
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
8
Example 8.1:
Parameters, Statistics,
and Point Estimates
 In estimating the mean number of goals scored in a
Division 1 (M’s) hockey game for a given
year, the
n
1
statistic used is a sample mean
X

X

n
i
X is a POINT ESTIMATOR for the mean goals/game 
i 1
n
1
ˆ  X  X

i
n i 1
 The 59 Division 1 Men’s teams play about a thousand
games per year and scores from a sample of 34 are
as shown on the following page
 So for this particular sample the value of 3.23 goals
per per game could be taken as a POINT ESTIMATE
ˆ  ˆ  x  3.23goals / game
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
9
Example 8.1:
Parameters, Statistics,
and Point Estimates(continued)
goals/game
goals/game
0
0
4
3
2
3
3
4
6
3
2
5
4
1
6
7
6
L Berkley Davis
Copyright 2009
xi
2
3
4
4
4
3
5
4
2
3
4
0
1
6
4
0
2
1 n
X   Xi
n i 1
S2 
1 n
( X i  X )2

n  1 i 1
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
Confidence Level(95.0%)
3.235294118
0.321701203
3
4
1.87582424
3.518716578
-0.483954805
-0.070283187
7
0
7
110
34
0.654506538
ˆ  ˆ  x  3.23goals / game
ˆ  ˆ  s  1.88 goals / game
10
How Good is the Estimate
 Estimators should
generate estimates
that are close in
value to θ
 This is expected if
the Estimator has
the properties
 ˆ is UNBIASED for θ
 ˆ has small variance
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
11
Unbiased Estimator Definition
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
12
Unbiased Estimators for Population
Parameters
 Let X be the Random Variable of a
Population
 Population Mean: 
2
 Population Variance: 
 Unbiased Estimators of the Population
Mean and Variance for a Random Sample
of Size n
1 n
1 n
2
2
X
X

n
i 1
i
E (ˆ  x )  
L Berkley Davis
Copyright 2009
S 
(X

n 1
i 1
i
 X)
E (ˆ 2  s 2 )   2
MER301: Engineering Reliability
Lecture 8
13
How Good is the Estimate
 Estimators should
generate estimates
that are close in
value to θ
 This is expected if
the Estimator has
the properties
 ˆ is UNBIASED for θ
 ˆ has small variance
L Berkley Davis
Copyright 2009
-Standard Error of the Mean
-How Samples are Drawn
MER301: Engineering Reliability
Lecture 8
14
Central Limit Theorem and the Standard
Error of the Mean…
16 Sample Data Sets- mean =48, standard deviation= 3
Set 1
Set 2
Set 3
Set 4
Set 6
Set 7
Set 8
Set 9
1
47.1
44.17
48.73
51.83
51.6
53.2
41.45
47.3
51.29
2
44.74
45.93
42.93
42.46
Set 5
45.07
45.68
41.65
46.3
46.79
3
48.4
46.9
47.02
46.89
52.03
47.74
47.44
46.46
53.92
4
50.6
55.13
46.04
52.98
43.16
49.62
50.71
53.76
47.75
5
46.43
50.03
46.86
50.27
43.67
45.46
43.44
46.91
47.9
6
48.08
47.03
54.58
42.77
45.79
40.27
52.34
44.16
46.04
7
50.27
49.4
50.62
49.79
43.88
44.65
50.08
48.97
45.18
8
47.28
48.39
49.67
48.42
45.27
53.65
49.46
48.22
50.49
9
10
11
12
13
50.59
52.33
49.33
46.64
48.13
46.09
51.91
49.85
46.43
46.04
45.23
48.34
48.64
50.55
46.35
51.33
48.01
44.92
49.54
50.55
44.4
49.36
51.71
46.18
50.41
43.32
47.92
47.07
51.91
49.37
50.13
44.84
45.48
42.72
50.07
49.92
42.68
45.54
49.65
52.89
54.62
50.48
46.71
47.65
48.91
14
15
16
49.77
46.25
47.3
53.56
49.6
56.51
46.99
49.64
51.76
51.11
47.05
50.64
48.43
46.68
52
51.42
43.9
48.56
47.56
53.98
49.63
45.66
46.3
47.25
51.23
48.26
44.34
Distribution of X
Distribution of
X
  3 / 16  0.75
 3
  48
Standard Error of the Mean:
L Berkley Davis
Copyright 2009
X 16Sample  / 16
48.3275
49.18563
48.37188
48.66
47.4775
47.73375
47.56125
47.62313
48.8475
0.502336
0.890323
0.702219
0.783431
0.817835
0.940054
0.969401
0.727962
0.737096
X  / n
15
Standard Error
X 
L Berkley Davis
Copyright 2009

S
ˆ X 
n
n
MER301: Engineering Reliability
Lecture 8
16
Example 8.1(continued)
goals/game
S
ˆ X  goals/game
n
x
n
1
xi

n i 1
L Berkley Davis
Copyright 2009
0
0
4
3
2
3
3
4
6
3
2
5
4
1
6
7
6
xi
2
3
4
4
4
3
5
4
2
3
4
0
1
6
4
0
2
3.235294118
0.321701203
3
4
1.87582424
3.518716578
-0.483954805
-0.070283187
27
0
7
110
34
0.654506538
1
S 
(Xi  X )

n 1
2
n
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
n
Range
Minimum
Maximum
i 1
Sum
Count
Confidence Level(95.0%)
ˆ  ˆ  x  3.23goals / game
ˆ  ˆ  S  1.88 goals / game
17
How Good is the Estimate
 Estimators should
generate estimates
that are close in
value to θ
 This is expected if
the Estimator has
the properties
 ˆ is UNBIASED for θ
 ˆ has small variance
L Berkley Davis
Copyright 2009
-Standard Error of the Mean
-How Samples are Drawn
MER301: Engineering Reliability
Lecture 8
18
Example 7.3: Throwing Dice..
ˆ X  S  1.7
ˆ X  S  0.67
ˆ X  S  1.2
ˆ X  S  0.5
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
19
Example 7.3: Throwing Dice..
Samples
Roll
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
3
2
3
1
1
3
4
1
6
6
1
3
6
2
3
6
1
2
1
2
6
3
3
5
6
4
6
1
3
2
6
1
1
2
1
6
Single
die 4
5
4
Average
of 25
6
6
Average
of 55
5
5
1
1
2
5
6
4
5
3
2
3
2
4
1
6
3
5
14
12
4
1.6
6
3
4
5
5
6
5
3
2
3
5
4
1
3
2
5
6
5
62
3.51
1
3.62
3
1
1
6
5
1
3
5
2
1
5
5
5
2
Average of 10
3
2.5
3
3.6
Data Sets
4
2.5
3.4
Statistics3.5
3.4
Mean
Standard Error
̂ X  S
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
L Berkley Davis
Copyright 2009
6
4
4
3Single die
4.1
4
0.53748385
4.5
5
6
3.5
3.2
Average
4 of 2
1
3
1.5
3
2.8
4.4
of 5
2.9Average3.3
2.85
3.26
3.45
0.38042374
0.258284942
0.125830574
2.75
2.5
3.3
4
3.45
3.6
1.69967317 1.2030055 0.816768701
2.888888889 1.447222222
-0.834953508 -0.268898214
-0.509147659 0.201032008
5
4
1
1
6
5
40
10
6
1
5
2
4
2.6
Average of
3.6
3.110
28.5
10
0.397911213
0.667111111
0.631032628
-0.673670743
2.8
1.6
4.4
0.158333333
-0.690937871
0.317447173
1.2
2.9
4.1
32.6
10
34.5
10
20
Example 7.3: Throwing Dice..
ˆ X  1.7 / 10  0.54
ˆ X  0.67 / 10  0.21
ˆ X  1.2 / 10  0.38
ˆ X  0.5 / 10  0.16
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
21
Example 7.3: Throwing Dice..
Samples
Roll
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
3
2
3
1
1
3
4
1
6
6
1
3
6
2
3
6
1
2
1
2
6
3
3
5
6
4
6
1
3
2
1
2
6
4
5
6
5
5
1
1
3 2
2.5 5
6
3 4
5
3
2
31
2
1
4
1.6
1
6
3
5
4
2
4
6
3
4
5
5
6
5
3
2
3
5
64
1
3.5
3
3.6
2
5
6
5
2
1
1
2
3
1
1
6
5
1
3
5
2
1
65
3.55
5
3.22
6
1
1
Single die5
Average of42
6
Average of55
Average of 10
ˆ X  S / n
3.6
ˆ X  1.7 / 10  0.54
ˆ X  1.2 / 10  0.38
ˆ X  0.67 / 10  0.21
ˆ X  0.5 / 10  0.16
L Berkley Davis
Copyright 2009
Data Sets
4
2.5
Statistics
3.4
3.4 Mean3.5
6
4
Single die
4
3
4.1
4
1
3
1.5
3
Average of 22.8 Average
4.4 of 5
42.85 2.9
3.3
3.26
6
1
5
2
4 Average
2.6of 10
3.6
3.1
3.45
Standard Error 0.5374838 0.38042 0.258285 0.12583057
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
4.5
5
2.75
2.5
3.3
4
1.69967317 1.2030055 0.816768701
2.888888889 1.447222222
-0.834953508 -0.268898214
-0.509147659 0.201032008
5
4
1
1
6
5
40
28.5
10
10
0.667111111
0.631032628
-0.673670743
2.8
1.6
4.4
32.6
10
3.45
3.6
0.397911213
0.158333333
-0.690937871
0.317447173
1.2
2.9
4.1
34.5
10
22
Relationship Between
Population and Sample
 Parameters of
populations are
what we want to
2
know  , 
 Estimates from
samples are
2
what we get
 Need to estimate
parameters from
the samples
 ,
x, s
L Berkley Davis
Copyright 2009
2
X,S2
x, s
MER301: Engineering Reliability
Lecture 8
2
23
L Berkley Davis
Copyright 2009
24
Hypothesis Testing
 What is it?
 Methods for making conclusions about the values
of population parameters by using sample data
and statistical techniques
 Why are Engineers interested?
 For manufacturing processes , lines are sampled
and the results assessed using hypothesis
testing to establish whether the processes are
producing product with the required CTQ’s
 In statistical design, hypothesis testing is used
to compare the performance of two different
designs to see whether there are real differences
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
25
Hypothesis Testing
 The hypothesis consists of an either/or statement
about parameters of the populations,for example
this is a two sided hypothesis
H 0 :   0
 Null Hypothesis
 Alternative Hypothesis H :   
1
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
0
26
Hypotheses for Two Sided and One
Sided Hypothesis Tests
 Two –Sided Test
will detect the
differences on
either tail of the
distribution
H 0 :   0
H1 :    0
L Berkley Davis
Copyright 2009
 Right-tailed test
 Upper tail
region
 H0: µ=µ0
 H1: µ>µ0
 Left-tailed test
 Lower tail
region
 H0: µ=µ0
 H1: µ<µ0
MER301: Engineering Reliability
Lecture 8
27
Hypothesis Testing
Hypotheses are always statements about
the population, not the sample from the
population
 The sample is being used to estimate parameters
of the population
 Values of population parameters are obtained
from previous knowledge, from theoretical
calculations, or are customer requirements
Testing procedures use samples drawn
from the population to make statistical
predictions as to whether the hypothesis
is true.
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
28
Hypothesis Testing
 Data from the Sample will be used to
calculate a Test Statistic. This may be
of the form
X  0
Z
/ n
 A Critical Region for the test is
established based on the purpose for
the test. For example, power plants
performance tests have a tolerance
band- outside of that band is the critical
region
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
29
L Berkley Davis
Copyright 2009
30
Hypothesis Testing Error
(4-7)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
31
Summary of Hypothesis Testing
 Comments on Hypothesis Testing
 Null Hypothesis is what is tested
 Rejection of the Null Hypothesis always
leads to accepting the Alternative
Hypothesis
 Test Statistic is computed from Sample data
 Critical region is the range of values for the
test statistic where we reject the Null
Hypothesis in favor of the Alternative
Hypothesis
 Rejecting H when it is true is a Type I error
0
 Failing to reject H when it is false is a
0
Type II error
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
32
Example 8.2 :Hypothesis Testing(1)
Suppose we are interested in the burning rate of a solid
propellant used to power aircrew escape systems.
•
The burning rate of a solid propellant during manufacture
is a random variable that,over the long term with a
controlled manufacturing process, is characterized by
population parameters mean  0and standard deviation  0
•
Our interest focuses on the mean burning rate (a
parameter of this population), a critical performance CTQ
•
Specifically, we are interested in deciding whether or not
the mean burning rate of current production is kept stable
at the population mean of 50 centimeters per second.
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
33
Example 8.2 :Hypothesis Testing(2)
 The mean burning rate of each production lot
needs to be maintained within a narrow range
of the population mean for the system to
function properly. The manufacturing process
can be designed to maintain the mean burning
range for each lot within engineering
tolerances
X   0  3%
 The Hypotheses are given by
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
34
Example 8.2 :Hypothesis Testing(3)
Critical/Acceptance Regions
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
35
Example 8.2 :Hypothesis Testing(4)
 The mean and standard deviation of
the population are to be 50cm/sec
and 2.5cm/sec, respectively.
 A sample of ten(10) specimens from
the propellant is tested and the mean
sample burning rate calculated (there
will be Gage R&R considerations, ie
how well can the burning rate be
measured- with what accuracy and
precision?)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
36
Example 8.2 :Hypothesis Testing(5)
 The Z values corresponding to the critical
values are
Z1 
48.5  50
2.5 / 10
 1.90
Z2 
51.5  50
2.5 / 10
 1.90
 The probability of a Type I error is given by
  PX  48.5when  50  PX  51.5when  50
 The probability of a Type II error calculation is
done for a specific alternative mean burning
rate. The fraction of a distribution centered
about that mean that is within the acceptable
region for the H 0 hypothesis is the Type II error
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
37
Example 8.2 :Hypothesis Testing(6)
Type I Error
 The Type I error is the
probability that a
sample will be rejected
when in fact the true
mean of the product
being sampled is really
within the acceptance
region
L Ber kle y Davis
Cop yrig ht 20 09
  PZ  1.90  PZ  1.90
or
  0.0288  0.0288  0.0576
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 8
38
Example 8.2 :Hypothesis Testing(7)
Type II Error
 The probability of a
Type II error is shown
by the shaded region
  P48.5  X  51.5when  52
Z1 
48.5  52
2.5 / 10
 4.43 Z 2 
51.5  52
2.5 / 10
 0.63
  P 4.43  Z  0.63  0.2643
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Lecture 8
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Summary of Errors
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 The probability of making a Type I error- rejecting a
good sample or false negative- is mostly under the
direct control of the experimenter. There is a direct
effect of both sample size and the size of the
acceptable tolerance band
 The probability of making a Type II error-accepting a
bad sample or a false positive-is affected by the
same variables but is also strongly influenced by the
exact values of the test variables
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Lecture 8
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Sensitivity of a Statistical Test
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 This is 1-P(Type II error) and is descriptive of the
sensitivity of a statistical test, ie, of the ability of
the test to detect differences. The planning of the
test- how samples will be drawn, the number of
samples, establishment of the critical regions- is
one factor that influences Power
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Lecture 8
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Hypothesis Testing Procedure
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Summary
 Statistical Inference
 Estimates of Population Parameters
 Point Estimation
 Standard error
 Hypothesis Testing
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MER301: Engineering Reliability
Lecture 8
43