L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 10 1 MER301: Engineering Reliability LECTURE 10: Chapter 4: Decision Making for

L Berkley DavisCopyright 2009

MER301: Engineering ReliabilityLecture 10

1

MER301: Engineering Reliability

LECTURE 10:

Chapter 4:Decision Making for a Single Sample, part 3



2

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


Inference on the Mean of a Population, Variance Unknown- the t-test



4

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 t-distribution 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

The t-distribution applies for small sample sizes, in fact for n greater than or equal to 2

nS

Xt

/0

0



5

Inference on the Mean of a Population, Variance Unknown



6

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

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

1 nk

Normal distribution

k

k

4-15


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 48.42 45.27 53.65 49.46 48.22 50.49

9 50.59 46.09 45.23 51.33 44.4 43.32 50.13 49.92 54.6210 52.33 51.91 48.34 48.01 49.36 47.92 44.84 42.68 50.4811 49.33 49.85 48.64 44.92 51.71 47.07 45.48 45.54 46.7112 46.64 46.43 50.55 49.54 46.18 51.91 42.72 49.65 47.6513 48.13 46.04 46.35 50.55 50.41 49.37 50.07 52.89 48.91

14 49.77 53.56 46.99 51.11 48.43 51.42 47.56 45.66 51.23

15 46.25 49.6 49.64 47.05 46.68 43.9 53.98 46.3 48.2616 47.3 56.51 51.76 50.64 52 48.56 49.63 47.25 44.34

16/

480s

xt

16/3

48x

z

Cumulative Dist

Z to z-dist t-dist48.3275 0.5023358 0.43667 0.65195 0.66882 0.733645

49.185625 0.8903234 1.58083 1.33168 0.94304 0.890171248.371875 0.7022192 0.49583 0.52957 0.68999 0.694607

48.66 0.783431 0.88 0.84245 0.81057 0.787997247.4775 0.8178353 -0.6967 -0.6389 0.24301 0.2703779

47.73375 0.940054 -0.355 -0.2832 0.36129 0.392096447.56125 0.9694005 -0.585 -0.4526 0.27927 0.3314294

47.623125 0.7279616 -0.5025 -0.5177 0.30766 0.309333348.8475 0.7370965 1.13 1.14978 0.87076 0.8582784

16/SSampleX 16

Comparison of normal (Z) and t-distributions

0t


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 48.42 45.27 53.65 49.46 48.22 50.49

9 50.59 46.09 45.23 51.33 44.4 43.32 50.13 49.92 54.6210 52.33 51.91 48.34 48.01 49.36 47.92 44.84 42.68 50.4811 49.33 49.85 48.64 44.92 51.71 47.07 45.48 45.54 46.7112 46.64 46.43 50.55 49.54 46.18 51.91 42.72 49.65 47.6513 48.13 46.04 46.35 50.55 50.41 49.37 50.07 52.89 48.91

14 49.77 53.56 46.99 51.11 48.43 51.42 47.56 45.66 51.23

15 46.25 49.6 49.64 47.05 46.68 43.9 53.98 46.3 48.2616 47.3 56.51 51.76 50.64 52 48.56 49.63 47.25 44.34

Cumulative Dist

Z to z-dist t-dist48.3275 0.5023358 0.43667 0.65195 0.66882 0.733645

49.185625 0.8903234 1.58083 1.33168 0.94304 0.890171248.371875 0.7022192 0.49583 0.52957 0.68999 0.694607

48.66 0.783431 0.88 0.84245 0.81057 0.787997247.4775 0.8178353 -0.6967 -0.6389 0.24301 0.2703779

47.73375 0.940054 -0.355 -0.2832 0.36129 0.392096447.56125 0.9694005 -0.585 -0.4526 0.27927 0.3314294

47.623125 0.7279616 -0.5025 -0.5177 0.30766 0.309333348.8475 0.7370965 1.13 1.14978 0.87076 0.8582784

16/SSampleX 16

Xbar z-dist t-dist49.19 0.943 0.8948.85 0.871 0.85848.66 0.811 0.78848.37 0.69 0.69548.33 0.669 0.734

48 0.5 0.547.73 0.361 0.39247.62 0.308 0.30947.56 0.279 0.33147.48 0.243 0.27

16/

480s

xt

16/3

48x

z

Comparison of Z and t Distributions

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

47 47.5 48 48.5 49 49.5

Sample Mean Xbar

cum

ula

tive

dis

trib

uti

on

phi(z)

t-dist

distzdistt

Comparison of normal (Z) and t-distributions



9

Percentage Points of a t-distribution

The t-distribution is symmetric about zero so that

and for all = k

,,1 tt

0.0,5.0 t

4-16



10

,,1 tt

1 nk

,t

0.0,5.0 t

ns

xt

/0

0



11

Hypotheses TestingVariance Unknown



12

Hypothesis Testing with the t-distribution

4-19



13

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

The desired significance level is so that

82.0:

82.0:

1

0

H

H

05.0761.114,05.0 t

Restitution Coeff0.8411

0.858

0.80420.81910.8532

0.873

0.81820.84830.82820.81250.82760.83590.875

0.7983

0.866



14

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 clarifywhat is being asked is often

very helpful….



15

,,1 tt

1 nk

72.20 t

0.0,5.0 t

ns

xt

/0

0

005.001.0 p

4-16



16

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 a quantity d (the number of standard deviations between the two means)is calculated where for a two sided test

The quantity , the required level of significance and the number of samples n determine

0

d



17

Type II error for a t-test The equations for have been integrated

numerically for selected values of and n and the results are in the Appendix

d,


Type II error for a t-test

nconstn

One sided

Two sided

05.0

01.0

05.0

18



19

Confidence IntervalVariance Unknown



20

Confidence IntervalVariance Unknown

82608.0

0

1

0

82.0:

82.0:

Hreject

H

H



21

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





23




24

The Chi-Squared Distribution

4-21

4-22


The Chi-Squared Distribution

25

2

220

)1(

Sn

20



26

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 or if

where the hypothesis is given as

21,2/

20 n 2

1,2/12

no

4-23



27

Hypothesis Testing on Variance



28

Confidence Limit on Variance



29

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



30

Example 10.3 Data

Displacement(mm)6.2 4.2 2.54.6 4.2 1.53.5 3.7 4.94.1 2.6 1.31.4 3.2 3.74.8 4.4 3.91.9 1.1

L Berkley DavisCopyright 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


32



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

The results can be analyzed using a normal approximation if np>5 and n(1-p)>5

)1( pnp

npXZ

np )1(2 pnp

xnx ppxnx

nxXP

)1(

)!(!

!)(

33


Population Proportion and the Binomial Distribution with 500<n<2500 and 0.4<p<0.6


34

Values of X vs Z for p=0.5

n/Z -3 -2 -1 0 1 2 3500 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

Values of Standard Deviationn/p 0.4 0.5 0.6500 11 11 111000 15 16 151600 20 20 202000 22 22 222500 24 25 24

for p=0.5

0.044

0.032

0.025

0.022

0.02

np/np

)1( pnp

)1( pnp

npXZ



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

The Z term can be written as

npp

pp

npp

pnX

pnp

pnXn

pnp

npXZ

oo

o

/)1(

ˆ

/)1(

/

)1(

)/(

)1(

35

n

Xp


Hypothesis Testing on a Binomial Proportion


36

npp

ppZ

oo

o

/)1(

ˆ

or


Inference on Population Proportion - Confidence Interval on a Binomial Proportion

37

%4.4ˆ0438.0ˆ499

)50.01(50.096.1ˆ

)ˆ1(ˆˆ 2/

ppp

n

ppZp


Inference on Population Proportion - Choice of Sample Size

38



39

Summary of topics Inference on the Mean of a Population,

Variance Unknown

Confidence Interval,Variance Unknown



Documents

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 10 1 MER301: Engineering Reliability LECTURE 10: Chapter 4: Decision Making for