Lesoon. 4 - 1

Statistics for Management

Confidence Interval Estimation

Lesoon. 4 - 2

Lesson Topics

•Confidence Interval Estimation for the Mean

(Known)•Confidence Interval Estimation for the Mean

(Unknown)•Confidence Interval Estimation for the

Proportion

•The Situation of Finite Populations

•Sample Size Estimation

Lesoon. 4 - 3

Mean, , is unknown

Population Random SampleI am 95%

confident that is between 40 &

60.

Mean X = 50

Estimation Process

Sample

Lesoon. 4 - 4

Estimate PopulationParameter...

with SampleStatistic

Mean

Proportion p ps

Variance s2

Population Parameters Estimated

2

Difference - 1 2

x - x 1 2

X_

__

Lesoon. 4 - 5

• Provides Range of Values Based on Observations from 1 Sample

• Gives Information about Closeness to Unknown Population Parameter

• Stated in terms of Probability Never 100% Sure

Confidence Interval Estimation

Lesoon. 4 - 6

Confidence Interval Sample Statistic

Confidence Limit (Lower)

Confidence Limit (Upper)

A Probability That the Population Parameter Falls Somewhere Within the Interval.

Elements of Confidence Interval Estimation

Lesoon. 4 - 7

Parameter = Statistic ± Its Error

© 1984-1994 T/Maker Co.

Confidence Limits for Population Mean

X Error

= Error = X

XX

XZ

xZ

XZX

Error

Error

X

Lesoon. 4 - 8

90% Samples

95% Samples

x_

Confidence Intervals

xx .. 64516451

xx 96.196.1

xx .. 582582 99% Samples

nZXZX X

X_

Lesoon. 4 - 9

• Probability that the unknown

population parameter falls within the

interval

• Denoted (1 - ) % = level of confidence

e.g. 90%, 95%, 99%

Is Probability That the Parameter Is Not Within the Interval

Level of Confidence

Lesoon. 4 - 10

Confidence Intervals

Intervals Extend from

(1 - ) % of Intervals Contain .

% Do Not.

1 - /2/2

X_

x_

Intervals & Level of Confidence

Sampling Distribution of

the Mean

toXZX

XZX

X

Lesoon. 4 - 11

• Data Variation

measured by

• Sample Size

• Level of Confidence

(1 - )

Intervals Extend from

© 1984-1994 T/Maker Co.

Factors Affecting Interval Width

X - Z to X + Z xx

n/XX

Lesoon. 4 - 12

Mean

Unknown

ConfidenceIntervals

Proportion

FinitePopulation Known

Confidence Interval Estimates

Lesoon. 4 - 13

• Assumptions Population Standard Deviation Is Known Population Is Normally Distributed If Not Normal, use large samples

• Confidence Interval Estimate

Confidence Intervals (Known)

nZX /

2

n

ZX /

2

Lesoon. 4 - 14

Mean

Unknown

ConfidenceIntervals

Proportion

FinitePopulation Known

Confidence Interval Estimates

Lesoon. 4 - 15

• Assumptions Population Standard Deviation Is Unknown Population Must Be Normally Distributed

• Use Student’s t Distribution

• Confidence Interval Estimate

Confidence Intervals (Unknown)

n

StX n,/ 12

n

StX n,/ 12

Lesoon. 4 - 16

Zt

0

t (df = 5)

Standard Normal

t (df = 13)Bell-Shaped

Symmetric

‘Fatter’ Tails

Student’s t Distribution

Lesoon. 4 - 17

• Number of Observations that Are Free to Vary After Sample Mean Has Been Calculated

• Example Mean of 3 Numbers Is 2

X1 = 1 (or Any Number)

X2 = 2 (or Any Number)

X3 = 3 (Cannot Vary)

Mean = 2

degrees of freedom = n -1 = 3 -1= 2

Degrees of Freedom (df)

Lesoon. 4 - 18

Upper Tail Area

df .25 .10 .05

1 1.000 3.078 6.314

2 0.817 1.886 2.920

3 0.765 1.638 2.353

t0

Assume: n = 3 df = n - 1 = 2

= .10 /2 =.05

2.920t Values

/ 2

.05

Student’s t Table

Lesoon. 4 - 19

A random sample of n = 25 has = 50 and

s = 8. Set up a 95% confidence interval estimate for .

. .46 69 53 30

X

Example: Interval EstimationUnknown

n

StX n,/ 12

n

StX n,/ 12

25

80639250 . 25

80639250 .

Lesoon. 4 - 20

Mean

Unknown

ConfidenceIntervals

Proportion

FinitePopulation Known

Confidence Interval Estimates

Lesoon. 4 - 21

• Assumptions Sample Is Large Relative to Population

n / N > .05

• Use Finite Population Correction Factor

• Confidence Interval (Mean, X Unknown)

X

Estimation for Finite Populations

n

StX n,/ 12

n

StX n,/ 12

1

N

nN

1

N

nN

Lesoon. 4 - 22

Mean

Unknown

ConfidenceIntervals

Proportion

FinitePopulation Known

Confidence Interval Estimates

Lesoon. 4 - 23

• Assumptions Two Categorical Outcomes Population Follows Binomial Distribution Normal Approximation Can Be Used n·p 5 & n·(1 - p) 5

• Confidence Interval Estimate

Confidence Interval Estimate Proportion

n

)p(pZp ss

/s

1

2 pn

)p(pZp ss

/s

1

2

Lesoon. 4 - 24

A random sample of 400 Voters showed 32 preferred Candidate A. Set up a 95%

confidence interval estimate for p.

p .053 .107

Example: Estimating Proportion

n

)p(pZp ss

/s

1

2 p

n

)p(pZp ss

/s

1

2

400

0810896108

).(...

400

0810896108

).(...

p

Lesoon. 4 - 25

Sample Size

Too Big:•Requires toomuch resources

Too Small:•Won’t do the job

Lesoon. 4 - 26

What sample size is needed to be 90% confident of being correct within ± 5? A pilot study suggested that the standard

deviation is 45.

nZ

Error

2 2

2

2 2

2

1645 45

5219 2 220

..

Example: Sample Size for Mean

Round Up

Lesoon. 4 - 27

What sample size is needed to be within ± 5 with 90% confidence? Out of a population of 1,000, we randomly selected 100 of which 30 were defective.

Example: Sample Size for Proportion

Round Up

322705

7030645112

2

2

2

..

))(.(..

error

)p(pZn

228

Lesoon. 4 - 28

What sample size is needed to be 90% confident of being correct within ± 5? Suppose the population size N = 500.

Example: Sample Size for Mean Using fpc

Round Up

615215002219

5002219

10

0 .)(.

.

)N(n

Nnn

where 22192

22

0 .error

Zn

153

Lesoon. 4 - 29

Lesson Summary

•Discussed Confidence Interval Estimation for

the Mean (Known)•Discussed Confidence Interval Estimation for

the Mean (Unknown)•Addressed Confidence Interval Estimation for

the Proportion •Addressed the Situation of Finite Populations

•Determined Sample Size