[--------------------- ---------------------] Chapter 8 Interval Estimation n Interval Estimation...

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Chapter 8Chapter 8Interval EstimationInterval Estimation

Interval Estimation of a Population Mean:Interval Estimation of a Population Mean:

Large-Sample CaseLarge-Sample Case Interval Estimation of a Population Mean: Interval Estimation of a Population Mean:

Small-Sample CaseSmall-Sample Case Determining the Sample SizeDetermining the Sample Size Interval Estimation of a Population ProportionInterval Estimation of a Population Proportion

Interval Estimation of a Population Mean:Interval Estimation of a Population Mean:Large-Sample CaseLarge-Sample Case

Sampling ErrorSampling Error Probability Statements about the Sampling Probability Statements about the Sampling

ErrorError Interval Estimation: Interval Estimation: AssumedAssumedKnownKnown Interval Estimation: Interval Estimation: Estimated by Estimated by ss

Sampling ErrorSampling Error

The absolute value of the difference between The absolute value of the difference between an unbiased point estimate and the population an unbiased point estimate and the population parameter it estimates is called the parameter it estimates is called the sampling sampling errorerror..

For the case of a sample mean estimating a For the case of a sample mean estimating a population mean, the sampling error ispopulation mean, the sampling error is

Sampling Error =Sampling Error =| |x | |x

Probability StatementsProbability StatementsAbout the Sampling ErrorAbout the Sampling Error

Knowledge of the sampling distribution of Knowledge of the sampling distribution of enables us to make probability statements enables us to make probability statements about the sampling error even though the about the sampling error even though the population mean population mean is not known. is not known.

A probability statement about the sampling A probability statement about the sampling error is a error is a precision statementprecision statement..

xx

Precision StatementPrecision Statement

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

Probability StatementsProbability StatementsAbout the Sampling ErrorAbout the Sampling Error

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

z x /2z x /2Sampling distribution of

Sampling distribution of xx

xx

Assumed KnownAssumed Known

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 /2 in the upper tail of the standard standard

normal probability distributionnormal probability distribution

is the population standard is the population standard deviationdeviation

nn is the sample size is the sample size

Interval Estimate of a Population Mean:Interval Estimate of a Population Mean:Large-Sample Case (Large-Sample Case (nn >> 30) 30)

x zn

/2x zn

/2

xx

Interval Estimate of a Population Mean:Interval Estimate of a Population Mean:Large-Sample Case (Large-Sample Case (nn >> 30) 30)

Estimated by Estimated by ss

In most applications the value of the In most applications the value of the population standard deviation is unknown. We population standard deviation is unknown. We simply use the value of the sample standard simply use the value of the sample standard deviation, deviation, ss, as the point estimate of the , as the point estimate of the population standard deviation.population standard deviation.

x zsn

/2x zsn

/2

National Discount has 260 retail outlets National Discount has 260 retail outlets throughout the United States. National throughout the United States. National evaluates each potential location for a new retail evaluates each potential location for a new retail outlet in part on the mean annual income of the outlet in part on the mean annual income of the individuals in the marketing area of the new individuals in the marketing area of the new location.location.

Sampling can be used to develop an interval Sampling can be used to develop an interval estimate of the mean annual income for estimate of the mean annual income for individuals in a potential marketing area for individuals in a potential marketing area for National Discount. National Discount.

A sample of size A sample of size nn = 36 was taken. The sample = 36 was taken. The sample mean, , is $21,100 and the sample standard mean, , is $21,100 and the sample standard deviation, deviation, ss, is $4,500. We will use .95 as the , is $4,500. We will use .95 as the confidence coefficient in our interval estimate.confidence coefficient in our interval estimate.

Example: National Discount, Inc.Example: National Discount, Inc.

x

Precision StatementPrecision Statement

There is a .95 probability that the value There is a .95 probability that the value of a sample mean for National Discount will of a sample mean for National Discount will provide a sampling error of $1,470 or less……. provide a sampling error of $1,470 or less……. determined as follows:determined as follows:

95% of the sample means that can be 95% of the sample means that can be observed are within observed are within ++ 1.96 of the 1.96 of the population mean population mean . .

If If , then 1.96 = , then 1.96 = 1,470.1,470.

x x

75036

500,4 n

sx 750

36500,4

ns

x x x

Example: National Discount, IncExample: National Discount, Inc..

Interval Estimate of Population Mean: Interval Estimate of Population Mean: Estimated Estimated by by ss

Interval Estimate of Interval Estimate of is: is:

$21,100 $21,100 ++ $1,470 $1,470oror

$19,630 to $22,570$19,630 to $22,570

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

population mean.population mean.


Interval Estimation of a Population Mean:Interval Estimation of a Population Mean:Small-Sample Case (Small-Sample Case (nn < 30) < 30)

Population is Not Normally DistributedPopulation is Not Normally Distributed

The only option is to increase the sample size to The only option is to increase the sample size to

nn >> 30 and use the large-sample interval- 30 and use the large-sample interval-estimationestimation

procedures.procedures.


Population is Normally Distributed: Population is Normally Distributed: Assumed Assumed KnownKnown

The large-sample interval-estimation procedure The large-sample interval-estimation procedure cancan

be used.be used. x zn

/2x zn

/2

Population is Normally Distributed: Population is Normally Distributed: Estimated Estimated by by ss

The appropriate interval estimate is based on aThe appropriate interval estimate is based on a

probability distribution known as the probability distribution known as the t t distributiondistribution..


tt Distribution Distribution

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

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

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

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

The mean of the The mean of the tt distribution is zero. distribution is zero.


Standard normal Standard normal distributiondistribution

tt distribution distribution(20 degrees of freedom)(20 degrees of freedom)

tt distribution distribution(10 degrees of freedom)(10 degrees of freedom)

00zz, , tt

/2 Area or Probability in the Upper Tail/2 Area or Probability in the Upper Tail


00

/2/2

tttt/2/2tt/2/2

Interval EstimateInterval Estimate

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

tt/2 /2 == the the tt value providing an area value providing an area of of /2 /2 in the upper tail of a in the upper tail of a t t distributiondistribution

with with nn - 1 degrees of freedom - 1 degrees of freedom

ss = the sample standard deviation = the sample standard deviation

Interval Estimation of a Population Mean:Interval Estimation of a Population Mean:Small-Sample Case (Small-Sample Case (nn < 30) and < 30) and

Estimated by Estimated by ss

x tsn

/2x tsn

/2

Example: Apartment RentsExample: Apartment Rents


Small-Sample Case (Small-Sample Case (nn < 30) with < 30) with Estimated Estimated by by ss

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

article on the cost of off-campus housing. A article on the cost of off-campus housing. A sample of 10 one-bedroom units within a half-sample of 10 one-bedroom units within a half-mile of campus resulted in a sample mean of mile of campus resulted in a sample mean of $550 per month and a sample standard $550 per month and a sample standard deviation of $60.deviation of $60.



Small-Sample Case (Small-Sample Case (nn < 30) with < 30) with Estimated Estimated by by ss

Let us provide a 95% confidence interval Let us provide a 95% confidence interval estimate of the mean rent per month for the estimate of the mean rent per month for the population of one-bedroom units within a half-population of one-bedroom units within a half-mile of campus. We’ll assume this population mile of campus. We’ll assume this population to be normally distributed.to be normally distributed.

tt Value Value

At 95% confidence, 1 - At 95% confidence, 1 - = .95, = .95, = .05, and = .05, and /2 /2 = .025.= .025.

tt.025.025 is based on n - 1 = 10 - 1 = 9 degrees of is based on n - 1 = 10 - 1 = 9 degrees of freedom.freedom.

In the In the tt distribution table we see that distribution table we see that tt.025.025 = = 2.262.2.262.

Degrees Area in Upper Tail

of Freedom .10 .05 .025 .01 .005

. . . . . .

7 1.415 1.895 2.365 2.998 3.499

8 1.397 1.860 2.306 2.896 3.355

9 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169

. . . . . .

Degrees Area in Upper Tail

of Freedom .10 .05 .025 .01 .005

. . . . . .

7 1.415 1.895 2.365 2.998 3.499

8 1.397 1.860 2.306 2.896 3.355

9 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169

. . . . . .


Interval Estimation of a Population Mean:Interval Estimation of a Population Mean:Small-Sample Case (Small-Sample Case (nn < 30) with < 30) with Estimated Estimated by by ss

550 550 ++ 42.92 42.92

or $507.08 to $592.92or $507.08 to $592.92

We are 95% confident that the mean rent per We are 95% confident that the mean rent per month for the population of one-bedroom units month for the population of one-bedroom units within a half-mile of campus is between within a half-mile of campus is between $507.08 and $592.92.$507.08 and $592.92.

x tsn

.025x tsn

.025

10

60262.2550

10

60262.2550


Summary of Interval Estimation Summary of Interval Estimation ProceduresProcedures

for a Population Meanfor a Population Mean

n n >> 30 ? 30 ?

known ?known ?

Popul. Popul. approx.approx.normal normal

?? known ?known ?

Use Use ss to toestimate estimate

Use Use ss to toestimate estimate

Increase Increase nnto to >> 30 30

YesYes

YesYes

YesYes

YesYes

NoNo

NoNo

NoNo

NoNo

/ 2x zn

/ 2x z

n

/ 2

sx z

n / 2

sx z

n / 2

sx t

n / 2

sx t

n/ 2x tn

/ 2x t

n

Let Let EE = the maximum sampling error = the maximum sampling error mentioned in the precision statement.mentioned in the precision statement.

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

EE is often referred to as the is often referred to as the margin of error.margin of error.

Sample Size for an Interval EstimateSample Size for an Interval Estimateof a Population Meanof a Population Mean

Sample Size for an Interval EstimateSample Size for an Interval Estimateof a Population Meanof a Population Mean

Margin of ErrorMargin of Error

Necessary Sample SizeNecessary Sample Size

E zn

/2E zn

/2

nz

E( )/ 2

2 2

2n

z

E( )/ 2

2 2

2


Sample Size for an Interval Estimate of a Sample Size for an Interval Estimate of a Population MeanPopulation Mean

Suppose that National’s management Suppose that National’s management team wants an estimate of the population team wants an estimate of the population mean such that there is a .95 probability that mean such that there is a .95 probability that the sampling error is $500 or less.the sampling error is $500 or less.

How large a sample size is needed to How large a sample size is needed to meet the required precision?meet the required precision?

Sample Size for Interval Estimate of a PopulationSample Size for Interval Estimate of a Population MeanMean

At 95% confidence, zAt 95% confidence, z.025.025 = 1.96. Recall that = 1.96. Recall that = = 4,500.4,500.

We need to sample 312 to reach a desired We need to sample 312 to reach a desired precision ofprecision of

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


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

Normal Approximation of Sampling Distribution Normal Approximation of Sampling Distribution of When of When npnp >> 5 and 5 and nn(1 – (1 – pp) ) >> 5 5

Interval EstimationInterval Estimationof a Population Proportionof a Population Proportion

pp

/2/2 /2/2

Samplingdistribution of

Samplingdistribution of pp

(1 )p

p p

n

(1 )p

p p

n

pp

/ 2 pz / 2 pz pp

/ 2 pz / 2 pz

Interval EstimationInterval Estimationof a Population Proportionof a Population Proportion

Interval EstimateInterval Estimate

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

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

/2 in the upper tail of the /2 in the upper tail of the standard standard normal probability normal probability distributiondistribution

is the sample proportionis the sample proportion

p zp pn

/

( )2

1p z

p pn

/

( )2

1

pp

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

Interval Estimation of a Population ProportionInterval Estimation of a Population Proportion

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

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

Interval Estimate of a Population ProportionInterval Estimate of a Population Proportion

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

.44 + .0435.44 + .0435

PSI is 95% confident that the proportion of all PSI is 95% confident that the proportion of all votersvoters

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

p zp pn

/( )

21

p zp pn

/( )

21

pp

. .. ( . )

44 1 9644 1 44500

. .. ( . )

44 1 9644 1 44500


Let Let EE = the maximum sampling error = the maximum sampling error mentioned in the precision statement.mentioned in the precision statement.

Margin of ErrorMargin of Error

Necessary Sample SizeNecessary Sample Size

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

E zp pn

/

( )2

1E z

p pn

/

( )2

1

nz p p

E

( ) ( )/ 22

2

1n

z p p

E

( ) ( )/ 22

2

1

Sample Size for an Interval Estimate of a Sample Size for an Interval Estimate of a Population ProportionPopulation Proportion

Suppose that PSI would like a .99 Suppose that PSI would like a .99 probability that the sample proportion is within probability that the sample proportion is within + .03 of the population proportion.+ .03 of the population proportion.

How large a sample size is needed to How large a sample size is needed to meet the required precision?meet the required precision?


Sample Size for Interval Estimate of a Sample Size for Interval Estimate of a Population ProportionPopulation Proportion

At 99% confidence, zAt 99% confidence, z.005.005 = 2.576. = 2.576.2 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


Sample Size for Interval Estimate of a Sample Size for Interval Estimate of a Population ProportionPopulation Proportion

Note: We used .44 as the best estimate of Note: We used .44 as the best estimate of pp in in thethepreceding expression. If no information is preceding expression. If no information is availableavailableabout about pp, then .5 is often assumed because it , then .5 is often assumed because it providesprovidesthe highest possible sample size. If we had the highest possible sample size. If we had usedusedpp = .5, the recommended = .5, the recommended nn would have been would have been 1843.1843.


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