Confidence Intervals for means and proportions by nthanht777

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									Statistics for Managers Using Microsoft Excel
Chapter 7 Confidence Interval Estimation: Single Sample Means and Proportions
© 1999 Prentice-Hall, Inc. Chap. 7 - 1

Chapter 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
© 1999 Prentice-Hall, Inc. Chap. 7 - 2

Estimation Process
Population
Mean, , is unknown

Random Sample
Mean X = 50

I am 95% confident that  is between 40 & 60.

Sample

© 1999 Prentice-Hall, Inc.

Chap. 7 - 3

Population Parameters Estimated
Estimate Population Parameter... Mean  Proportion Variance Difference
© 1999 Prentice-Hall, Inc.

with Sample Statistic _
X

p 
1
2

ps s
2 1
2

 - 

_ _ x - x

2
Chap. 7 - 4

Confidence Interval Estimation
• 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
© 1999 Prentice-Hall, Inc. Chap. 7 - 5

Elements of Confidence Interval Estimation
A Probability That the Population Parameter Falls Somewhere Within the Interval. Sample Confidence Interval Statistic

Confidence Limit (Lower)
© 1999 Prentice-Hall, Inc.

Confidence Limit (Upper)
Chap. 7 - 6

Confidence Limits for Population Mean
Parameter = Statistic ± Its Error
X 

X

Error
X

= Error = 
X  

Z 





Error

X

X

Error  Z 

x

  X  Z X
© 1984-1994 T/Maker Co. © 1999 Prentice-Hall, Inc. Chap. 7 - 7

Confidence Intervals
X  Z  X  X  Z 


n
_ x
_ X

  1.645 x
  1 .96 x

  1.645 x
  1 .96 x

90% Samples

95% Samples

  2.58 x
99% Samples
© 1999 Prentice-Hall, Inc.

  2.58 x
Chap. 7 - 8

Level of Confidence is an EXPECTED RELATIONSHIP
• Probability that the unknown population parameter is in the confidence interval in 100 trials. Denoted (1 - ) % = level of confidence
e.g. 90%, 95%, 99%


•

Is Probability That the Parameter Is Not Within the Interval in 100 trials (NOT THIS TRIAL ALONE!)
Chap. 7 - 9

© 1999 Prentice-Hall, Inc.

Intervals & Level of Confidence
Sampling Distribution of the Mean /2

_ 1 -

x

/2

Intervals Extend from

X  

_
X
(1 - ) % of Intervals Contain . % Do Not.

X  ZX
to

X  ZX
Confidence Intervals
© 1999 Prentice-Hall, Inc.

Chap. 7 - 10

Factors Affecting Interval Width
• • • Data Variation measured by  Sample Size X  X / n Level of Confidence (1 - )
© 1984-1994 T/Maker Co. © 1999 Prentice-Hall, Inc. Chap. 7 - 11

Intervals Extend from X - Z
x

to X + Z 

x

Confidence Interval Estimates
Confidence Intervals

Mean

Proportion

 Known

Finite Population

© 1999 Prentice-Hall, Inc.

Chap. 7 - 12

Confidence Intervals (Known - this is hardly ever true)
• Assumptions


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

 

•

Confidence Interval Estimate

    X  Z / 2  n
© 1999 Prentice-Hall, Inc.

 X  Z / 2  n
Chap. 7 - 13

Confidence Interval Estimates
Confidence Intervals

Mean

Proportion

 Known

Finite Population

© 1999 Prentice-Hall, Inc.

Chap. 7 - 14

Confidence Intervals (Unknown)
• Assumptions
Population Standard Deviation Is Unknown  Sample size must be large enough for central limit theorem or Population Must Be Normally Distributed


•
•

Use Student’s t Distribution

Confidence Interval Estimate S S    X t X  t / 2 ,n1   / 2 ,n1  n n
© 1999 Prentice-Hall, Inc. Chap. 7 - 15

Student’s t Distribution
Standard Normal Bell-Shaped Symmetric ‘Fatter’ Tails t (df = 13) t (df = 5)

0
© 1999 Prentice-Hall, Inc.

Z t
Chap. 7 - 16

Degrees of Freedom (df)
• 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

© 1999 Prentice-Hall, Inc.

Chap. 7 - 17

Student’s t Table
/2
Upper Tail Area df .25 .10
Assume: n = 3 =n-1=2  = .10 /2 =.05 df

.05

1 1.000 3.078 6.314

2 0.817 1.886 2.920
3 0.765 1.638 2.353

.05

0
t Values
© 1999 Prentice-Hall, Inc.

2.920

t

Chap. 7 - 18

Example: Interval Estimation Unknown
A random sample of n = 25 has X = 50 and s = 8. Set up a 95% confidence interval estimate for . S S X  t / 2 ,n1     X  t / 2 ,n1  n n
50  2 . 0639  8 25


  

50  2 . 0639 

8 25

46 . 69
© 1999 Prentice-Hall, Inc.

53 . 30
Chap. 7 - 19

Confidence Interval Estimates
Confidence Intervals

Mean

Proportion

 Known

Finite Population

© 1999 Prentice-Hall, Inc.

Chap. 7 - 20

Confidence Interval Estimate Proportion
• Assumptions  Two Categorical Outcomes  Population Follows Binomial Distribution  Normal Approximation Can Be Used


n·p 5

&

n·(1 - p)  5

•

Confidence Interval Estimate ps ( 1  ps ) ps ( 1  ps ) ps  Z / 2   p  ps  Z / 2  n n
Chap. 7 - 21

© 1999 Prentice-Hall, Inc.

Example: Estimating Proportion
A random sample of 400 Voters showed 32 preferred Candidate A. Set up a 95% confidence interval estimate for p.
ps ( 1  ps ) ps  Z / 2  n

 p  ps  Z / 2  ps ( 1  ps )
n

.08( 1  .08 )  p  .08  1.96  .08( 1  .08 ) .08  1.96  400 400
.053  p  .107
© 1999 Prentice-Hall, Inc. Chap. 7 - 22

Sample Size
Too Big: •Requires too much resources Too Small: •Won’t do the job

© 1999 Prentice-Hall, Inc.

Chap. 7 - 23

Does the CI Contain the True Mean?
Click to try a couple

© 1999 Prentice-Hall, Inc.

Chap. 7 - 24

Example: Sample Size for Mean
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 
2
2

Error

2



1645 . 5

2 2

45

2

 219.2  220
Round Up

© 1999 Prentice-Hall, Inc.

Chap. 7 - 25

Example: Sample Size for Proportion
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.
Z 2 p ( 1  p ) 1 . 645 2 (. 30 )(. 70 ) n   227 . 3 2 2 error . 05

 228
Round Up
© 1999 Prentice-Hall, Inc. Chap. 7 - 26

Chapter Summary
•Discussed Confidence Interval Estimation for the Mean (Known) •Discussed Confidence Interval Estimation for the Mean (Unknown) •Addressed Confidence Interval Estimation for the Proportion

•Determined Sample Size
© 1999 Prentice-Hall, Inc. Chap. 7 - 27


								
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