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MER301: Engineering
Reliability
LECTURE 10:
Chapter 4:
Decision Making for a Single Sample,
part 3
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
1
Summary of topics
 Inference on the Mean of a Population,
Variance Unknown
 Confidence Interval,Variance Unknown
 Inference on the Variance of a Normal
Population
 Inference on Population Proportion
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
2
Inference on the Mean of a Population,
Variance Unknown- the t-test
L Berkley Davis
Copyright 2009
Inference on the Mean of a
Population, Variance Unknown
 For cases where both the mean and variance
of a population are unknown AND the
population is normally distributed, then the tdistribution can be used for hypothesis testing.
The Test Statistic is the same form as the Z
based statistic but the underlying distribution
used to interpret the results is different
X  0
t0 
S/ n
 The t-distribution applies for small sample
sizes, in fact for n greater than or equal to 2
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
4
Inference on the Mean of a
Population, Variance Unknown
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
5
The t-distribution for several degrees
of freedom

The number of degrees of
freedom k or
is equal to
n-1 where n is the number
of samples

Normal distribution
k 
  k  n 1

For n>30,the t-distribution
approaches the standard
normal distribution

For small k or n,the tails of
the t-distribution include a
greater proportion of the
area.
The t-distribution is
symmetric about zero

L Berkley Davis
Copyright 2009
k 
4-15
MER301: Engineering Reliability
Lecture 10
6
Comparison of normal (Z) and t-distributions
16 Sample Data Sets- mean =48, standard deviation= 3
Set 1
Set 2
Set 3
Set 4
Set 5
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
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
49.46
48.22
50.49
9
10
11
12
13
14
15
16
50.59
52.33
49.33
46.64
48.13
49.77
46.25
47.3
46.09
51.91
49.85
46.43
46.04
53.56
49.6
56.51
48.34
48.64
50.55
46.35
46.99
49.64
51.76
50.13
44.84
45.48
42.72
50.07
47.56
53.98
49.63
49.92
42.68
45.54
49.65
52.89
45.66
46.3
47.25
54.62
50.48
46.71
47.65
48.91
51.23
48.26
44.34
x  48
t0 
s / 16
L Berkley Davis
Copyright 2009
48.42
45.27
53.65
X
S
/
16
16
Sample
45.23
51.33
44.4
43.32
48.01
48.3275
44.92
49.54
49.185625
50.55
48.371875
51.11
47.05
48.66
50.64
47.4775
47.73375
47.56125
47.623125
48.8475
49.36
47.92
0.5023358
51.71
47.07
46.18
51.91
0.8903234
50.41
49.37
0.7022192
48.43
51.42
46.68
43.9
0.783431
52
48.56
0.8178353
0.940054
0.9694005
0.7279616
0.7370965
Z
0.43667
1.58083
0.49583
0.88
-0.6967
-0.355
-0.585
-0.5025
1.13
tto0
x  48
z
3 / 16
Cumulative Dist
0.65195
1.33168
0.52957
0.84245
-0.6389
-0.2832
-0.4526
-0.5177
1.14978
z-dist
t-dist
0.66882
0.94304
0.68999
0.81057
0.24301
0.36129
0.27927
0.30766
0.87076
0.733645
0.8901712
0.694607
0.7879972
0.2703779
0.3920964
0.3314294
0.3093333
0.8582784
Comparison of normal (Z) and t-distributions
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
45.07
45.68
41.65
46.3
46.79
47.74
47.44
46.46
53.92
49.62
50.71
53.76
45.46
43.44
46.91
48.4
46.9
47.02
46.89
52.03
4
50.6
55.13
46.04
52.98
43.16
5
46.43
50.03
46.86
50.27
43.67
6
48.08
47.03
54.58
42.77
45.79
7
50.27
8
9
10
11
12
13
14
15
16
X 16 Sample S / 16
Z40.27
49.4
50.62 0.5023358
49.79
43.88
44.65
48.3275
0.43667
47.28
48.39
49.67
48.42
45.27
53.65
1.58083
50.5949.185625
46.09
45.23 0.8903234
51.33
44.4
43.32
52.3348.371875
51.91
48.34 0.7022192
48.01
49.36
47.92
0.49583
49.33
49.85
48.64
44.92
51.71
47.07
46.64
46.43
50.55 0.783431
49.54
46.18
51.91
48.66
0.88
48.13
46.04
46.35
50.55
50.41
49.37
-0.6967
49.77 47.4775
53.56
46.99 0.8178353
51.11
48.43
51.42
46.25
49.6
49.64
47.05
46.68
43.9
47.73375
0.940054
-0.355
47.3
56.51
51.76
50.64
52
48.56
47.56125
0.9694005 -0.585
47.623125 0.7279616 -0.5025
48.8475
0.7370965
1.13
to
52.34
47.75
Cumulative
Dist
47.9
z-dist
44.16
46.04
50.08
48.97
45.18
0.65195
0.66882
49.46
48.22
50.49
1.33168
0.94304
50.13
49.92
54.62
44.84
42.68
50.48
0.52957
0.68999
45.48
45.54
46.71
42.72
49.65
47.65
0.84245
0.81057
50.07
52.89
48.91
-0.6389
0.24301
47.56
45.66
51.23
53.98
46.3
48.26
-0.2832
0.36129
49.63
47.25
44.34
-0.4526 0.27927
-0.5177 0.30766
1.14978 0.87076
Xbar
49.19
48.85
48.66
48.37
48.33
48
47.73
47.62
47.56
47.48
x  48
t0 
s / 16
L Berkley Davis
Copyright 2009
t-dist
0.733645
0.8901712
0.694607
0.7879972
0.2703779
0.3920964
0.3314294
0.3093333
0.8582784
z-dist
0.943
0.871
0.811
0.69
0.669
0.5
0.361
0.308
0.279
0.243
t-dist
0.89
0.858
0.788
0.695
0.734
0.5
0.392
0.309
0.331
0.27
x  48
z
3 / 16
 t dist   z dist
Com parison of Z and t Distributions
1
0.9
0.8
cumulative distribution
3
Set 5
0.7
0.6
0.5
phi(z)
0.4
t-dist
0.3
0.2
0.1
0
47
47.5
48
48.5
Sam ple Mean Xbar
49
49.5
Percentage Points of a t-distribution
 The t-distribution is symmetric about zero
so that
t1 ,  t ,
and t 0.5,
 0.0 for all 
=k
4-16
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
9
t 0.5,  0.0


Percentage
Points
of a t-distribution
t1 ,  t ,
t ,
  k  n 1
x  0
t0 
s/ n
L Berkley Davis
Copyright 2009
10
Hypotheses Testing
Variance Unknown
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
11
Hypothesis Testing with the
t-distribution
4-19
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
12
Text Example 4-7 :Hypothesis
Testing with the t-distribution
 15 golf clubs were tested to establish the ratio
of the outgoing velocity of the golf ball to its
incoming velocity(coefficient of restitution). A
high coefficient is good. The designers want to
know if the mean coefficient of restitution
exceeds 0.82. The Test Hypotheses are
Restitution Coeff
0.8411
0.858
0.8042
0.8191
0.8532
0.873
0.8182
0.8483
0.8282
0.8125
0.8276
0.8359
0.875
0.7983
0.866
H 0 :   0.82
H 1 :   0.82
 The desired significance level is
that t 0.05,14  1.761
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
  0.05
so
13
P-value for a t-test

P-value is the smallest
level of significance for
which the null hypothesis
would be rejected
 Tail area beyond the
value of the test
statistic
 For a two sided test
this value is doubled
Drawing a sketch to clarify
what is being asked is often
very helpful….
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
14
t 0.5,  0.0

Percentage
Points
of a t-distribution
4-16
0.01  p  0.005

t0  2.72
t1 ,  t ,
  k  n 1
x  0
t0 
s/ n
L Berkley Davis
Copyright 2009
15
Type II error for a t-test
 The Type II Error for a t-test is the probability that
the Null Hypothesis is accepted when it is false. To
compare the Null Hypothesis to an Alternative
Hypothesis where the true mean is given by    0  
a quantity d (the number of standard deviations
between the two means)is calculated where for a
two sided test
 The quantity d , the required level of significance
and the number of samples n determine 
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10

16
Type II error for a t-test
 The equations for  have been integrated
numerically for selected values of  , d and n
and the results are in the Appendix
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
17
Type II error for a t-test
  0.05
  0.05

Two sided
  0.01
L Berkley Davis
Copyright 2009
n
n  const
One sided
18
Confidence Interval
Variance Unknown
L Berkley Davis
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MER301: Engineering Reliability
Lecture 10
19
Confidence Interval
Variance Unknown
H 0 :   0.82
H1 :   0.82
reject  H 0
  0.82608
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
20
Example 10.2
 Sulfur dioxide and nitrogen oxide are both
products of fossil fuel consumption. These
compounds can be carried long distances
and converted to acid before being
deposited in the form of “acid rain.”
 Data are obtained on the sulfur dioxide
concentration (in micrograms per cubic
meter) in the Adirondacks
 Determine the 95% confidence interval on
the mean sulfur dioxide concentration in
this forest
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
21
Inference on the Variance
of a Normal Population
L Berkley Davis
Copyright 2009
Inference on the Variance of a
Normal Population
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
23
The Chi-Squared Distribution
4-22
4-21
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
24
 
2
0


(n  1)  S 2

2
The Chi-Squared
Distribution
 02
L Berkley Davis
Copyright 2009
25
Inference on the Variance of a
Normal Population
 Hypothesis Tests on the variance can be two sided
or one sided. For the Two Sided Test the hypothesis
would be rejected if  02   2 / 2,n 1 or if  o2   12 / 2,n 1
where the hypothesis is given as
4-23
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
26
Hypothesis Testing on Variance
L Berkley Davis
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MER301: Engineering Reliability
Lecture 10
27
Confidence Limit on Variance
L Berkley Davis
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MER301: Engineering Reliability
Lecture 10
28
Example 10.3
 One random variable studied while designing the
front wheel drive half shaft of a new model
automobile is the displacement (in millimeters) of
the constant velocity (CV) joints. With the joint
angle fixed at 12o, twenty simulations were
conducted.
 Engineers claim that the standard deviation in the
displacement of the CV shaft is less than 1.5mm.
 Do these data support the contention of the
Engineers?
 Estimate the Confidence Interval on the standard
deviation for this data set
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
29
Example 10.3 Data
Displacement(mm)
6.2
4.2
4.6
4.2
3.5
3.7
4.1
2.6
1.4
3.2
4.8
4.4
1.9
1.1
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
2.5
1.5
4.9
1.3
3.7
3.9
30
L Berkley Davis
Copyright 2009
31
Many See Economy as Top Problem-
Population Proportion and Political Sampling





By ALAN FRAM,
AP
Posted: 2007-10-10 12:47:36
WASHINGTON (Oct. 10) - A growing number of Americans say the economy is the nation's top problem, with the less educated
among the most worried, an Associated Press-Ipsos poll showed Tuesday.
Yet even with a credit crunch and soft housing market, economic angst remains well behind war and domestic issues among the
public's chief concerns, according to survey results.
Given an open-ended opportunity to name the major problem facing the U.S., 15 percent volunteered the economy. That was six
percentage points more than named it when the AP-Ipsos poll last asked the question in July.
"They talk about a big surge in Iraq; well, there hasn't been a big surge over here," said Sadruddin El-Amin, 55, a truck driver in
Hanahan, South Carolina, who named the economy as the top problem. "The job market isn't getting any better, not for the working
class."
Twenty-two percent of those with a high school education or less named the economy
as the country's worst problem, compared to eight percent with college degrees. In
addition, 20 percent of minorities cited the economy as the top issue, compared to nine
percent who did so in July. There was no real difference between Republicans and Democrats, with just under a fifth
of each naming the economy as biggest worry.
Foreign affairs was considered the top problem by 42 percent, down from 49 percent in July.
Within that category, concern over the Iraq war and other conflicts was named most frequently - by 30 percent - and showed little
change since the summer, while fewer people chose immigration as the top issue. Democrats were nearly twice as likely as
Republicans to mention war as the primary concern.
Domestic issues were named by 33 percent in this month's poll, about the same as the 29 percent who cited them in July. That
included eight percent who named morality as the major problem, up from two percent in the earlier survey.
The poll was taken Oct. 1-3 and involved telephone interviews
with 499 adults. It had a margin of sampling error of plus or minus
4.4 percentage points
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
32
Inference on Population Proportion
 In political surveys(and many engineering/manufacturing
problems)there are “yes/no” answers and a fixed number
“n” of trials. Assuming constant probability, these can be
treated as binomial distribution problems
n!
P( X  x) 
p x (1  p) n  x
x!(n  x)!
  np
 2  np(1  p)
 The results can be analyzed using a normal
approximation if np>5 and n(1-p)>5
X  np
Z
np(1  p)
L Berkley Davis
Copyright 2009
33
Population Proportion and the Binomial Distribution
with 500<n<2500 and 0.4<p<0.6
  np
  np(1  p)
Values of Standard Deviation
n/p
0.4
0.5
0.6
500
11
11
11
1000
15
16
15
1600
20
20
20
2000
22
22
22
2500
24
25
24
 / np
for p=0.5
0.044
0.032
0.025
0.022
0.02
Values of X vs Z for p=0.5
X  np
Z
np(1  p)
L Berkley Davis
Copyright 2009
n/Z
-3
-2
-1
0
1
2
3
500 217 228 239 250 261 272 283
1000 452 468 484 500 516 532 548
1600 740 760 780 800 820 840 860
2000 934 956 978 1000 1022 1044 1066
2500 1175 1200 1225 1250 1275 1300 1325
MER301: Engineering Reliability
Lecture 10
34
Inference on Population Proportion
 In political surveys(and many engineering/manufacturing
problems)there are “yes/no” answers and a fixed number
“n” of trials. Assuming constant probability, these can be
treated as binomial distribution problems with the
unknown being p
X
p
n
 The Z term can be written as
X  np
n  ( X / n  p)
Z


np(1  p)
np(1  p)
L Berkley Davis
Copyright 2009
X /n p

p(1  p) / n
pˆ  po
po (1  po ) / n
35
Hypothesis Testing on a Binomial Proportion
or Z 
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
pˆ  po
po (1  po ) / n
36
Inference on Population Proportion
- Confidence Interval on a Binomial Proportion
pˆ  (1  pˆ )
0.50  (1  0.50)
pˆ  Z / 2 
 pˆ  1.96 
 pˆ  0.0438  pˆ  4.4%
n
499
L Berkley Davis
Copyright 2009
37
Inference on Population Proportion
- Choice of Sample Size
L Berkley Davis
Copyright 2009
38
Summary of topics
 Inference on the Mean of a Population,
Variance Unknown
 Confidence Interval,Variance Unknown
 Inference on the Variance of a Normal
Population
 Inference on Population Proportion
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 10
39