Applied Statistics and Probability for Engineers

226 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

estimate. Since many point estimators are normally distributed (or approximately so) for large

n, this is a very useful result. Even in cases in which the point estimator is not normally

distributed, we can state that so long as the estimator is unbiased, the estimate of the parameter

will deviate from the true value by as much as four standard errors at most 6 percent of the time.

Thus a very conservative statement is that the true value of the parameter differs from the point

estimate by at most four standard errors. See Chebyshev’s inequality in the CD only material.

EXAMPLE 7-2 An article in the Journal of Heat Transfer(Trans. ASME, Sec. C, 96, 1974, p. 59) described

a new method of measuring the thermal conductivity of Armco iron. Using a temperature of

100 F and a power input of 550 watts, the following 10 measurements of thermal conductiv-

ity (in Btu/hr-ft- F) were obtained:

A point estimate of the mean thermal conductivity at and 550 watts is the sample mean or

The standard error of the sample mean is , and since is unknown, we may replace

it by the sample standard deviation to obtain the estimated standard error of as

Notice that the standard error is about 0.2 percent of the sample mean, implying that we have ob-

tained a relatively precise point estimate of thermal conductivity. If we can assume that thermal

conductivity is normally distributed, 2 times the standard error is  0.1796,

and we are highly confident that the true mean thermal conductivity is with the interval

, or between 41.744 and 42.104.

7-2.5 Bootstrap Estimate of the Standard Error (CD Only)

7-2.6 Mean Square Error of an Estimator

Sometimes it is necessary to use a biased estimator. In such cases, the mean square error of the

estimator can be important. The mean square errorof an estimator is the expected squared

difference between ˆand .

ˆ

41.924 0.1756

2 ˆX 21 0.0898 2

ˆX

s

1 n



0.284

110

0.0898

s0.284 X

X^1 n 

x41.924 Btu/hr-ft- F

100 F

41.60, 41.48, 42.34, 41.95, 41.86,

42.18, 41.72, 42.26, 41.81, 42.04

The mean square errorof an estimator of the parameter is defined as

MSE 1 ˆ 2 E 1 ˆ  22 (7-3)

ˆ

Definition

The mean square error can be rewritten as follows:

V 1 ˆ 2  1 bias 22

MSE 1 ˆ 2 E 3 ˆ E 1 ˆ 242  3 E 1 ˆ 242

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