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Chapter 11

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Operations Management and Information System

OMIS 2010

Alan Marshall

Fall

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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. Alm ost every
chapter following this one uses the concepts, terminology, and notation discussed here. As a result, your
understanding of the m aterial in this chapter is criti cal to your m astery of the subject. W e cannot state
this too strongly. If you do not have a strong grasp of this m aterial, you will not be capable of applying
statistical techniques with any degree of successW e recommend that you spend as m uch 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
121 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 ?
11.3 Testing the Population Mean When the Population Standard
Deviation Is Known
This section is extremely important because it de monstrates how to test a hypothesis and because
the method described here is repeated throughout Chapters 12 to 24. There are three cr itical 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 alterna tive hypothesis to answer
the question. Hence, in order to specify the alternative hypothesis, you must determine what the question
asks. If it ask s (either implicitly or explicitly) if th ere is su fficient evidence to conclude that is not
equal to (or is different from) a specific value (say, 100), then
H 1 100
and of course it automatically follows that
H 0 = 100
122 If the question asks if there is sufficient evidence to conclude that is greater than 100, then
H 1 > 100
and
H 0: = 100
If the question asks if there is sufficient evidence to conclude that is less than 100, then
H 1: < 100
and
H 0: = 100
It should be noted that even t hough in the form al 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 m ore than arithmetic to determine the value of the test statistic,
which is
x
z = / n
The value of 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 . That means that in a two-tail test the rejection region
is
z > z or z < z
/ 2 / 2
In a one-tail test with
H 0: = 100
H 1: > 100
the rejection region is
z > z
Notice that we (rather than / 2) because the entire rejection region is located in only one tail of the
sampling distribution. Similarly, if we test
H : = 100
0
H 1: < 100
123 the rejection region is
z < z
Dont forget the minus signthis is a very common mistake.
p-Value of the Test
The p-value is th e probability of observing a v alue at least as extreme as the value of the test
statistic given that the null hypothesis is true. We can compute the p-value of a test manually only when
the sampling distribution is normal. For all other tests we need the computer to produce the p-value.
The interpretation of the p-value is quite straightforward. The smaller the p-value, the greater the
statistical evidence to reject the null hypothesis. That means that if a statistician judges that a p-value is
small then the conclusion of the test is to reject the null hypothesis. The definition of a small p-value is
subjective. That is, everyone must judge for himself whether a p-value is small enough. That choice usu-
ally is somewhere between 1% and 10%. For example, if the p-value of the test is .0372 and the statisti-
cian decides that for this test anything less than .05 is small enough, he would reject the null hypothesis.
Another statistician who decides that the p-value must be less than .01 to reject would not reject the null
hypothesis.
The calculation of the p-value is quite sim ilar to the norm al probability questions encountered in
Chapter 8. If th e test statistic is normally distributed (z-statistic), the p-value is simply the probability
that appears in one or bot h tails of the standard normal curve. Whether its one-tail or two-tail (and if
one-tail, which one) depends on t he rejection region. For example, in a two-tail test where the test
statistic is z = 2.0 and the rejection region is z / 2or z < z / 2
p-value = P(z > 2.0) + P(z < 2.0) = .0228 + .0228 = .0456
If the rejection region is z >zand z = 2.0, then
p-value = P(z > 2.0) = .0228
Finally, if the rejection region is z < az and z = 2.0, then
p-value = P(z < 2.0) = .9772
If this last calcu lation seems strange, remember that the direction of the inequality in the probability
statement is based completely on the rejection region.
124

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