Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson...

Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 8-1

Business Statistics, 3eby Ken Black

Chapter 8

Statistical Inference: Estimation for

Single Populations

Discrete Distributions


Learning Objectives• Know the difference between point and interval

estimation.• Estimate a population mean from a sample mean for

large sample sizes.• Estimate a population mean from a sample mean for

small sample sizes.• Estimate a population proportion from a sample

proportion.• Estimate the population variance from a sample variance.• Estimate the minimum sample size necessary to achieve

given statistical goals.


Statistical Estimation

• Point estimate -- the single value of a statistic calculated from a sample

• Interval Estimate -- a range of values calculated from a sample statistic(s) and standardized statistics, such as the Z. – Selection of the standardized statistic is

determined by the sampling distribution.– Selection of critical values of the standardized

statistic is determined by the desired level of confidence.


Confidence Interval to Estimate when n is Large

• Point estimate• Interval Estimate

XX

n

X Zn

or

X Zn

X Zn


Distribution of Sample Meansfor (1-)% Confidence

X

Z0

2Z

2Z

2

2



X

Z0

2Z

2Z

2

2

.52

.52



X

Z0

2Z

2Z

2

2 1

2 1

2


Probability Interpretation of the Level of Confidence

Pr [ ]ob Xn

XnZ Z

2 21


Distribution of Sample Means for 95% Confidence

.4750 .4750

X

95%.025.025

Z1.96-1.96 0


95% Confidence Interval for

X and n 4 26 11 60. , . , .

X Zn

X Zn

4 26 1961160

4 26 1961160

4 26 0 28 4 26 0 283 98 4 54

. ..

. ..

. . . .. .


95% Confidence Intervals for

X

95%

XX

X

XX

X


95% Confidence Intervals for

X

95%

XX

X

XX

X

Is our interval,

3.98 4.54, in the

red?


Demonstration Problem 8.1

X Zn

X Zn

10 455 16457 744

10 455 16457 744

10 455 191 10 455 1918 545 12 365

. ..

. ..

. . . .. .

X and nZ

10 455 7 7 4490% 1645

. , . , ..

confidence

Pr [ . . ] .ob 8 545 12 365 0 90


Demonstration Problem 8.2

X Zn

N nN

X Zn

N nN

1 1

34 3 2 33850

800 50800 1

34 3 2 33850

800 50800 1

34 3 2 554 34 3 2 5543175 36 85

. . . .

. . . .. .

X N and nZ

34 3 8 800 5098% 2 33

. , , ..

= confidence


Confidence Interval to Estimate when n is Large and is Unknown

X ZSn

or

X ZSn

X ZSn

2

2 2


Z Values for Some of the More Common Levels of Confidence

90%

95%

98%

99%

Confidence Level Z Value

1.645

1.96

2.33

2.575


Estimating the Mean of a Normal Population: Small n and Unknown

• The population has a normal distribution.• The value of the population standard

deviation is unknown.• The sample size is small, n < 30.• Z distribution is not appropriate for these

conditions• t distribution is appropriate


The t Distribution

• Developed by British statistician, William Gosset

• A family of distributions -- a unique distribution for each value of its parameter, degrees of freedom (d.f.)

• Symmetric, Unimodal, Mean = 0, Flatter than a Z

• t formula t XSn


Comparison of Selected t Distributions to the Standard Normal

-3 -2 -1 0 1 2 3

Standard Normalt (d.f. = 25)t (d.f. = 5)t (d.f. = 1)


Table of Critical Values of t

df t0.100 t0.050 t0.025 t0.010 t0.0051 3.078 6.314 12.706 31.821 63.6562 1.886 2.920 4.303 6.965 9.9253 1.638 2.353 3.182 4.541 5.8414 1.533 2.132 2.776 3.747 4.6045 1.476 2.015 2.571 3.365 4.032

23 1.319 1.714 2.069 2.500 2.80724 1.318 1.711 2.064 2.492 2.79725 1.316 1.708 2.060 2.485 2.787

29 1.311 1.699 2.045 2.462 2.75630 1.310 1.697 2.042 2.457 2.750

40 1.303 1.684 2.021 2.423 2.70460 1.296 1.671 2.000 2.390 2.660

120 1.289 1.658 1.980 2.358 2.6171.282 1.645 1.960 2.327 2.576

t


Confidence Intervals for of a Normal Population: Small n and Unknown

X t Sn

or

X t Sn

X t Sn

df n

1


Solution for Demonstration Problem 8.3

X tSn

X tSn

2 14 3012129

142 14 3012

12914

2 14 104 2 14 104110 318

. ..

. ..

. . . .. .

X S n df n

t

214 129 14 1 13

21 99

20 005

3012005 13

. , . , ,.

.

.. ,



X tSn

X tSn

2 14 3012129

142 14 3012

12914

2 14 104 2 14 104110 318

. ..

. ..

. . . .. .

Pr [ . . ] .ob 110 318 0 99


Confidence Interval to Estimate the Population Proportion

:

ppqn

P ppqn

wherepq pPn

Z Z 2 2

1 = sample proportion= - = population proportion = sample size



. .( . )( . )

. .( . )( . )

. . . .. .

p Zpqn

P p Zpqn

P

PP

016 1645016 084

212016 1645

016 084212

016 0 04 016 0 04012 0 20

n X pXn

q pConfidence Z

212 3434212

016

1 1 016 0 8490% 1645

, , .

. ..

= -

Pr [ . . ] .ob P012 0 20 0 90


Population Variance• Variance is an inverse measure of the group’s

homogeneity.• Variance is an important indicator of total quality

in standardized products and services. Managers improve processes to reduce variance.

• Variance is a measure of financial risk. Variance of rates of return help managers assess financial and capital investment alternatives.

• Variability is a reality in global markets. Productivity, wages, and costs of living vary between regions and nations.


Estimating the Population Variance

• Population Parameter

• Estimator of

• formula for Single Variance

2

2

1SX Xn

22

21

1

n S

ndegrees of freedom = -


Confidence Interval for 2

n n

df n

S S

1 1

11

2

2

22

2

12

2

level of confidence


Selected 2 Distributions

df = 3

df = 5

df = 10

0


2 Table

df 0.975 0.950 0.100 0.050 0.0251 9.82068E-04 3.93219E-03 2.70554 3.84146 5.023902 0.0506357 0.102586 4.60518 5.99148 7.377783 0.2157949 0.351846 6.25139 7.81472 9.348404 0.484419 0.710724 7.77943 9.48773 11.143265 0.831209 1.145477 9.23635 11.07048 12.832496 1.237342 1.63538 10.6446 12.5916 14.44947 1.689864 2.16735 12.0170 14.0671 16.01288 2.179725 2.73263 13.3616 15.5073 17.53459 2.700389 3.32512 14.6837 16.9190 19.0228

10 3.24696 3.94030 15.9872 18.3070 20.4832

20 9.59077 10.8508 28.4120 31.4104 34.169621 10.28291 11.5913 29.6151 32.6706 35.478922 10.9823 12.3380 30.8133 33.9245 36.780723 11.6885 13.0905 32.0069 35.1725 38.075624 12.4011 13.8484 33.1962 36.4150 39.364125 13.1197 14.6114 34.3816 37.6525 40.6465

70 48.7575 51.7393 85.5270 90.5313 95.023180 57.1532 60.3915 96.5782 101.8795 106.628590 65.6466 69.1260 107.5650 113.1452 118.1359

100 74.2219 77.9294 118.4980 124.3421 129.5613

0 5 10 15 20

0.10

df = 5

9.23635


Two Table Values of 2

0 2 4 6 8 10 12 14 16 18 20

df = 7

.05

.05

.95

2.16735 14.0671

df 0.950 0.0501 3.93219E-03 3.841462 0.102586 5.991483 0.351846 7.814724 0.710724 9.487735 1.145477 11.070486 1.63538 12.59167 2.16735 14.06718 2.73263 15.50739 3.32512 16.9190

10 3.94030 18.3070

20 10.8508 31.410421 11.5913 32.670622 12.3380 33.924523 13.0905 35.172524 13.8484 36.415025 14.6114 37.6525


90% Confidence Interval for 2

n nS S

1 1

8 1 002212514 0671

8 1 00221252 16735

001101 007146

2

2

22

2

12

2

2

2

..

..

. .

2

2

212

2

05

2

12

2

112

2

95

2

0022125 8 1 7 10

14 0671

2 16735

S n df n

. , , , .

.

.

..

..

Pr [ . . ] .ob 0 001101 0 007146 0 902



n nS S

1 1

25 1 02539 3641

25 1 02512 4011

347 540 1103 176

2

2

22

2

12

2

2

2

(570, ).

(570, ).

, , ,

2

2

2052

2

025

2

12

2

1052

2

975

2

570 025 25 1 24 05

39 3641

12 4011

S n df n

, , , , .

.

.

..

..


Determining Sample Size when Estimating

• Z formula

• Error of Estimation (tolerable error)

• Estimated Sample Size

• Estimated

Z X

n

E X

nZ

EZ

E

2

2 2

2

2

2

14

range


Sample Size When Estimating Example

nZ

E

2

2 2

2

2 2

2

1645 41

4330 44

( . ) ( )

. or

EZ

1 490% 1645

,.

confidence



n ZE

2 2

2

2 2

2

196 6252

37 52 38

( . ) ( . )

. or

E rangeZ

estimated range

2 2595% 196

14

14

25 6 25

,.

: .

confidence


Determining Sample Size when Estimating P

• Z formula• Error of Estimation (tolerable

error)• Estimated Sample Size

Z p PP Q

n

E p P

n PQZE

2

2



nPQ

or

ZE

2

2

2

2

2 33003

0 40 0 60

1 447 7 1 448

( . ).

. .

, . ,

EConfidence Z

estimated PQ P

0 0398% 2 33

0 401 0 60

..

..


Determining Sample Size when Estimating P with No Prior Information

P

n

0

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Z = 1.96E = 0.05

nZE

2

2

14

P

0.5

0.4

0.3

0.2

0.1

PQ

0.25

0.24

0.21

0.16

0.09



nPQ

or

ZE

2

2

2

2

164505

0 50 050

270 6 271

( . ).

. .

.

EConfidence Z

with no prior of P use PQ P

0 0590% 1645

0501 050

.., .

.

estimate

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