© 2002 thomson / south-western slide 8-1 chapter 8 estimation with single samples

© 2002 Thomson / South-Western Slide 8-1

Chapter 8

Estimation with Single

Samples


Learning ObjectivesLearning 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 minimum sample size necessary

to achieve given statistical goals.


Statistical EstimationStatistical 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

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


Z Scores for Confidence Intervals in Relation to

Z Scores for Confidence Intervals in Relation to

X

Z0

2Z

2Z

2

2

.52

.52




X

Z0

2Z

2Z

2

2 1

2 1

2


Probability Interpretation of the Level of ConfidenceProbability Interpretation of the Level of Confidence

2 2

Pr ob[ ] 1X Xn nZ Z


Distribution of Sample Means for 95% Confidence

Distribution of Sample Means for 95% Confidence

.4750 .4750

X

95%.025.025

Z1.96-1.96 0


Example: 95% Confidence Interval for

Example: 95% Confidence Interval for

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

X Zn

X Zn

4 26 19611

604 26 196

11

604 26 0 28 4 26 0 28

3 98 4 54

. ..

. ..

. . . .

. .


95% Confidence Intervals for 95% Confidence Intervals for

X

95%

XX

X

X

X

X


95% Confidence Intervals for 95% Confidence Intervals for

X

95%

XX

X

X

X

X

Is our interval,

3.98 4.54, in the

red?


Demonstration Problem 8.1Demonstration Problem 8.1

X Zn

X Zn

10 455 16457 7

4410 455 1645

7 7

4410 455 191 10 455 191

8 545 12 365

. ..

. ..

. . . .

. .

X and n

Z

10 455 7 7 44

90% 1645

. , . , .

.

confidence

Pr [ . . ] .ob 8545 12 365 0 90


Demonstration Problem 8.2Demonstration Problem 8.2

1 1

8 800 50 8 800 5034.3 2.33 34.3 2.33

800 1 800 150 5034.3 2.55 34.3 2.55

31.75 36.85

N n N nX Z X Z

N Nn n

X N and n

Z

34 3 8 800 50

98% 2 33

. , , .

.

=

confidence


Confidence Interval to Estimate when n is Large and is UnknownConfidence Interval to Estimate when n is Large and is Unknown

X ZS

nor

X ZS

nX Z

S

n

2

2 2


Z Values for Some of the More Common Levels of ConfidenceZ 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

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 DistributionThe t Distribution

• 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 formulatXSn


Comparison of Selected t Distributions to the Standard Normal

Comparison of Selected t Distributions to the Standard Normal

-3 -2 -1 0 1 2 3

Standard Normal

t (d.f. = 25)

t (d.f. = 5)

t (d.f. = 1)


Table of Critical Values of tTable 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

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

X tS

nor

X tS

nX t

S

ndf n

1


Solution for Demonstration Problem 8.3

X tS

nX t

S

n

2 14 3 012129

142 14 3 012

129

142 14 104 2 14 104

110 318

. ..

. ..

. . . .

. .

X S n df n

t

214 129 14 1 13

2

1 99

20 005

3012005 13

. , . , ,

..

.. ,


Solution for Demonstration Problem 8.3Solution for Demonstration Problem 8.3

X tS

nX t

S

n

214 3012129

14214 3012

129

14214 104 214 104

110 318

. ..

. ..

. . . .

. .

Pr [ . . ] .ob 110 318 0 99


Confidence Interval to Estimate the Population Proportion

Confidence Interval to Estimate the Population Proportion

:

ppq

nP p

pq

nwhere

p

q p

P

n

Z Z 2 2

1

= sample proportion

= -

= population proportion

= sample size


Solution for Demonstration Problem 8.5Solution for Demonstration Problem 8.5

. .( . )( . )

. .( . )( . )

. . . .

. .

p Zpq

nP p Z

pq

n

P

P

P

016 1645016 0 84

212016 1645

016 0 84

212016 0 04 016 0 04

012 0 20

n X pX

nq p

Confidence Z

212 3434

212016

1 1 016 0 84

90% 1645

, , .

. .

.

= -

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


Determining Sample Size when Estimating

Determining Sample Size when Estimating

• Z formula

• Error of Estimation (tolerable error)

• Estimated Sample Size

• Estimated

ZX

n

E X

nZ

EZ

E

2

2 2

2

2

2

1

4range


Example: Sample Size when Estimating

nZ

E

2

2 2

2

2 2

2

1645 41

4330 44

( . ) ( )

. or

E

Z

1 4

90% 1645

,

.

confidence



n ZE

2 2

2

2 2

2

196 6252

37 52 38

( . ) ( . )

. or

E range

Z

estimated range

2 25

95% 196

1

4

1

425 6 25

,

.

: .

confidence


Determining Sample Size when Estimating P

Determining Sample Size when Estimating P

• Z formula

• Error of Estimation (tolerable error)

• Estimated Sample Size

Zp P

P Qn

E p P

nPQZE

2

2



nPQ

or

ZE

2

2

2

2

2 33003

0 40 0 60

1 447 7 1 448

( . ).

. .

, . ,

E

Confidence Z

estimated P

Q P

0 03

98% 2 33

0 40

1 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 0 50

270 6 271

( . ).

. .

.

E

Confidence Z

with no prior of P use P

Q P

0 05

90% 1645

050

1 050

.

.

, .

.

estimate

© 2002 thomson / south-western slide 8-1 chapter 8 estimation with single samples

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