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252 Chapter 7/Point Estimation of Parameters and Sampling Distributions

Standard Error of
an Estimator
ˆ
Θ
ˆ
The standard error of an estimator Θ is its standard deviation given by σ = V ( ).
ˆ
Θ
If the standard error involves unknown parameters that can be estimated, substitution
of those values into σ ˆ produces an estimated standard error, denoted by ˆ σ .
Θ
ˆ
Θ
ˆ
Sometimes the estimated standard error is denoted by s ˆ or se( )Θ .
Θ
2
Suppose that we are sampling from a normal distribution with mean μ and variance σ . Now
2
the distribution of X is normal with mean μ and variance σ /n, so the standard error of X is
σ
σ =
X
n
If we did not know σ but substituted the sample standard deviation S into the preceding equa-
tion, the estimated standard error of X would be
SE( ) = σ = S
ˆ
X
X
n
When the estimator follows a normal distribution as in the preceding situation, we can be rea-

sonably conident that the true value of the parameter lies within two standard errors of the esti-
mate. Because many point estimators are normally distributed (or approximately so) for large n,
this is a very useful result. Even when 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 conserv-
ative 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 supplemental material on the Web site.
Example 7-5 Thermal Conductivity 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 conductivity (in
Btu / hr-ft-°F) were obtained:
.
.
.
.
.
,
,
,
41 60 41 48 42 34 41 95 41 86,
,
.
,
.
.
,
,
.
.
42 18 41 72 42 26 41 81, 442 04
A point estimate of the mean thermal conductivity at 100°F and 550 watts is the sample mean or
x = 41 .924 Btu / hr-ft- °F
The standard error of the sample mean is σ = σ / n , and because σ is unknown, we may replace it by the sample
X
0
standard deviation s = .284 to obtain the estimated standard error of X as
∧
SE( ) = σ = s = . 0 284 = . 0 0898
X
X
n 10
Practical Interpretation: Notice that the standard error is about 0.2 percent of the sample mean, implying that we
have obtained a relatively precise point estimate of thermal conductivity. If we can assume that thermal conductivity is
normally distributed, 2 times the standard error is 2σ = ( .
∧
2 0 0898) = 0.1796, and we are highly coni dent that the true
mean thermal conductivity is within the interval 41 924. X ± 0 1796 or between 41.744 and 42.104.
.
7.3.4 Bootstrap Standard Error
In some situations, the form of a point estimator is complicated, and standard statistical
methods to i nd its standard error are difi cult or impossible to apply. One example of
these is S, the point estimator of the population standard deviation σ. Others occur with