statistics for managers using microsoft excel · 08/03/2018 · © 1999 prentice-hall, inc. chap. 7...

© 1999 Prentice-Hall, Inc. Chap. 7 - 1

Statistics for Managers

Using Microsoft Excel

Chapter 7

Confidence Interval Estimation


Chapter Topics

•Confidence Interval Estimation for the Mean

(s Known)

•Confidence Interval Estimation for the Mean

(s Unknown)

•Confidence Interval Estimation for the

Proportion

•The Situation of Finite Populations

•Sample Size Estimation


Mean, m, is

unknown

Population Random Sample I am 95%

confident that m

is between 40 &

60.

Mean

X = 50

Estimation Process

Sample


Estimate Population

Parameter...

with Sample

Statistic

Mean m

Proportion p p s

Variance s 2

Population Parameters

Estimated

s 2

Difference m - m 1 2

x - x 1 2

X

_

_ _


• 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


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


Parameter =

Statistic ± Its Error

© 1984-1994 T/Maker Co.

Confidence Limits for

Population Mean

Xm Error

= Error = Xm

XX

XZ

ss

m

xZ s

XZX sm

Error

Error

mX


90% Samples

95% Samples

s x _

Confidence Intervals

xx .. smsm 64516451

xx smsm 96.196.1

xx .. smsm 582582 99% Samples

nZXZX X

ss

X

_


• Probability that the unknown

population parameter falls within the

interval

• Denoted (1 - a) % = level of confidence

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

a Is Probability That the Parameter Is Not

Within the Interval

Level of Confidence


Confidence Intervals

Intervals

Extend from (1 - a) % of

Intervals

Contain m.

a % Do Not.

1 - a a /2 a /2

X _

s x _

Intervals &

Level of Confidence

Sampling

Distribution of

the Mean

to XZX s

XZX s

mm X


• Data Variation

measured by s

• Sample Size

• Level of Confidence

(1 - a)

Intervals Extend from

© 1984-1994 T/Maker Co.

Factors Affecting

Interval Width

X - Zs to X + Z s x x

n/XX ss


Mean

s Unknown

Confidence

Intervals

Proportion

Finite

Population s Known

Confidence Interval Estimates


• Assumptions

Population Standard Deviation Is Known

Population Is Normally Distributed

If Not Normal, use large samples

• Confidence Interval Estimate

Confidence Intervals (s Known)

nZX /

s a 2

mn

ZX /

s a 2


Mean

s Unknown

Confidence

Intervals

Proportion

Finite

Population s Known



• Assumptions

Population Standard Deviation Is Unknown

Population Must Be Normally Distributed

• Use Student’s t Distribution


Confidence Intervals (s Unknown)

n

StX n,/ a 12

mn

StX n,/ a 12


Z

t 0

t (df = 5)

Standard

Normal

t (df = 13) Bell-Shaped

Symmetric

‘Fatter’ Tails

Student’s t Distribution


• 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)


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

t 0

Assume: n = 3 df

= n - 1 = 2

a = .10

a/2 =.05

2.920 t Values

a / 2

.05

Student’s t Table


A random sample of n = 25 has = 50 and

s = 8. Set up a 95% confidence interval

estimate for m.

m . . 46 69 53 30

X

Example: Interval Estimation

s Unknown

n

StX n,/ a 12 m

n

StX n,/ a 12

25

80639250 . m 25

80639250 .


Mean

s Unknown

Confidence

Intervals

Proportion

Finite

Population s Known



• Assumptions

Sample Is Large Relative to Population

n / N > .05

• Use Finite Population Correction Factor

• Confidence Interval (Mean, sX Unknown)

X m

Estimation for Finite Populations

n

StX n,/ a 12

n

StX n,/ a 12

1

N

nN

1

N

nN


Mean

s Unknown

Confidence

Intervals

Proportion

Finite

Population s Known



• Assumptions

Two Categorical Outcomes

Population Follows Binomial Distribution

Normal Approximation Can Be Used

n·p 5 & n·(1 - p) 5


Confidence Interval Estimate

Proportion

n

)p(pZp ss

/s

a

12 p

n

)p(pZp ss

/s

a

12


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

a

12

pn

)p(pZp ss

/s

a

12

400

0810896108

).(...

400

0810896108

).(...

p


Sample Size

Too Big:

•Requires too

much resources

Too Small:

•Won’t do

the job


What sample size is needed to be 90%

confident of being correct within ± 5? A

pilot study suggested that the standard

deviation is 45.

n Z

Error @

2 2

2

2 2

2

1 645 45

5 219 2 220

s . .

Example: Sample Size

for Mean

Round Up


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.


for Proportion

Round Up

322705

7030645112

2

2

2

..

))(.(..

error

)p(pZn

228 @


What sample size is needed to be 90%

confident of being correct within ± 5?

Suppose the population size N = 500.


for Mean Using fpc

Round Up

615215002219

5002219

10

0 .)(.

.

)N(n

Nnn

where 22192

22

0 .error

Zn

s

153 @


Chapter Summary

•Discussed Confidence Interval Estimation for

the Mean (s Known)

•Discussed Confidence Interval Estimation for

the Mean (s Unknown)

•Addressed Confidence Interval Estimation for

the Proportion

•Addressed the Situation of Finite Populations

•Determined Sample Size

statistics for managers using microsoft excel · 08/03/2018 · © 1999 prentice-hall, inc. chap. 7...

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