introduction to Estimation

The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.. E.g., the sample mean is employed to estimate the popu

CHAPTER 9

In almost all realistic situations

parameters are unknown.

We will use the sampling distribution to draw inferences about the unknown

population parameters.

Statistical inference is the process by which we

acquire information and draw conclusions about populations from samples.

There are two procedures for making inferences:

• Estimation.

• Hypotheses testing.

Parameter

Population

Sample

Statistic

Inference

Data

Statistics

Information

The objective of estimation is to determine the

approximate value of a population parameter on the basis of a sample statistic.

E.g., the sample mean ( ) is employed to

estimate the population mean ( ).

There are two types of estimators:

Point Estimator

Interval Estimator

A point estimator draws inferences about a

population by estimating the value of an

point.

We saw earlier that point probabilities in

continuous distributions were virtually zero The probability of the point estimator being correct is zero

An interval estimator draws inferences

about a population by estimating the value

That is we say (with some _% certainty) that the population parameter of interest is between some lower and upper bounds.

For example, suppose we want to estimate the mean summer income of a class of business students For

n = 25 students, is calculated to be 400 $/week.

point estimate interval estimate

An alternative statement is:

The mean income is between 380 and 420 $/week.

ESTIMATING WHEN IS KNOWN…

From Chapter 9, the sampling distribution

of is approximately normal with mean µ and standard deviation

Thus

is (approximately) standard normally

distributed

From Chapter 8,

α

−

=

<

<

− Zα Z Zα ) 1 (

X

n /

σ

n /

X Z

σ

µ

−

=

Thus, substituting Z produces

With a little bit of algebra,

With a little bit of different algebra we have

α

−

=

<

σ

µ

−

<

n /

x z

(

α

−

=











µ − σ < < µ + σ

α

n

z

x n

z

P /2 /2

α

−

=











 − σ < µ < + σ

α

n

z

x n

z x

The confidence interval

Lower confidence limit (LCL) =

Upper confidence limit (UCL) =

The probability 1 – α is the confidence level , which is a measure of how frequently the

interval will actually include µ.











α

n

z

x /2











 + σ

α

n z

x /2

Four commonly used confidence levels

Confidence level α α/2

0.98 0.02 0.01 2.33

0.99 0.01 0.005 2.575

zα/

2

Example 2: Doll Computer Comp found

that the demand over the lead time is

normally distributed with a standard

deviation of 75 Estimate the expected demand over the lead time at 95%

confidence level Assume N=25 and x = 370 16

[ 340 76 , 399 56 ]

40 29 16

.

370 25

75 96

1 16

370

25

75 z

16

370 n

z

=

±

=

±

=

±

= σ

Comparing two confidence intervals with the same

level of confidence, the narrower interval provides

more information than the wider interval

The width of the confidence interval is calculated by

and therefore is affected by

• the population standard deviation (s)

• the confidence level (1-a)

• the sample size (n).

n

Z

2 n

z

x n

z

x 2 2 = /2 σ







−

−







1- α

Confidence level

α /2

α /2

n

5

1 z

2 /2 σ

α

n

z

2 /2 σ

α

If the standard deviation grows larger, a longer

confidence interval is needed to maintain the

confidence level

Note what happens when σ increases to 1.5 σ

Example 1: Estimate the mean value of the distribution resulting from the 100 repeated throws of the die It is known that σ = 1.71

Use 90% confidence level:

Use 95% confidence level:

=

σ

n

z

100

71

1 645

1

=

σ

n

z

100

71

1 96

1

90%

Confidence level

n

) 96 1 (

2 n

z

α /2 = 2.5%

n

) 645 1 (

2 n

z

2 .05 σ = σ

α /2 = 5%

α /2 = 5%

Larger confidence level requires longer confidence

interval

By increasing the sample size we can decrease

the width of the confidence interval while the

confidence level can remain unchanged.

n

z 2 width

α

There is an inverse relationship between the width of

the interval and the sample size

The phrase “estimate the mean to within

W units”, translates to an interval

estimate of the form

where W is the margin of error.

W

n

W

W

x ±

The required sample size to estimate

the mean is

2 2

α

W

σ

z











=

2 2

α

W

σ

z











=

Example 4: To estimate the amount of lumber that can be harvested in a tract of land, the

mean diameter of trees in the tract must be

estimated to within one inch with 99%

confidence What sample size should be taken for the margin of error +/-1 inch? (assume

diameters are normally distributed with σ = 6

inches).

The confidence level 99% leads to α = 01,

thus zα/2 = z.005 = 2.575

239 1

2.575(6) W

σ

z n

2

2 2

α

=









=













=