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222 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

As an example, suppose that the random variable X is normally distributed with an un-
known mean . The sample mean is a point estimator of the unknown population mean .
That is, ˆ X . After the sample has been selected, the numerical value is the point estimate
x
of . Thus, if x 25, x 30, x 29 1 2 3 , and x 31 , the point estimate of is

4
25 30 29 31
x 28.75
4
Similarly, if the population variance 2 is also unknown, a point estimator for 2 is the sample
2
variance S 2 , and the numerical value s 6.9 calculated from the sample data is called the
point estimate of 2 .
Estimation problems occur frequently in engineering. We often need to estimate
The mean of a single population
2
The variance (or standard deviation ) of a single population
The proportion p of items in a population that belong to a class of interest
The difference in means of two populations, 2
1
The difference in two population proportions, p p 2
1
Reasonable point estimates of these parameters are as follows:

For , the estimate is ˆ x, the sample mean.
2
2
For , the estimate is ˆ s 2 , the sample variance.
For p, the estimate is p ˆ x n , the sample proportion, where x is the number of items
in a random sample of size n that belong to the class of interest.
For 2 , the estimate is ˆ ˆ x x 2 , the difference between the sample
1
1
1
2
means of two independent random samples.
For p p 2 , the estimate is p ˆ p ˆ 2 , the difference between two sample proportions
1
1
computed from two independent random samples.
We may have several different choices for the point estimator of a parameter. For ex-
ample, if we wish to estimate the mean of a population, we might consider the sample
mean, the sample median, or perhaps the average of the smallest and largest observations
in the sample as point estimators. In order to decide which point estimator of a particular
parameter is the best one to use, we need to examine their statistical properties and develop
some criteria for comparing estimators.

7-2 GENERAL CONCEPTS OF POINT ESTIMATION

7-2.1 Unbiased Estimators

An estimator should be “close” in some sense to the true value of the unknown parameter.
ˆ
ˆ
Formally, we say that is an unbiased estimator of if the expected value of is equal to .
ˆ
This is equivalent to saying that the mean of the probability distribution of (or the mean of
ˆ

the sampling distribution of ) is equal to .