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Chapter 11

OMIS 2010 Chapter Notes - Chapter 11: Null Hypothesis, Type I And Type Ii Errors, Test Statistic

Department
Operations Management and Information System
Course Code
OMIS 2010
Professor
Alan Marshall
Chapter
11

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Chapter 11: Introduction to Hypothesis Testing
11.1 Introduction
This chapter introduced hypothesis testing, the central topic of a statistics course. Almost every
chapter following this one uses the concepts, terminology, and notation discussed here. As a result, your
understanding of the material in this chapter is critical to your mastery of the subject. We cannot state
this too strongly. If you do not have a strong grasp of this material, you will not be capable of applying
statistical techniques with any degree of success. We recommend that you spend as much time as
possible on this chapter. You will find it to be a worthwhile investment.
At the completion of this chapter you are expected to know the following:
1. Understand the fundamental concepts of hypothesis testing.
2. How to test hypotheses about the population mean.
3. How to set up the null and alternative hypotheses.
4. How to interpret the results of a test of hypothesis.
5. How to compute the p-value of a test when the sampling distribution is normal.
6. How to interpret the p-value of a test.
7. How to calculate the probability of a Type II error and interpret the results.
8. That there are five problem objectives and three types of data addressed in the book and that for
each combination there are one or more statistical techniques that can be employed.
9. Understand that the format of the statistical techniques introduced in subsequent chapters is
identical to those presented in this chapter and that the real challenge of this subject lies in
identifying the correct statistical technique to use.
11.2 Concepts of Hypothesis Testing
In this section, we presented the basic concepts of hypothesis testing. You are expected to know that
meaning of the new terms introduced in the section.
EXERCISES
11.1 Define the following terms:
a) Type I error
b) Type II error
c) Rejection region
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11.2 What do we call the probability of committing a Type I error?
11.3 What do we call the probability of committing a Type II error?
11.4 What is the significance level of a test?
11.5 What is the basis of the test statistic for a test of Âµ?
11.3 Testing the Population Mean When the Population Standard
Deviation Is Known
This section is extremely important because it demonstrates how to test a hypothesis and because
the method described here is repeated throughout Chapters 12 to 24. There are three critical elements
that you must understand:
1. How to set up the null and alternative hypotheses.
2. How to perform the required calculations, which include the determination of the rejection region
and the computation of the value of the test statistic.
3. How to interpret the results of the test.
Null and Alternative Hypotheses
Of these elements, the first is usually the most difficult to grasp. As we repeatedly point out in the
text, the null hypothesis must always specify that the parameter is equal to some particular value. Since
we cannot establish equality by using statistical methods, it falls to the alternative hypothesis to answer
the question. Hence, in order to specify the alternative hypothesis, you must determine what the question
asks. If it asks (either implicitly or explicitly) if there is sufficient evidence to conclude that Âµ is not
equal to (or is different from) a specific value (say, 100), then
H1: Âµ â‰  100
and of course it automatically follows that
H0: Âµ = 100
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If the question asks if there is sufficient evidence to conclude that Âµ is greater than 100, then
H1: Âµ > 100
and H0: Âµ = 100
If the question asks if there is sufficient evidence to conclude that Âµ is less than 100, then
H1: Âµ < 100
and H0: Âµ = 100
It should be noted that even though in the formal hypothesis test the null hypothesis precedes the
alternative hypothesis, we determine the alternative hypothesis first, and the null hypothesis automati-
cally follows.
The second element requires little more than arithmetic to determine the value of the test statistic,
which is
z=
x
âˆ’Âµ
Ïƒ/n
The value of Âµ in the test statistic comes from the null hypothesis.
Rejection Region
Some care must be exercised in setting up the rejection region. Bear in mind that the probability that
the test statistic falls into the rejection region is Î±. That means that in a two-tail test the rejection region
is
z >
z
Î±/2
or z < âˆ’
z
Î±/2
In a one-tail test with
H
0: Âµ = 100
H
1: Âµ > 100
the rejection region is
z >
z
Î±
Notice that we
z
Î± (rather than
z
Î±/2
) because the entire rejection region is located in only one tail of the
sampling distribution. Similarly, if we test
H
0: Âµ = 100
H
1: Âµ < 100
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