統計學 fall 2003

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統計學 Fall 2003

授課教師：統計系余清祥日期： 2003 年 12 月 16日

第十四週：區間估計

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Chapter 8Chapter 8Interval EstimationInterval Estimation

Interval Estimation of a Population Mean: Large-Sample Interval Estimation of a Population Mean: Large-Sample CaseCase

Interval Estimation of a Population Mean: Small-Sample Interval Estimation of a Population Mean: Small-Sample CaseCase

Determining the Sample SizeDetermining the Sample Size Interval Estimation of a Population ProportionInterval Estimation of a Population Proportion

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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 Constructing an Interval Estimate:Constructing an Interval Estimate:

Large-Sample Case with Large-Sample Case with KnownKnown Calculating an Interval Estimate:Calculating an Interval Estimate:

Large-Sample Case with Large-Sample Case with UnknownUnknown

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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

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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

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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 /2

Sampling distribution of

Sampling distribution of xx

xx

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

With With KnownKnown

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

x zn

/2x zn

/2

xx

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

With With UnknownUnknown

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

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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

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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..

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Interval Estimate of the Population Mean: Interval Estimate of the Population Mean: UnknownUnknown

Interval Estimate of Interval Estimate of is: is:

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

or $19,630 to $22,570or $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.

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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-estimation 30 and use the large-sample interval-estimation

procedures.procedures. Population is Normally Distributed and Population is Normally Distributed and is Known is Known

The large-sample interval-estimation procedure canThe large-sample interval-estimation procedure can

be used.be used. Population is Normally Distributed and Population is Normally Distributed and is Unknown is Unknown

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..

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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.

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

UnknownUnknown 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

x tsn

/2x tsn

/2

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Example: Apartment RentsExample: Apartment Rents

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

Small-Sample Case (Small-Sample Case (nn < 30) with < 30) with Unknown Unknown

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.

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 to mile of campus. We’ll assume this population to be normally distributed.be normally distributed.

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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

. . . . . .


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

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


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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. We haveWe have

Solving for Solving for nn we have we have

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

E zn

/2E zn

/2

nz

E

( )/ 22 2

2n

z

E

( )/ 22 2

2

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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?

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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. = 1.96.

Recall that 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

(500)n

2 2

2

(1.96) (4,500)311.17

(500)n

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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

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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 position political office seekers informed of their position in a race. Using telephone surveys, 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.

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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 votersPSI is 95% confident that the proportion of all voters

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 44

500

. .

. ( . )44 1 96

44 1 44500


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

We haveWe have


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

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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?


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Sample Size for Interval Estimate of a Population Sample Size for Interval Estimate of a Population ProportionProportion

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

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

nz p p

E

( ) ( ) ( . ) (. )(. )

(. )/ 2

2

2

2

2

1 2 576 44 56

031817n

z p p

E

( ) ( ) ( . ) (. )(. )

(. )/ 2

2

2

2

2

1 2 576 44 56

031817


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End of Chapter 8End of Chapter 8

統計學 fall 2003

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