07_business statistics:sampling and sampling distributions

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Chapter 8

Confidence Interval Estimation

Statistics for Managers Using Microsoft Excel, 5e

2008 Pearson Prentice-Hall, Inc.Chap 8-1


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Learning ObjectivesIn this chapter, you will learn:

To construct and interpret confidence intervalestimates for the mean and the proportion

How to determine the sample size necessaryto develop a confidence interval for the meanor proportion

How to use confidence interval estimates inauditing


2008 Pearson Prentice-Hall, Inc. Chap 8-2


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Chapter Outline1. Confidence Intervals for the Population

Mean,

when Population Standard Deviation is Known

when Population Standard Deviation is Unknown

2. Confidence Intervals for the PopulationProportion,

3. Determining the Required Sample Size




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Point Estimates Apoint estimate is a single number. For thepopulation mean (and population standarddeviation), a point estimate is the sample mean(and sample standard deviation).

A confidence interval provides additionalinformation about variability



Point Estimate

Lower Confidence

Limit

Upper Confidence

Limit

Width ofconfidence interval


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Confidence Interval EstimatesA confidence interval gives a range estimate of

values:

Takes into consideration variation in samplestatistics from sample to sample

Based on all the observations from 1 sample

Gives information about closeness to unknownpopulation parameters

Stated in terms of level of confidence

Ex. 95% confidence, 99% confidence

Can never be 100% confident




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Confidence Interval Estimates The general formula for all

confidence intervals is:



Point Estimate (Critical Value) (Standard Error)


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Confidence Level Confidence Level

Confidence in which the interval will containthe unknown population parameter

A percentage (less than 100%)




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Confidence Level Suppose confidence level = 95%

Also written (1 - ) = .95

A relative frequency interpretation: In the long run, 95% of all the confidence

intervals that can be constructed will containthe unknown true parameter

A specific interval either will contain or willnot contain the true parameter




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Confidence Interval for

( Known)Assumptions Population standard deviation is known

Population is normally distributed

If population is not normal, use large sampleConfidence interval estimate:

(where Z is the standardized normal distribution criticalvalue for a probability of/2 in each tail)



n

ZX


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Finding the Critical Value, ZConsider a 95% confidence interval:



Z= -1.96 Z= 1.96

.951

.0252

.025

2

LowerConfidenceLimit

UpperConfidenceLimit

Z units:

X units: Point Estimate

0


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Finding the Critical Value, ZCommonly used confidence levels are 90%, 95%, and99%



Confidence

Level

Confidence

Coefficient Z value

1.28

1.645

1.96

2.33

2.58

3.08

3.27

.80

.90

.95

.98

.99

.998

.999

80%

90%

95%

98%

99%

99.8%

99.9%


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Intervals and Level of Confidence



x

Confidence Intervals

Intervalsextend from

to

(1-)x100%

of intervals

constructedcontain ;

()x100% do

not.

Sampling Distribution

of the Mean

n

ZX

n

ZX

x

x1

x2

/2 /21


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( Known) ExampleA sample of 11 circuits from a large normal

population has a mean resistance of 2.20

ohms. We know from past testing that thepopulation standard deviation is .35 ohms.

Determine a 95% confidence interval for the

true mean resistance of the population.




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( Known) Example

2.4068),(1.9932

.20682.20

)11(.35/1.962.20

n

ZX



We are 95% confident that the true mean resistance is between

1.9932 and 2.4068 ohms

Although the true mean may or may not be in this interval, 95%

of intervals formed in this manner will contain the true mean


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( Unknown) If the population standard deviation is

unknown, we can substitute the sample

standard deviation, S This introduces extra uncertainty, since S is

variable from sample to sample

So we use the t distribution instead of the

normal distribution




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( Unknown)Assumptions Population standard deviation is unknown

Population is normally distributed

If population is not normal, use large sample

Use Students t Distribution

Confidence Interval Estimate:

(where t is the critical value of the t distribution withn-1 d.f. and an area of/2 in each tail)



n

StX

1-n


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Students t Distribution The t value depends on degrees of freedom (d.f.)

Number of observations that are free to vary aftersample mean has been calculated

d.f. = n - 1




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Degrees of FreedomIdea: Number of observations that are free to varyafter sample mean has been calculated

Example: Suppose the mean of 3 numbers is 8.0

Let X1 = 7

Let X2 = 8

What is X3?



If the mean of these three

values is 8.0,

then X3 must be 9

(i.e., X3 is not free to vary)

Here, n = 3, so degrees of freedom = n1 = 31 = 2

(2 values can be any numbers, but the third is not free to vary for a

given mean)


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Students t Distribution



t0

t (df= 5)

t (df= 13)t-distributions are bell-shapedand symmetric, but havefatter tails than the normal

Standard

Normal(t with df = )

Note: t Z as n increases


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Students t Table



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 2.920

The body of the table contains t

values, not probabilities

Let: n = 3

df = n - 1 = 2

= .10/2 =.05

/2 = .05


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( Unknown) ExampleA random sample of n = 25 has X = 50 and S = 8. Form a

95% confidence interval for

d.f. = n 1 = 24, so

The confidence interval is



25

8(2.0639)50

n

S1-n/2, tX

(46.698 , 53.302)


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Confidence Intervals for the

Population Proportion, An interval estimate for the population

proportion ( ) can be calculated by adding

an allowance for uncertainty to the sampleproportion ( p )




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Population Proportion, Recall that the distribution of the sample proportion isapproximately normal if the sample size is large, withstandard deviation

We will estimate this with sample data:



n

p)p(1

n)(1p


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Population Proportion, Upper and lower confidence limits for the populationproportion are calculated with the formula

where

Z is the standardized normal value for the level ofconfidence desired

p is the sample proportion

n is the sample sizeStatistics for Managers Using Microsoft Excel, 5e


np)p(1Zp


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Population Proportion, ExampleA random sample of 100 people shows that 25 haveopened IRAs this year. Form a 95% confidence intervalfor the true proportion of the population who have

opened IRAs.



00.25(.75)/196.125/100

p)/np(1p

Z

0.3349),(0.1651

(.0433)1.96.25


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Population Proportion, ExampleWe are 95% confident that the true

percentage of left-handers in the populationis between 16.51% and 33.49%. Although theinterval from .1651 to .3349 may or may notcontain the true proportion, 95% of intervalsformed from samples of size 100 in thismanner will contain the true proportion.




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Determining Sample Size The required sample size can be found to reach a

desired margin of error (e) with a specified level ofconfidence (1 - )

The margin of error is also called sampling error

the amount of imprecision in the estimate of thepopulation parameter

the amount added and subtracted to the point estimateto form the confidence interval




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Determining Sample Size To determine the required sample size for the

mean, you must know:

The desired level of confidence (1 - ), whichdetermines the critical Z value

The acceptable sampling error (margin oferror), e

The standard deviation,



n

Ze

2

22

e

Zn Now solve

for n to get


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Determining Sample SizeIf = 45, what sample size is needed to estimatethe mean within 5 with 90% confidence?



219.195

(45)(1.645)

e

Zn

2

22

2

22

So the required sample size is n = 220


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Determining Sample Size If unknown, can be estimated when using

the required sample size formula

Use a value for that is expected to be at leastas large as the true

Select a pilot sample and estimate with the

sample standard deviation, S




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Determining Sample SizeTo determine the required sample size for the proportion,you must know:

The desired level of confidence (1 - ), which

determines the critical Z value The acceptable sampling error (margin of error), e

The true proportion of successes,

can be estimated with a pilot sample, if necessary (or

conservatively use = .50)



2

2 )1(

e

Zn

Now solve

for n to getn

Ze)1(


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Determining Sample SizeHow large a sample would be necessary toestimate the true proportion defective in a largepopulation within 3%, with 95% confidence?

(Assume a pilot sample yields p = .12)




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Determining Sample Size



Solution:

For 95% confidence, use Z = 1.96

e = .03

p = .12, so use this to estimate

So use n = 451

450.74(.03)

.12)(.12)(1(1.96))1(2

2

2

2

e

Z

n


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Applications in Auditing Six advantages of statistical sampling in

auditing

Sample result is objective and defensible

Based on demonstrable statistical principles

Provides sample size estimation in advance onan objective basis

Provides an estimate of the sampling error




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Applications in Auditing Can provide more accurate conclusions on the

population Examination of the population can be time consuming and

subject to more nonsampling error

Samples can be combined and evaluated by differentauditors Samples are based on scientific approach

Samples can be treated as if they have been done by a single

auditor Objective evaluation of the results is possible

Based on known sampling error




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Population Total Amount Point estimate:

Confidence interval estimate:



XNtotalPopulation

1N

nN

n

S

)t(NXN 1n

(This is sampling without replacement, so use the finite population correction

in the confidence interval formula)


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Population Total Amount A firm has a population of 1000 accounts and

wishes to estimate the total population value.

A sample of 80 accounts is selected with averagebalance of $87.6 and standard deviation of $22.3.

Find the 95% confidence interval estimate of thetotal balance.




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Population Total Amount



The 95% confidence interval for the population total balance

is $82,837.52 to $92,362.48

4,762.4887,600

11000

801000

80

22.3905)(1000)(1.96)(1000)(87.

1NnN

nS)(tNXN 1n

22.3S87.6,X80,n1000,N


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Total Difference Point estimate:

Where the mean difference, , is:



DNDifferenceTotal

valueoriginal-valueD

D

i

1

auditedwhere

n

D

n

i

i

D


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Total DifferenceConfidence interval estimate:

where



1N

nN

n

S)t(NDN D1n

1n

)DD(

S

n

1i

2i

D


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One Sided Confidence IntervalsApplication: find the upper bound for the proportion ofitems that do not conform with internal controls

where Z is the standardized normal value for the level of

confidence desired p is the sample proportion of items that do not conform n is the sample size N is the population size



1NnN

np)p(1ZpboundUpper


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Ethical IssuesA confidence interval (reflecting sampling

error) should always be reported along with apoint estimate

The level of confidence should always bereported

The sample size should be reported

An interpretation of the confidence intervalestimate should also be provided




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Chapter Summary Introduced the concept of confidence intervals

Discussed point estimates

Developed confidence interval estimates

Created confidence interval estimates for themean ( known)

Determined confidence interval estimates for the

mean ( unknown)

Created confidence interval estimates for theproportion



In this chapter, we have


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Chapter Summary Determined required sample size for mean and

proportion settings

Developed applications of confidence intervalestimation in auditing

Confidence interval estimation for populationtotal

Confidence interval estimation for total difference

in the population One sided confidence intervals

Addressed confidence interval estimation andethical issues


In this chapter, we have

07_business statistics:sampling and sampling distributions

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