statistics sampling intervals for a single sample contents, figures, and exercises come from the...

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


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


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


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

);,...,,( 21 nXXXg

2

LC

);,...,,( 21 nXXXg

);,...,,( 21 nXXXg

)//()();,...,,( 21 nXXXXg n

UC

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 .


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


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 .


1

Example 8-4 Mercury Contamination◦ 53 measurements: 1.230, 0.490, …◦ , , , .◦ Find a 95% CI for .


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).


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


CDF of distributiont

The probability density function

is the number of degrees of freedom Mean : Variance : for

◦Percentage points



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


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


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


1x

nj /)5.0(

)()(5.0

jj zzZPn

j

2x nx )1(x )2(x )(nx

)1(x )2(x

)( jx

jz )( jx


Percent-percent plot

Example 8-5 Alloy Adhesion◦ The load at specimen failure: 19.8, 10.1, …◦ , , .◦ Find a 95% CI on .


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?


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


PDF of distribution2


CDF of distribution2

The probability density function

is the number of degrees of freedom Mean : Variance :

◦Percentage points



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


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


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.


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 .


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


PMF of binomial distribution

To construct the confidence interval on ,


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


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


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.


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?


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.


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)

statistics sampling intervals for a single sample contents, figures, and exercises come from the...

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