statistics sampling intervals for a single sample contents, figures, and exercises come from the...
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StatisticsStatisticsSampling 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.
If , ,…, are normally and independently distributed with unknown mean and known variance has a standard
normal distribution
Confidence Interval on the Confidence Interval on the Mean of a Normal Mean of a Normal Distribution, Variance KnownDistribution, Variance Known
1X 2X nX
n
XZ
/
2
1/
2/2/ zn
XzP
12/2/
nzX
nzXP
Confidence interval on the mean, variance known
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.
nzx
nzx
2/2/
From , we have
If is used as an estimate of , we can be confident that the error will not exceed a specified amount when the sample size is
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.
nzx
nzx
2/2/
nzx
2/||
2
2/
||
x
zn
x
)%1(100 || xE
2
2/
E
zn
One-sided confidence bounds on the mean, variance known◦A upper-confidence bound
for is
◦A lower-confidence bound for is
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.
nzxu
)%1(100
ln
zx
)%1(100
General method to derive a confidence interval ◦We find a statistic
that 1. depends on both
the sample and 2. The probability distribution of
does not depend on and any other unknown parameter
For example,
◦Find constants and so that
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LC
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);,...,,( 21 nXXXg
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1]);,...,,([ 21 UnL CXXXgCP
1)],...,,(),...,,([ 2121 nn XXXUXXXLP
Large-sample confidence interval on the mean When is large, the quantity
has an approximate standard normal distribution. Consequently,
is a large-sample confidence interval for , with confidence level of approximately .
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.
n
szx
n
szx 2/2/
nnS
X
/
)%1(100
Large-sample approximate confidence interval If the quantity
has an approximate standard normal distribution. Consequently,
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.
ˆ2/ˆ2/
ˆˆ zz
ˆ
ˆ
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 . 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.
1
Example 8-4 Mercury Contamination◦ 53 measurements: 1.230, 0.490, …◦ , , , .◦ 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.
53n 5250.0x 3486.0 96.1025.0 z
Exercise 8-14◦ The life in hours of a 75-watt light bulb is known to
be normally distributed with hours. A random sample of 20 bulbs has a mean life of 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.
25
1014x
DistributionLet , ,…, are normally
and independently distributed with unknown mean and unknown variance . The random variable
has a distribution with degrees of freedom.
Confidence Interval on the Confidence Interval on the Mean of a Normal Mean of a Normal Distribution, Variance Distribution, Variance UnknownUnknown
1X 2X nX
nS
XT
/
2
t
t
1n
From Wikipedia, http://www.wikipedia.org.
PDF of distributiont
From Wikipedia, http://www.wikipedia.org.
CDF of distributiont
The probability density function
is the number of degrees of freedom Mean : Variance : for
◦Percentage points
is a large-sample confidence interval for , with confidence level of approximately .
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.
kt ,
xkxkk
kxf
k ,
]1)/[(
1
)2/(
]2/)1[()(
2/)1(2
k
)( ,ktTP
nn tt ,,1
t
0
)2/( kk 2k
confidence interval 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.
t
1)( 1,2/1,2/ nn tTtP
1)
/( 1,2/1,2/ nn t
nS
XtP
1)( 1,2/1,2/n
StX
n
StXP nn
Confidence interval on the mean, variance unknown◦ If and are the mean and standard deviation of a random
sample from a normal distribution with unknown variance , a confidence interval on is given by
◦ where is the upper percentage point of the distribution with 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.
x
n
StX
n
StX nn 1,2/1,2/
s
2 )%1(100
1,2/ nt 2/100t 1n
Normal probability plot◦ The sample , ,…, is arranged as , ,…, ,where is
the smallest observation, is the second-smallest observation, and so forth.
◦ The ordered observations are then plotted against their observed cumulative frequency on the appropriate probability paper.
◦ Or, plot the standardized normal scores against , where
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.
1x
nj /)5.0(
)()(5.0
jj zzZPn
j
2x nx )1(x )2(x )(nx
)1(x )2(x
)( jx
jz )( jx
From Wikipedia, http://www.wikipedia.org.
Percent-percent plot
Example 8-5 Alloy Adhesion◦ The load at specimen failure: 19.8, 10.1, …◦ , , .◦ 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.
71.13x55.3s 22n
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.
DistributionLet , ,…, are normally
and independently distributed mean and variance , and let be the sample variance. The random variable
has a chi-square distribution with degrees of freedom.
Confidence Interval on the Confidence Interval on the Variance and Standard Variance and Standard Deviation of a Normal Deviation of a Normal DistributionDistribution
1X 2X nX
2
22 )1(
Sn
X
2
2
1n
2S
2
From Wikipedia, http://www.wikipedia.org.
PDF of distribution2
From Wikipedia, http://www.wikipedia.org.
CDF of distribution2
The probability density function
is the number of degrees of freedom Mean : Variance :
◦Percentage points
is a large-sample confidence interval for , with confidence level of approximately .
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.
2,k
0 x,)2/(2
1)( 2/1)2/(
2/
xk
kex
kxf
k
2,
)()( 2,
2
k
duufXP k
k2
2
k
◦ Since
◦ is chi-square with degrees of freedom, we have
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.
1n
2
22 )1(
Sn
X
1)( 21,2/
221,2/1 nn XP
1))1(
( 21,2/2
22
1,2/1 nn
SnP
1))1()1(
(2
1,2/1
22
21,2/
2
nn
SnSnP
Confidence interval on the variance◦ If is the sample variance from a random sample of observations
from a normal distribution with unknown variance , then a confidence interval on is
◦ Where and are the upper and lower percentage points of the chi-square distribution with
◦ 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.
2s
21,2/1
22
21,2/
2 )1()1(
nn
SnSn
1n
n
2 )%1(100 2
21,2/ n
21,2/1 n
One-sided confidence bounds on the variance◦ The lower and upper confidence bounds on
are
◦ 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.
21,1
222
21,
2 )1( and
)1(
nn
SnSn
)%1(100 2
Example 8-6 Detergent Filling◦ , .◦ Find a 95% upper confidence bound on and .
Exercise 8-44 ◦ A rivet is to be inserted into a hole. A random sample of
parts is selected, and the hole diameter is measured. The sample standard deviation of the hole diameter measurements is millimeters. Construct a 99% lower confidence bound 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.
25.132 s 20n
15n
008.0s2
Normal approximation for a binomial proportionIf is large, the distribution of
is approximately standard normal.
Large-Sample Confidence Large-Sample Confidence Interval for a population Interval for a population proportionproportion
npp
pp
pnp
npXZ
)1(
ˆ
)1(
n
From Wikipedia, http://www.wikipedia.org.
PMF of binomial distribution
To construct the confidence interval 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.
1)( 2/2/ zZzP
p
1))1(
ˆ( 2/2/ z
npp
ppzP
1
)1(ˆ
)1(ˆ 2/2/ n
ppzpp
n
ppzpP
1
)ˆ1(ˆˆ
)ˆ1(ˆˆ 2/2/ n
ppzpp
n
ppzpP
◦Approximate confidence interval on a binomial proportion If is the proportion of observations in a random sample of size
that belongs to a class of interest, an approximate confidence interval on the proportion of the population that belongs to this class is
where is the upper percentage of the standard normal distribution.
Required: and
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.
p̂
n
ppzpp
n
ppzp
)ˆ1(ˆˆ
)ˆ1(ˆˆ 2/2/
n)%1(100
p
2/z 2/
5np 5)1( pn
◦Sample size for a specified error on a binomial proportion Set Then
Or
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.
nppzppE /)1(|ˆ| 2/
)1(2
2/ ppE
zn
)25.0(2
2/
E
zn
◦Approximate one-sided confidence bounds on a binomial proportion The approximate lower and upper
confidence bounds are
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.
n
ppzppp
n
ppzp
)ˆ1(ˆˆ and
)ˆ1(ˆˆ
)%1(100
◦Example 8-7 Crankshaft Bearings , , and Find a 95% two-sided confidence interval for .
◦Example 8-8 Crankshaft Bearings How large a sample is required if we want to be 95% confident that the error in
using to estimate 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.
85n 10x 12.085/10/ˆ nxp
p
p̂ p
◦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.
is a single future observation
Then
has a standard normal distribution and
Has a distribution with degrees of freedom.
Tolerance and Prediction Tolerance and Prediction IntervalsIntervals
n
XXZ n
11
1
0)( 1 XXE n
1nX
1nt
)1
1()( 22
21 nnXXV n
nS
XXT n
11
1
Prediction intervalA prediction interval (PI) on a
single future observation from a normal distribution is given by
nstxX
nstx nnn
11
11 1,2/11,2/
)%1(100
Tolerance intervalA tolerance interval for capturing at least
of the values in a normal distribution with confidence level is
where 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.
ksxksx ,
)%1(100
%
k
Example 8-9 Alloy Adhesion , , andFind a 95% prediction interval on the load
at failure for a new specimen.
Example 8-10 Alloy AdhesionFind a tolerance interval for the load at
failure that includes 90% of the values in the population with 95% confidence.
22n 71.13x 55.3s
Exercise 8-39(a)