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Introduction toPredictionIntroduction toPrediction

Introduction

Statistical inference is the process bywhich we acquire information aboutpopulations from samples.

There are two types of inference:

Estimation

Hypotheses testing

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Concepts of Estimation

The objective of estimation is todetermine the value of a populationparameter on the basis of a samplestatistic.

There are two types of estimators:Point EstimatorInterval estimator

Point Estimator

A point estimator draws inference about apopulation by estimating the value of anunknown parameter using a single valueor point.

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Population distribution

Point Estimator

Parameter

?

Sampling distribution

A point estimator draws inference about apopulation by estimating the value of anunknown parameter using a single valueor point.

Point estimator

An interval estimator draws inferencesabout a population by estimating the valueof an unknown parameter using aninterval.

Interval estimator

Population distribution

Sample distribution

Parameter

Interval Estimator

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Selecting the right sample statistic to estimatea parameter value depends on thecharacteristics of the statistic.

Estimators Characteristics

Estimators desirable characteristics:Unbiasedness: An unbiased estimator is one whose

expected value is equal to the parameter it estimates.Consistency: An unbiased estimator is said to be

consistent if the difference between the estimator andthe parameter grows smaller as the sample sizeincreases.

Relative efficiency:For two unbiased estimators, the onewith a smaller variance is said to be relatively efficient.

Estimating the Population Mean whenthe Population Variance is Known

How is an interval estimator producedfrom a sampling distribution?

A sample of size n is drawn from thepopulation, and its mean is calculated.

By the central limit theorem is normallydistributed (or approximately normallydistributed.), thus

x

x

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n

xZ

=

We have established beforethat

=

+

1)nzxnz(P22

Estimating the Population Mean whenthe Population Variance is Known

=

+

1)n

zxn

zx(P 22

This leads to the followingequivalent statement

The Confidence Interval for ( isknown)

The confidence interval

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Interpreting the Confidence Interval for

1 of all the values of obtained in repeated

sampling from a given distribution, construct an interval

that includes (covers) the expected value of thepopulation.

1 of all the values of obtained in repeated

sampling from a given distribution, construct an interval

that includes (covers) the expected value of thepopulation.

x

+

nzx,

nzx 22

x

nz2 2

nzx 2

nzx 2

+

Lower confidence limit Upper confidence limit

1 -

Confidence level

Graphical Demonstration of the

Confidence Interval for

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The Confidence Interval for ( isknown)

Four commonly used confidence levels

Confidence

level /2/ 2/ 2/ 2

0.90 0.10 0.05 1.645

0.95 0.05 0.025 1.96

0.98 0.02 0.01 2.33

0.99 0.01 0.005 2.575

Confidence

level /2/ 2/ 2/ 2

0.90 0.10 0.05 1.645

0.95 0.05 0.025 1.96

0.98 0.02 0.01 2.33

0.99 0.01 0.005 2.575

z/2/2/2/2

Example: Estimate the mean value of the distribution resulting from

the throw of a fair die. It is known that = 1.71. Use a 90%

confidence level, and 100 repeated throws of the die

Solution: The confidence interval is

The Confidence Interval for ( isknown)

=

n

zx 2 28.x100

71.1645.1x =

The mean values obtained in repeated draws of samples of size100 result in interval estimators of the form

[sample mean - .28, Sample mean + .28],

90% of which cover the real mean of the distribution.

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The Confidence Interval for ( isknown)

Recalculate the confidence interval for 95% confidence level.

Solution: =

n

zx 2 34.x100

71.196.1x =

34.x +34.x

.95

.90

28.x +28.x

The Confidence Interval for ( isknown)

The width of the 90% confidence interval = 2(.28) = .56

The width of the 95% confidence interval = 2(.34) = .68 The width of the 90% confidence interval = 2(.28) = .56

The width of the 95% confidence interval = 2(.34) = .68

Because the 95% confidence interval is wider, it is

more likely to include the value of. Because the 95% confidence interval is wider, it is

more likely to include the value of.

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Example Doll Computer Company delivers computers

directly to its customers who order via theInternet.

To reduce inventory costs in its warehousesDoll employs an inventory model, thatrequires the estimate of the mean demandduring lead time.

It is found that lead time demand is normally

distributed with a standard deviation of 75computers per lead time.

Estimate the lead time demand with 95%confidence.

The Confidence Interval for ( isknown)

Example 10.1 Solution

The parameter to be estimated is , themean demand during lead time.

We need to compute the interval estimation

for .

From the data provided in file Xm10-01, the

sample mean is

The Confidence Interval for ( isknown)

.16.370=x

[ ]56.399,76.34040.2916.37025

7596.116.370

25

75z16.370

nzx

025.2

===

=

Since 1 - =.95, = .05.

Thus /2 = .025. Z.025 = 1.96

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Using Excel

Tools > Data Analysis Plus > Z Estimate:Mean

The Confidence Interval for ( isknown)

Wide interval estimator provides littleinformation. Where is ? ? ? ?

??? ??? ??? ??? ??? ??? ??? ??? ??? ??? ??? ??? ??? ??? ???

Information and the Width of the Interval

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Here is a much narrower interval.

If the confidence level remains

unchanged, the narrower interval

provides more meaningful

information.

Here is a much narrower interval.

If the confidence level remains

unchanged, the narrower interval

provides more meaningful

information.

Wide interval estimator provides littleinformation. Where is ? ? ? ?

Ahaaa!

Information and the Width of the Interval

The width of the confidence interval isaffected by

the population standard deviation () the confidence level (1-) the sample size (n).

The Width of the Confidence Interval

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90%

Confidence level

To maintain a certain level of confidence, a larger

standard deviation requires a larger confidence interval.To maintain a certain level of confidence, a larger

standard deviation requires a larger confidence interval.

n)645.1(2

nz2 05.

=

/2 = .05/2 = .05

n

5.1)645.1(2

n

5.1z2

05.

=

Suppose the standard

deviation has increased

by 50%.

The Affects of on the interval width

n)96.1(2

nz2

025.

=

/2 = 2.5%/2 = 2.5%

/2 = 5%/2 = 5%

n)645.1(2

nz2 05.

=

Confidence level

90%95%

Let us increase the

confidence level

from 90% to 95%.

Larger confidence level produces a wider confidence intervalLarger confidence level produces a wider confidence interval

The Affects of Changing the ConfidenceLevel

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90%

Confidence level

n)645.1(2

nz2 05.

=

Increasing the sample size decreases the width of theconfidence interval while the confidence level can remain

unchanged.

Increasing the sample size decreases the width of theconfidence interval while the confidence level can remain

unchanged.

The Affects of Changing the SampleSize

Selecting the Sample size

We can control the width of the confidenceinterval by changing the sample size.

Thus, we determine the interval width first, andderive the required sample size.

The phrase estimate the mean to within Wunits, translates to an interval estimate of the

form

wx

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The required sample size to estimate themean is

2

2

w

zn

=

Selecting the Sample size

Example

To estimate the amount of lumber thatcan be harvested in a tract of land, themean diameter of trees in the tractmust be estimated to within one inchwith 99% confidence.

What sample size should be taken?Assume that diameters are normallydistributed with = 6 inches.

Selecting the Sample size

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Solution

The estimate accuracy is +/-1 inch. That isw = 1.

The confidence level 99% leads to = .01,thus z/2 = z.005 = 2.575.

We compute

2391

)6(575.2

w

zn

22

2=

=

=

If the standard deviation is really 6 inches,

the interval resulting from the random sampling

will be of the form . If the standard deviationis greater than 6 inches the actual interval will

be wider than +/-1.

If the standard deviation is really 6 inches,

the interval resulting from the random sampling

will be of the form . If the standard deviation

is greater than 6 inches the actual interval will

be wider than +/-1.

1x

Selecting the Sample size