1 Introduction to Estimation Chapter 10. 2 10.1 Introduction Statistical inference is the process by...

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Introduction to EstimationIntroduction to Estimation

Chapter 10

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10.1 Introduction

Statistical inference is the process by which we acquire information about populations from samples. There are two types of inference:

Estimation Hypotheses testing

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10.2 Concepts of Estimation

The objective of estimation is to determine the value of a population parameter on the basis of a sample statistic. There are two types of estimators:Point EstimatorInterval estimator

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Point Estimator

A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or point.

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Population distribution

Point Estimator

Parameter

?

Sampling distribution

A point estimator draws inference about a population by estimating the value of an unknown parameter using a single value or point.

Point estimator

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An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval.

Interval estimator

Population distribution

Sample distribution

Parameter

Interval Estimator

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Selecting the right sample statistic to estimate a parameter value depends on the characteristics of the statistic.

Estimator’s Characteristics

Estimator’s 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 and the parameter grows smaller as the sample size increases.

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

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10.3 Estimating the Population Mean when the Population Variance is Known

How is an interval estimator produced from a sampling distribution?A sample of size n is drawn from the population, and

its mean is calculated.By the central limit theorem is normally distributed

(or approximately normally distributed.), thus…x

x

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nx

Z

We have established before that

1)n

zxn

z(P 22

10.3 Estimating the Population Mean when the Population Variance is Known

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1)n

zxn

zx(P 22

This leads to the following equivalent statement

The Confidence Interval for ( is known)

The confidence interval

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Interpreting the Confidence Interval for

1 – of all the values of obtained in repeatedsampling from a given distribution, construct an interval

that includes (covers) the expected value of thepopulation.

1 – of all the values of obtained in repeatedsampling from a given distribution, construct an interval

that includes (covers) the expected value of thepopulation.

x

nzx,

nzx 22

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x

nz2 2

nzx 2

n

zx 2

Lower confidence limit Upper confidence limit

1 -

Confidence level

Graphical Demonstration of the Confidence Interval for

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Four commonly used confidence levels

Confidence level 0.90 0.10 0.05 1.6450.95 0.05 0.025 1.960.98 0.02 0.01 2.330.99 0.01 0.005 2.575

Confidence level 0.90 0.10 0.05 1.6450.95 0.05 0.025 1.960.98 0.02 0.01 2.330.99 0.01 0.005 2.575

z

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• 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


nzx 2 28.x

10071.1

645.1x

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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• Recalculate the confidence interval for 95% confidence level. • Solution:

n

zx 2 34.x10071.1

96.1x

34.x 34.x

.95

.90

28.x 28.x

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• 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 10.1Doll Computer Company delivers computers directly to

its customers who order via the Internet.To reduce inventory costs in its warehouses Doll

employs an inventory model, that requires the estimate of the mean demand during lead time.

It is found that lead time demand is normally distributed with a standard deviation of 75 computers per lead time.

Estimate the lead time demand with 95% confidence.


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Example 10.1 – SolutionThe parameter to be estimated is the mean

demand during lead time.We need to compute the interval estimation for From the data provided in file Xm10-01, the sample

mean is


.16.370x

56.399,76.34040.2916.37025

7596.116.370

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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


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Wide interval estimator provides little information.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 little information.Where is

Ahaaa!

Information and the Width of the Interval

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The width of the confidence interval is affected by

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

The width of the confidence interval is affected 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

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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 Confidence Level

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90%

Confidence level

n)645.1(2

nz2 05.

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

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

The Affects of Changing the Sample Size

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10.4 Selecting the Sample size

We can control the width of the confidence interval by changing the sample size.

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

The phrase “estimate the mean to within W units”, translates to an interval estimate of the form

wx wx

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The required sample size to estimate the mean is

Click to see how the formula is developed.

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w

zn

22

w

zn

10.4 Selecting the Sample size

Here

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Example 10.2To 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? Assume that diameters are normally distributed with = 6 inches.

Selecting the Sample size

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SolutionThe estimate accuracy is +/-1 inch. That is w = 1.The confidence level 99% leads to = .01, thus z/2

= z.005 = 2.575.We compute 239

1)6(575.2

w

zn

222

If the standard deviation is really 6 inches, the interval resulting from the random samplingwill be of the form . If the standard deviation is greater than 6 inches the actual interval willbe wider than +/-1.

If the standard deviation is really 6 inches, the interval resulting from the random samplingwill be of the form . If the standard deviation is greater than 6 inches the actual interval willbe wider than +/-1.

1x

Selecting the Sample size

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