6 Oct 2011

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Chapter 10: Introduction to Estimation

10.1 Introduction

This chapter introduced estimation, a form of statistical inference. Make certain that you understand

the sampling distribution of the mean before proceeding.

At the completion of this chapter, you are expected to know the following:

1. Understand the fundamental concepts of estimation.

2. How to produce interval estimates of the population mean.

3. How to interpret interval estimates.

4. How to determine the sample size to estimate a population mean.

10.2 Concepts of Estimation

This section dealt with the fundamental concepts of estimation. You are expected to know the mean-

ing of the terms introduced in this chapter and understand why estimation is necessary in most practical

situations.

EXERCISES

10.1 What is the difference between an estimator and an estimate?

10.2 Define each of the following terms:

a) Unbiased estimator

b) Consistent estimator

10.3 Explain why

x

is an unbiased estimator of µ.

10.4 Explain why s2 is an unbiased estimator of σ2.

10.5 Explain why

x

is a consistent estimator of µ.

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10.3 Estimating the Population Mean Value When

the Population Standard Deviation Is Known

We presented the technique to produce a interval estimate of a population mean when the

population variance is known. The interval estimator of µ is

x ±zα/2σ/n

It is assumed that the population variance is known. The confidence level is equal to 1 – α, and from this

value you need to be able to determine

z

α/2

. Other than plugging the numbers into the formula, no

other arithmetic skills are required.

Question: How can I tell when the population variance is known?

Answer: The question must state explicitly that the population variance σ2 is known,

and it must also specify the value.

Question: How do I find

z

α/2

?

Answer: The question will specify the confidence level 1 – α. From the confidence

level, determine a and divided it by 2. For example, if the confidence level is

.84, then since 1 – α = .84 it follows that α = .16 and α/2 = .08. To find

z

α/2 =

z

.08 , we need to find the value of z such that the area to its right is

.08.

Since Table 3 is designed to provide the areas between 0 and z, we need to

subtract .08 from .5 to produce .42. (Notice that we can get .42 more directly

by dividing the confidence level by 2.) Now in Table 3 find the area closest to

.42 (.4200). As you should be able to see, the area between 0 and 1.40 is

.4192 and the area between 0 and 1.41 is .4207. Since the latter area is closer

to .4200, we find z.08 = 1.41.

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