1 1 slide a point estimator cannot be expected to provide the a point estimator cannot be expected...

1 1 Slide Slide

A point estimator cannot be expected to provide theA point estimator cannot be expected to provide the exact value of the population parameter.exact value of the population parameter. A point estimator cannot be expected to provide theA point estimator cannot be expected to provide the exact value of the population parameter.exact value of the population parameter.

An An interval estimateinterval estimate can be computed by adding and can be computed by adding and subtracting a subtracting a margin of errormargin of error to the point estimate. to the point estimate.

An An interval estimateinterval estimate can be computed by adding and can be computed by adding and subtracting a subtracting a margin of errormargin of error to the point estimate. to the point estimate.

Point Estimate +/Point Estimate +/ Margin of Error Margin of Error

The purpose of an interval estimate is to provideThe purpose of an interval estimate is to provide information about how close the point estimate is toinformation about how close the point estimate is to the value of the parameter.the value of the parameter.

The purpose of an interval estimate is to provideThe purpose of an interval estimate is to provide information about how close the point estimate is toinformation about how close the point estimate is to the value of the parameter.the value of the parameter.

Margin of Error and the Interval EstimateMargin of Error and the Interval Estimate

Chapter 8 - Interval Estimation

2 2 Slide Slide

Interval Estimate of a Population Mean: Known

In order to develop an interval estimate of a In order to develop an interval estimate of a population mean, the margin of error must be population mean, the margin of error must be computed using either:computed using either:

• the population standard deviation the population standard deviation , or , or

• the sample standard deviation the sample standard deviation ss

is rarely known exactly, but often a good is rarely known exactly, but often a good estimate can be obtained based on historical estimate can be obtained based on historical data or other information.data or other information.

We refer to such cases as the We refer to such cases as the known known case. case.

The general form of an interval estimate of a The general form of an interval estimate of a population mean ispopulation mean is

The general form of an interval estimate of a The general form of an interval estimate of a population mean ispopulation mean is

Margin of Errorx Margin of Errorx

3 3 Slide Slide

There is a 1 There is a 1 probability that the value of a probability that the value of asample mean will provide a margin of error of sample mean will provide a margin of error of or less.or less.

z x /2z x /2

/2/2 /2/21 - of all values1 - of all valuesxx

Sampling distribution of

Sampling distribution of xx

xx

z x /2z x /2z x /2z x /2

Interval Estimate of a Population Mean:

Known

4 4 Slide Slide

/2/2 /2/21 - of all values1 - of all valuesxx

Sampling distribution of

Sampling distribution of xx

xx

z x /2z x /2z x /2z x /2

[------------------------- -------------------------][------------------------- -------------------------]

[------------------------- -------------------------][------------------------- -------------------------]

xx

intervalintervaldoes notdoes notinclude include

intervalintervalDoes not Does not includes includes


Known

xxxx

5 5 Slide Slide

Interval Estimate ofInterval Estimate of


Known

x zn

/2x zn

/2

where: is the sample meanwhere: is the sample mean 1 -1 - is the confidence coefficient is the confidence coefficient zz/2 /2 is the is the zz value providing an area of value providing an area of /2 in the upper tail of the standard /2 in the upper tail of the standard

normal probability distributionnormal probability distribution is the population standard deviationis the population standard deviation nn is the sample size is the sample size

xx

6 6 Slide Slide


Known Values of Values of zz/2/2 for the Most Commonly for the Most Commonly

Used Confidence LevelsUsed Confidence Levels

90% .10 .05 .9500 90% .10 .05 .9500 1.645 1.645

Confidence TableConfidence Table Level Level /2 Look-up Area /2 Look-up Area zz/2/2

95% .05 .025 .9750 95% .05 .025 .9750 1.960 1.960 99% .01 .005 .9950 99% .01 .005 .9950 2.576 2.576

7 7 Slide Slide

Meaning of Confidence

We say that this interval has been established at theWe say that this interval has been established at the

90% 90% confidence levelconfidence level..

We say that this interval has been established at theWe say that this interval has been established at the

90% 90% confidence levelconfidence level..

The value .90 is referred to as the The value .90 is referred to as the confidenceconfidence coefficientcoefficient.. The value .90 is referred to as the The value .90 is referred to as the confidenceconfidence coefficientcoefficient..

8 8 Slide Slide


Known Example: Discount SoundsExample: Discount Sounds

Discount Sounds has 260 retail outlets Discount Sounds has 260 retail outlets throughoutthroughout

the United States. The firm is evaluating a the United States. The firm is evaluating a potentialpotential

location for a new outlet, based in part, on the location for a new outlet, based in part, on the meanmean

annual income of the individuals in the annual income of the individuals in the marketingmarketing

area of the new location.area of the new location.

A sample of size A sample of size nn = 36 was taken; the = 36 was taken; the sample sample

mean income is $41,100. The population is mean income is $41,100. The population is notnot

believed to be highly skewed. The population believed to be highly skewed. The population standard deviation is estimated to be $4,500, standard deviation is estimated to be $4,500,

and theand theconfidence coefficient to be used in the confidence coefficient to be used in the

interval interval estimate is .95.estimate is .95.

9 9 Slide Slide

95% of the sample means that can be observed95% of the sample means that can be observed

are within are within ++ 1.96 of the population mean 1.96 of the population mean . . x xThe margin of error is: The margin of error is:

/ 2

4,5001.96 1,470

36z

n

/ 2

4,5001.96 1,470

36z

n Thus, at 95% confidence, the margin of errorThus, at 95% confidence, the margin of error is $1,470. is $1,470.


Known Example: Discount SoundsExample: Discount Sounds

10 10

Slide Slide

Interval estimate of Interval estimate of is: is:


Known

We are We are 95% confident95% confident that the interval contains the that the interval contains the

population mean.population mean.

$41,100 + $1,470or

$39,630 to $42,570

Example: Discount SoundsExample: Discount Sounds

11 11

Slide Slide


Known

90% 1234 $39,866 90% 1234 $39,866 to $42,334 to $42,334

Confidence MarginConfidence Margin Level of Error Interval EstimateLevel of Error Interval Estimate

95% 1470 $39,630 to 95% 1470 $39,630 to $42,570$42,570

99% 1932 $39,168 to 99% 1932 $39,168 to $43,032 $43,032


In order to have a higher degree of confidence,In order to have a higher degree of confidence,

the margin of error and thus the width of thethe margin of error and thus the width of the

confidence interval must be larger.confidence interval must be larger.

12 12

Slide Slide


Known Adequate Sample SizeAdequate Sample Size

In most applications, a sample size of In most applications, a sample size of nn = 30 is = 30 is adequate.adequate. In most applications, a sample size of In most applications, a sample size of nn = 30 is = 30 is adequate.adequate.

If the population distribution is highly skewed orIf the population distribution is highly skewed or contains outliers, a sample size of 50 or more iscontains outliers, a sample size of 50 or more is recommended.recommended.


If the population is not normally distributed but isIf the population is not normally distributed but is roughly symmetric, a sample size as small as 15 roughly symmetric, a sample size as small as 15 will suffice.will suffice.


If the population is believed to be at leastIf the population is believed to be at least approximately normal, a sample size of less than 15approximately normal, a sample size of less than 15 can be used.can be used.


13 13

Slide Slide


Unknown If an estimate of the population standard If an estimate of the population standard

deviation deviation cannot be developed prior to cannot be developed prior to sampling, we use the sample standard sampling, we use the sample standard deviation deviation ss to estimate to estimate . . This is the This is the unknown unknown case. case.

In this case, the interval estimate for In this case, the interval estimate for is based is based on the on the tt distribution. distribution.

(We’ll assume for now that the population is (We’ll assume for now that the population is normally distributed.)normally distributed.)

14 14

Slide Slide

The The t t distribution distribution is a family of similar probability distributions. is a family of similar probability distributions. The The t t distribution distribution is a family of similar probability distributions. is a family of similar probability distributions.

t Distribution

A specific A specific tt distribution depends on a parameter distribution depends on a parameter known as the known as the degrees of freedomdegrees of freedom.. A specific A specific tt distribution depends on a parameter distribution depends on a parameter known as the known as the degrees of freedomdegrees of freedom..

Degrees of freedom refer to the number of independent Degrees of freedom refer to the number of independent pieces of information that go into the computation of pieces of information that go into the computation of ss..Degrees of freedom refer to the number of independent Degrees of freedom refer to the number of independent pieces of information that go into the computation of pieces of information that go into the computation of ss..

A A tt distribution with more degrees of freedom has distribution with more degrees of freedom has less dispersion.less dispersion. A A tt distribution with more degrees of freedom has distribution with more degrees of freedom has less dispersion.less dispersion.

As the degrees of freedom increases, the difference between As the degrees of freedom increases, the difference between the the tt distribution and the standard normal probability distribution and the standard normal probabilitydistribution becomes smaller and smaller.distribution becomes smaller and smaller.

As the degrees of freedom increases, the difference between As the degrees of freedom increases, the difference between the the tt distribution and the standard normal probability distribution and the standard normal probabilitydistribution becomes smaller and smaller.distribution becomes smaller and smaller.

15 15

Slide Slide

t Distribution

StandardStandardnormalnormal

distributiondistribution

tt distributiondistribution(20 degrees(20 degreesof freedom)of freedom)

tt distributiondistribution(10 degrees(10 degrees

of of freedomfreedom))

00zz, , tt

For more than 100 degrees of freedom, the standard normal For more than 100 degrees of freedom, the standard normal zz value provides a good approximation to the value provides a good approximation to the tt value value

16 16

Slide Slide

Interval EstimateInterval Estimate

x tsn

/2x tsn

/2

where: 1 -where: 1 - = the confidence coefficient = the confidence coefficient

tt/2 /2 == the the tt value providing an area of value providing an area of /2/2 in the upper tail of a in the upper tail of a t t distribution distribution with with nn - 1 degrees of freedom - 1 degrees of freedom ss = the sample standard deviation = the sample standard deviation


Unknown

17 17

Slide Slide

A reporter for a student newspaper is writing anA reporter for a student newspaper is writing an

article on the cost of off-campus housing. A samplearticle on the cost of off-campus housing. A sample

of 16 efficiency apartments within a half-mile ofof 16 efficiency apartments within a half-mile of

campus resulted in a sample mean of $750 per campus resulted in a sample mean of $750 per monthmonth

and a sample standard deviation of $55.and a sample standard deviation of $55.


Unknown Example: Apartment RentsExample: Apartment Rents

Let us provide a 95% confidence interval Let us provide a 95% confidence interval estimateestimate

of the mean rent per month for the population of the mean rent per month for the population of of

efficiency apartments within a half-mile of efficiency apartments within a half-mile of campus.campus.

We will assume this population to be normallyWe will assume this population to be normally

distributed.distributed.

18 18

Slide Slide

At 95% confidence, At 95% confidence, = .05, and = .05, and /2 = .025./2 = .025.

In the In the tt distribution table we see that distribution table we see that tt.025.025 = 2.131. = 2.131.tt.025.025 is based on is based on nn 1 = 16 1 = 16 1 = 15 degrees of freedom. 1 = 15 degrees of freedom.


Unknown

Degrees Area in Upper Tail

of Freedom .20 .100 .050 .025 .010 .005

15 .866 1.341 1.753 2.131 2.602 2.947

16 .865 1.337 1.746 2.120 2.583 2.921

17 .863 1.333 1.740 2.110 2.567 2.898

18 .862 1.330 1.734 2.101 2.520 2.878

19 .861 1.328 1.729 2.093 2.539 2.861

. . . . . . .

19 19

Slide Slide

.025

sx t

n .025

sx t

n

We are 95% confident that the mean rent per monthWe are 95% confident that the mean rent per monthfor the population of efficiency apartments within afor the population of efficiency apartments within ahalf-mile of campus is between $720.70 and $779.30.half-mile of campus is between $720.70 and $779.30.



Unknown

55750 2.131 750 29.30

16

55750 2.131 750 29.30

16

MarginMarginof Errorof Error

20 20

Slide Slide


Unknown Adequate Sample SizeAdequate Sample Size



x tsn

/2x tsn

/2

21 21

Slide Slide


Unknown Adequate Sample Size (continued)Adequate Sample Size (continued)





22 22

Slide Slide

Summary of Interval Estimation Procedures

for a Population Mean

Can theCan thepopulation standardpopulation standard

deviation deviation be assumed be assumed known ?known ?

UseUse

YesYes NoNo

/ 2

sx t

n / 2

sx t

nUseUse/ 2x z

n

/ 2x zn

KnownKnownCaseCase

UnknownUnknownCaseCase

Use the sampleUse the samplestandard deviationstandard deviation

ss to estimate to estimate

23 23

Slide Slide

Let Let EE = the desired margin of error. = the desired margin of error. Let Let EE = the desired margin of error. = the desired margin of error.

EE is the amount added to and subtracted from the is the amount added to and subtracted from the point estimate to obtain an interval estimate.point estimate to obtain an interval estimate. EE is the amount added to and subtracted from the is the amount added to and subtracted from the point estimate to obtain an interval estimate.point estimate to obtain an interval estimate.

Sample Size for an Interval Estimateof a Population Mean

If a desired margin of error is selected prior toIf a desired margin of error is selected prior to sampling, the sample size necessary to satisfy thesampling, the sample size necessary to satisfy the margin of error can be determined.margin of error can be determined.

If a desired margin of error is selected prior toIf a desired margin of error is selected prior to sampling, the sample size necessary to satisfy thesampling, the sample size necessary to satisfy the margin of error can be determined.margin of error can be determined.

24 24

Slide Slide


E zn

/2E zn

/2

nz

E

( )/ 22 2

2n

z

E

( )/ 22 2

2

Margin of Error Margin of Error

Necessary Sample Size Necessary Sample Size

25 25

Slide Slide


The Necessary Sample Size equation requires aThe Necessary Sample Size equation requires a value for the population standard deviation value for the population standard deviation . . The Necessary Sample Size equation requires aThe Necessary Sample Size equation requires a value for the population standard deviation value for the population standard deviation . .

If If is unknown, a preliminary or is unknown, a preliminary or planning valueplanning value for for can be used in the equation. can be used in the equation. If If is unknown, a preliminary or is unknown, a preliminary or planning valueplanning value for for can be used in the equation. can be used in the equation.

1. Use the estimate of the population standard1. Use the estimate of the population standard deviation computed in a previous study.deviation computed in a previous study. 1. Use the estimate of the population standard1. Use the estimate of the population standard deviation computed in a previous study.deviation computed in a previous study.

2. Use a pilot study to select a preliminary study and2. Use a pilot study to select a preliminary study and use the sample standard deviation from the study.use the sample standard deviation from the study. 2. Use a pilot study to select a preliminary study and2. Use a pilot study to select a preliminary study and use the sample standard deviation from the study.use the sample standard deviation from the study.

3. Use judgment or a “best guess” for the value of 3. Use judgment or a “best guess” for the value of . . 3. Use judgment or a “best guess” for the value of 3. Use judgment or a “best guess” for the value of . .

26 26

Slide Slide

Recall that Discount Sounds is evaluating aRecall that Discount Sounds is evaluating apotential location for a new retail outlet, based inpotential location for a new retail outlet, based inpart, on the mean annual income of the individuals inpart, on the mean annual income of the individuals inthe marketing area of the new location.the marketing area of the new location.



Suppose that Discount Sounds’ Suppose that Discount Sounds’ management teammanagement team

wants an estimate of the population mean such wants an estimate of the population mean such thatthat

there is a .95 probability that the sampling there is a .95 probability that the sampling error iserror is

$500 or less.$500 or less.

How large a sample size is needed to meet How large a sample size is needed to meet thethe

required precision?required precision?

27 27

Slide Slide

At 95% confidence, At 95% confidence, zz.025.025 = 1.96. Recall that = 1.96. Recall that = 4,500.= 4,500.

zn

/2 500zn

/2 500

2 2

2

(1.96) (4,500)311.17 312

(500)n

2 2

2

(1.96) (4,500)311.17 312

(500)n


A sample of size 312 is needed to reach a desiredA sample of size 312 is needed to reach a desired

precision of precision of ++ $500 at 95% confidence. $500 at 95% confidence.

28 28

Slide Slide

Interval Estimateof a Population Proportion

The general form of an interval estimate of aThe general form of an interval estimate of a population proportion ispopulation proportion is

The general form of an interval estimate of aThe general form of an interval estimate of a population proportion ispopulation proportion is

Margin of Errorp Margin of Errorp

pp

pp

29 29

Slide Slide

/2/2 /2/2


Normal Approximation of Sampling Distribution Normal Approximation of Sampling Distribution of of

pp

Samplingdistribution of

Samplingdistribution of pp

(1 )p

p p

n

(1 )p

p p

n

pppp

/ 2 pz / 2 pz / 2 pz / 2 pz

1 - of all values1 - of all valuespp

30 30

Slide Slide



p zp pn

/

( )2

1p z

p pn

/

( )2

1

where: 1 -where: 1 - is the confidence coefficient is the confidence coefficient

zz/2 /2 is the is the zz value providing an area of value providing an area of

/2 in the upper tail of the standard/2 in the upper tail of the standard

normal probability distributionnormal probability distribution

is the sample proportionis the sample proportionpp

31 31

Slide Slide

Political Science, Inc. (PSI) specializes in voter polls and Political Science, Inc. (PSI) specializes in voter polls and surveys designed to keep political office seekers informed of surveys designed to keep political office seekers informed of their position in a race. Using telephone surveys, PSI their position in a race. Using telephone surveys, PSI interviewers ask registered voters who they would vote for if interviewers ask registered voters who they would vote for if the election were held that day. the election were held that day.


Example: Political Science, IncExample: Political Science, Inc..

In a current election campaign, PSI has just found In a current election campaign, PSI has just found that 220 registered voters, out of 500 contacted, that 220 registered voters, out of 500 contacted, favor a particular candidate. PSI wants to develop favor a particular candidate. PSI wants to develop a 95% confidence interval estimate for the a 95% confidence interval estimate for the proportion of the population of registered voters proportion of the population of registered voters that favor the candidate.that favor the candidate.

32 32

Slide Slide

p zp pn

/

( )2

1p z

p pn

/

( )2

1

where: where: nn = 500, = 220/500 = .44, = 500, = 220/500 = .44, zz/2 /2 = = 1.961.96pp


PSI is 95% confident that the proportion of all votersPSI is 95% confident that the proportion of all voters

that favor the candidate is between .3965 and .4835.that favor the candidate is between .3965 and .4835.PSI is 95% confident that the proportion of all votersPSI is 95% confident that the proportion of all voters

that favor the candidate is between .3965 and .4835.that favor the candidate is between .3965 and .4835.

.44(1 .44).44 1.96

500

.44(1 .44)

.44 1.96500

.44 .0435 .44 .0435

33 33

Slide Slide

Solving for the necessary sample size, we Solving for the necessary sample size, we getget

Margin of ErrorMargin of Error

Sample Size for an Interval Estimateof a Population Proportion

/ 2

(1 )p pE z

n

/ 2

(1 )p pE z

n

2/ 2

2

( ) (1 )z p pn

E

2

/ 22

( ) (1 )z p pn

E

However, will not be known until after we have However, will not be known until after we have selected the sample. We will use the planning valueselected the sample. We will use the planning value

pp** for . for .

pp

pp

34 34

Slide Slide

Sample Size for an Interval EstimateSample Size for an Interval Estimateof a Population Proportionof a Population Proportion

The planning value The planning value pp** can be chosen by: can be chosen by:

1. Using the sample proportion from a 1. Using the sample proportion from a previous sample of the same or similar previous sample of the same or similar units, orunits, or

2. Selecting a preliminary sample and 2. Selecting a preliminary sample and using theusing the

sample proportion from this sample.sample proportion from this sample.

3. Use judgment or a “best guess” for a 3. Use judgment or a “best guess” for a pp* * value.value.

4. Otherwise, use .50 as the 4. Otherwise, use .50 as the pp* value.* value.

2 * */ 2

2

( ) (1 )z p pn

E

2 * *

/ 22

( ) (1 )z p pn

E

Necessary Sample SizeNecessary Sample Size

35 35

Slide Slide

Suppose that PSI would like a .99 probability thatSuppose that PSI would like a .99 probability that

the sample proportion is within + .03 of thethe sample proportion is within + .03 of the

population proportion.population proportion.

How large a sample size is needed to meet theHow large a sample size is needed to meet the

required precision? (A previous sample of similarrequired precision? (A previous sample of similar

units yielded .44 for the sample proportion.)units yielded .44 for the sample proportion.)


Example: Political Science, IncExample: Political Science, Inc..

36 36

Slide Slide

At 99% confidence, At 99% confidence, zz.005.005 = 2.576. Recall that = 2.576. Recall that = .44.= .44.

pp

A sample of size 1817 is needed to reach a desiredA sample of size 1817 is needed to reach a desired

precision of precision of ++ .03 at 99% confidence. .03 at 99% confidence.

2 2/ 2

2 2

( ) (1 ) (2.576) (.44)(.56) 1817

(.03)

z p pn

E

2 2

/ 2

2 2

( ) (1 ) (2.576) (.44)(.56) 1817

(.03)

z p pn

E


/ 2

(1 ).03

p pz

n

/ 2

(1 ).03

p pz

n

1 1 slide a point estimator cannot be expected to provide the a point estimator cannot be expected...

Documents

point estimate

interval estimate chapter

good estimate

estimate margin of error

population mean

population parameter

interval estimation

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