Download Chapter 8

Statistics
Sampling Intervals for a Single Sample
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
Confidence Interval on the Mean of
a Normal Distribution,Variance
Known

If X1 , X 2 ,…, X n are normally and
independently distributed with unknown
mean  and known variance  2
X 
Z


has a standard normal
/ n
distribution


X 
P  z / 2 
 z / 2   1  
/ n



 

P  X  z / 2
   X  z / 2
  1
n
n


Confidence interval on the mean, variance
known
x  z / 2

n
   x  z / 2

n
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

From
x  z / 2

n
   x  z / 2
| x   | z / 2

n
, we have
n
 z / 2 

n  
| x   |


2
If x is used as an estimate of  , we can
be 100(1   )% confident that the error | x   |
will not exceed a specified amount E
when the sample size is
z  
n    /2 
 E 
2
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

One-sided confidence bounds on the
mean, variance known
◦ A 100(1   )% upper-confidence bound for 

is
 u  xz

n
◦ A 100(1   )% lower-confidence bound for 
is

x  z
n
l 
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

 method to derive a confidence
General
interval
◦ We find a statistic g ( X1 , X 2 ,..., X n ; ) that
 1. g ( X 1 , X 2 ,..., X n ; ) depends on both the sample
and 
g ( X 1 , X 2 ,..., X n ; )
 2. The probability distribution
 2 of
does not depend on  and any other unknown
parameter
 For example, g ( X1 , X 2 ,..., X n ; )  ( X   ) /( / n )
◦ Find constants CL and CU so that
P[CL  g ( X 1 , X 2 ,..., X n ; )  CU ]  1  
P[ L( X 1 , X 2 ,..., X n )    U ( X 1 , X 2 ,..., X n )]  1  

Large-sample confidence interval on the
mean
 When n is large, the quantity
X 
S/ n
 has an approximate standard normal distribution.
Consequently,
x  z / 2
s
s
   x  z / 2
n
n
 is a large-sample confidence interval for  , with
confidence level of approximately 100(1   )% .
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Large-sample approximate confidence
interval
 If the quantity
ˆ 

 ˆ
 has an approximate standard normal distribution.
Consequently,
ˆ  z / 2 ˆ    ˆ  z / 2 ˆ
 is a large-sample approximate confidence interval
for 
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Example 8-1 Metallic Material Transition
◦ Ten measurements: 64.1, 64.7, 64.5, 64.6, 64.5, 64.3,
64.6, 64.8, 64.2, 64.3
◦ Assume it is a normal distribution with   1 . Find a
95% CI for  .

Example 8-2 Metallic Material Transition
◦ Determine how many specimens must be tested to
ensure that the 95% CI for  has a length of at most
1.0.

Example 8-3 One-Sided Confidence
Bound
◦ Determine a lower, one-sided 95% CI for  .
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Example 8-4 Mercury Contamination
◦ 53 measurements: 1.230, 0.490, …
◦ n  53 , x  0.5250 ,   0.3486 , z0.025  1.96 .
◦ Find a 95% CI for  .

Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Exercise 8-14
◦ The life in hours of a 75-watt light bulb is known to
be normally distributed with   25 hours. A random
sample of 20 bulbshas a mean life of x  1014 hours.
◦ (a) Construct a 95% two-sided confidence interval on
the mean life.
◦ (b) Construct a 95% lower-confidence bound on the
mean life. Compare the lower bound of this

confidence interval with the one in part (a).

Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
Confidence Interval on the Mean of
a Normal Distribution,Variance
Unknown
Distribution
 Let X1 , X 2 ,…, X n are normally and
independently distributed with
unknown mean  and unknown
variance  2 . The random variable
 t


X 
T
S/ n
has a t distribution with
of freedom.
n 1
degrees

PDF of
t
distribution
From Wikipedia, http://www.wikipedia.org.

CDF of
t
distribution
From Wikipedia, http://www.wikipedia.org.

The
t
f ( x) 
probability density function
[( k  1) / 2]
1
 2
,   x  
( k 1) / 2
k (k / 2) [( x / k )  1]
 k is the number of degrees of freedom
 Mean : 0
 Variance : k /( k  2) for k  2
◦ Percentage points
t , k
P(T  t ,k )  

t1 ,n  t ,n
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
confidence interval on
 t

P(t / 2,n 1  T  t / 2,n 1 )  1  
P(t / 2,n 1 
P( X  t / 2,n 1
X 
 t / 2,n 1 )  1  
S/ n
S
S
   X  t / 2,n 1
)  1
n
n

Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Confidence interval on the mean, variance
unknown
◦ If x and s are the mean and standard deviation of
a random sample from a normal distribution with
unknown variance  2 , a 100(1   )% confidence
interval on  is given by
S
S
X  t / 2,n 1
   X  t / 2,n 1
n
n
◦ where t / 2 ,n 1 is the upper 100 / 2 percentage
point of the t distribution with n 1 degrees of
freedom
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Normal probability plot
◦ The sample x1 , x2 ,…, xn is arranged as x(1) , x( 2) ,…, x(n )
,where x(1) is the smallest observation, x( 2) is the
second-smallest observation, and so forth.
◦ The ordered observations x( j ) are then plotted
against their observed cumulative frequency ( j  0.5) / n
on the appropriate probability paper.
◦ Or, plot the standardized normal scores z j against x( j )
, where
j  0.5
 P( Z  z j )  ( z j )
n
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Percent-percent plot
From Wikipedia, http://www.wikipedia.org.

Example 8-5 Alloy Adhesion
◦ The load at specimen failure: 19.8, 10.1, …
◦ x  13.71 , s  3.55 , n  22 .
◦ Find a 95% CI on  .
Contents, figures,and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Exercise 8-41
◦ An article in Nuclear Engineering International
(February 1988, p. 33) describes several
characteristics of fuel rods used in a reactor owned
by an electric utility in Norway. Measurements on the
percentage of enrichment of 12 rods were reported
as follows: 2.94, 3.00, 2.90, 2.75, 3.00, 2.95, 2.90, 2.75,
2.95, 2.82, 2.81, 3.05.
◦ (a) Use a normal probability plot to check the
normality assumption.
◦ (b) Find a 99% two-sided confidence interval on the
mean percentage of enrichment. Are you comfortable
with the statement that the mean percentage of
enrichment is 2.95%? Why?
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
Confidence Interval on the Variance
and Standard Deviation of a Normal
Distribution
2


Distribution
 Let X1 , X 2 ,…, X n are normally and
independently distributed mean  and
variance  2 , and let S 2 be the sample
variance. The random variable


X 
2
(n  1) S 2
2
has a chi-square  2 distribution with
degrees of freedom.
n 1

PDF of  2 distribution
From Wikipedia, http://www.wikipedia.org.

CDF of  2 distribution
From Wikipedia, http://www.wikipedia.org.
2

 The
probability density function
1
f ( x)  k / 2
x ( k / 2)1e  x / 2 , x  0
2 (k / 2)
 k is the number of degrees of freedom
 Mean : k
 Variance : 2k
◦ Percentage points
2,k
P( X   ,k )  
2

2

2 ,k
f (u)du  
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
◦ Since
X2 
(n  1) S 2
2
◦ is chi-square with n 1 degrees of freedom, we have
P( 12 / 2,n1  X 2  2 / 2,n1 )  1  

P( 
2
1 / 2 , n 1
P(


(n  1) S 2
 / 2,n 1
2
(n  1) S 2
 2 / 2,n 1 )  1  
2
 2 
(n  1) S 2

2
1 / 2 , n 1
)  1
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Confidence interval on the variance
◦ If s 2 is the sample variance from a random sample of n
observations from a normal distribution with
unknown variance  2 , then a 100(1   )%
confidence interval on  2 is
2
(n  1) S 2
(
n

1
)
S
2


 2
2
 / 2,n 1
1 / 2,n 1
2
2


◦ Where  / 2,n1 and 1 / 2,n1 are the upper and lower
percentage points of the chi-square distribution with
◦ n 1 degrees of freedom, respectively. A confidence
interval for  has lower and upper limits that are the
square roots of the corresponding limits in the above
equation
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

One-sided confidence bounds on the
variance
◦ The 100(1   )% lower and upper confidence
bounds on  2 are
(n  1) S 2
 ,n 1
2

2
and  
2
(n  1) S 2
12 ,n 1
◦ respectively.

Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.

Example 8-6 Detergent Filling
◦ s 2  13.5 , n  20 .
◦ Find a 95% upper confidence bound on  2 and  .

Exercise 8-44
◦ A rivet is to be inserted into a hole. A random sample
of n  15 parts is selected, and the hole diameter is
measured. The sample standard deviation of the hole
diameter measurements is s  0.008 millimeters.
Construct a 99% lower confidence bound for  2 .

Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
Large-Sample Confidence Interval
for a population proportion

Normal approximation for a binomial
proportion
 If n is large, the distribution of
Z

X  np

np(1  p)
pˆ  p
p (1  p )
n
is approximately standard normal.

PMF of binomial distribution
From Wikipedia, http://www.wikipedia.org.
 To construct the confidence interval on p ,
P( z / 2  Z  z / 2 )  1  



P( z / 2 
pˆ  p
 z / 2 )  1  
p(1  p)
n

P pˆ  z / 2

p(1  p)
 p  pˆ  z / 2
n
p (1  p) 
  1

n


P pˆ  z / 2

pˆ (1  pˆ )
 p  pˆ  z / 2
n
pˆ (1  pˆ ) 
  1

n

Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
◦ Approximate confidence interval on a
binomial proportion
 If p̂ is the proportion of observations in a random
sample of size n that belongs to a class of interest,
an approximate 100(1   )% confidence interval on
the proportion p of the population that belongs to
this class is
pˆ  z / 2
pˆ (1  pˆ )
 p  pˆ  z / 2
n
pˆ (1  pˆ )
n
 where z / 2 is the upper  / 2 percentage of the
standard normal distribution.
 Required: np  5 and n(1  p)  5
Contents, figures,and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
◦ Sample size for a specified error on a binomial
proportion
 Set E | p  pˆ | z / 2 p(1  p) / n
 Then
2
z 
n    / 2  p (1  p )
 E 
 Or
2
z 
n    / 2  (0.25)
 E 
Contents, figures,and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
◦ Approximate one-sided confidence bounds on
a binomial proportion
 The approximate 100(1   )% lower and upper
confidence bounds are
pˆ  z
pˆ (1  pˆ )
 p and p  pˆ  z
n
pˆ (1  pˆ )
n
 respectively.



Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
◦ Example 8-7 Crankshaft Bearings
 n  85 , x  10 , and pˆ  x / n  10 / 85  0.12
 Find a 95% two-sided confidence interval for p .
◦ Example 8-8 Crankshaft Bearings
 How large a sample is required if we want to be
95% confident that the error in using p̂ to estimate p
is less than 0.05?
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
◦ Exercise 8-53
 The fraction of defective integrated circuits
produced in a photolithography process is being
studied. A random sample of 350 circuits is tested,
revealing 15 defectives.
 (a) Calculate a 95% two-sided CI on the fraction of
defective circuits produced by this particular tool.
 (b) Calculate a 95% upper confidence bound on the
fraction of defective circuits.
Contents, figures, and exercises come from the textbook: Applied Statistics and Probability
for Engineers, 5th Edition, by Douglas C. Montgomery, John Wiley & Sons, Inc., 2011.
Tolerance and Prediction Intervals
 X n1 is a single future observation
E ( X n 1  X )      0
2
1
V ( X n 1  X )   
  (1  )
n
n
X n1  X
Z

 Then
1
 1
n
 has a standard normal distribution and
2
2
X n1  X
T
1
S 1
n
 Has a t distribution with n  1 degrees of freedom.

Prediction interval

A 100(1   )% prediction interval (PI) on a single
future observation from a normal distribution is
given by
1
1
x  t / 2,n1s 1   X n1  x  t / 2,n1s 1 
n
n

Tolerance interval

A tolerance interval for capturing at least  % of
the values in a normal distribution with confidence
level 100(1   )% is
x  ks, x  ks

where k is a tolerance interval factor found in
Appendix Tabel XII.Values are given for  = 90%,
95%, and 99% and for 90%, 95%, and 99% confidence.

Example 8-9 Alloy Adhesion
n  22 , x  13.,71 and s  3.55
 Find a 95% prediction interval on the load at failure
for a new specimen.


Example 8-10 Alloy Adhesion

Find a tolerance interval for the load at failure that
includes 90% of the values in the population with
95% confidence.

Exercise 8-77

Consider the rainfall data in Exercise 8-33.
Compute a 95% tolerance interval that has
confidence level 95%. Compare the length of the
tolerance interval with the length of the 95% CI on
the population mean. Discuss the difference in
interpretation of these two intervals.