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Chapter 9
Fundamentals of Hypothesis Testing:
One-Sample Tests
9.1: Hypothesis Testing Methodology
• Confidence Intervals were our first Inference
• Hypothesis Tests are our second Inference
• “Methodology” implies a series of steps:
1. Develop hypotheses
2. Determine decision rule
3. Calculate test statistic
4. Compare results from 2 and 3: make a decision
5. Write conclusion
Step 1: Develop hypotheses
• You will need to develop 2
hypotheses:
1. Null hypothesis
2. Alternative hypothesis
– Hypotheses concern the population
parameter in question (ie “µ” or “π” or
other)
The Null Hypothesis
• A theory or idea about the population
parameter.
• Always contains some sort of
equality.
• Very often described as the
hypothesis of “no difference” or
“status quo.”
• H0: µ = 368
The Alternative Hypothesis
• An idea about a population
parameter that is the opposite of
the idea in the null hypothesis
• NEVER contains any sort of
equality!
• H1: µ  368 (sometimes use Ha:)
Hypotheses
• Null and alternative hypotheses are
mutually exclusive and collectively
exhaustive.
• Our sample either contains enough
information to reject the NULL hypothesis
OR the sample does not contain enough
information to reject the null hypothesis.
“Proof”
• There is no proof.
• There is only supporting
information.
Step 2: Decision Rule or
“Rejection Region”
• Always says something like “we shall reject H0
for some extreme value of the test statistic.”
• The “Rejection Region” is the range of the test
statistic that is extreme enough—so extreme that
the test statistic probably would not occur IF the
null hypothesis is true.
• Figure 9.1 shows the rejection region for a
hypothesis test of the mean.
• The “critical value” is looked up based on the
error rate that you are comfortable with.
Step 3: Calculating the Test
Statistic
• The test statistic depends on the sampling
distribution in use. This depends on the
parameter.
• This will be determined the same way it was in
chapter 8.
Step 4 & 5: Decision and
Conclusion
• The decision is always either (1) reject
H0 or (2) fail to reject H0. This is
determined by evaluating the decision
rule in step 2.
• The conclusion always says “At α =
0.05, there is (in)sufficient
information to say H1”
Alpha
• α is the probability of committing a Type I
error: erroneously rejecting a true Null
Hypothesis.
• α is called “The Level of Significance”
• α is determined before the sample results are
examined.
• α determines the critical value and rejection
region(s).
• α is set at an acceptably low level.
Beta
• Beta is the probability of committing
a Type II error: erroneously failing to
reject a false null hypothesis.
• Beta depends on several factors and it
cannot be arbitrarily set. Beta can be
indirectly influenced.
Compliments of Alpha and Beta
• (1-α) is called the confidence coefficient.
This is what we used in Chapter 8.
• (1-beta) is called the Power of the test.
Power is the chance of rejecting a null
hypothesis that ought to be rejected, ie a
false null. Bigger is better. Power cannot
be set directly.
9.2: z Test of Hypothesis for the mean
• Use this test ONLY for the mean
and only when σ is known.
• There are two approaches:
– critical value approach
– “p” value approach
Critical Value Approach
Remember your methodology (steps):
1. Create hypotheses
2. Create decision rule
– depends on α
– depends on distribution
3. Calculate test statistic
4. State the result
5. State the managerial conclusion
Hypotheses
• The discussion in 7.2 assumes a twotail test because the sample mean
might be extremely large or extremely
small.
– Either one would make you think the null
hypothesis is wrong.
Rejection Region
• The standard approach requires that the
value of α be divided evenly between
the tail areas.
• These tail areas are called the
“rejection region.”
Conclusion
• See Step 6 on pages 308-309.
“p-value” approach
• Rewrite the decision rule to say, “we will
reject the null hypothesis if the ‘p-value’ is
less than the value of α .”
• “p-value” definition, page 309.
• “p-value” is called the observed level of
significance.
• Excel--most statistical software--does a
good job of this (that’s why it’s a popular
approach).
Estimation and Hypothesis Testing
• The two inferences are closely related.
• Estimation answers the question “what is
it?”
• Hypothesis Testing answers the question “is
it ______ than some number?”
• See page 312.
9.3: One-Tail Tests
• The rejection region is one single area.
• Sometimes called a directional test.
• Mechanics:
– see the text example
• Problem identification:
– hypothesis test or interval estimation?
– One-tail or Two-tail?
– If One-tail, which is the null?
Text Example
• Page 314, the “milk problem.”
• Are we buying “watered-down milk” ?
– Watered-down milk freezes at a colder
temperature than normal milk.
– What are the null and alternative hypotheses?
– Hint: what do you want to conclude?
– Hint: what is the hypothesis of action?
– Hint: what is the hypothesis of status quo?
Mechanics of the One-tailed test
• Different hypotheses.
• Different decision rule/rejection
region.
• Different “p-value” or observed
significance of observed level of
significance.
Consider Problem 9.44, page 317
• Reading only the context, not the steps (a,
b, etc.), can you tell that the problem calls
for a hypothesis test?
– Knowing that a test of hypothesis is
called for, can you determine that a
one-tail test is appropriate?
• Knowing that a one-tail test is
to be used, can you set up the
hypotheses?
9.4: t test of Hypothesis for the
Mean (σ Unknown)
• When σ is unknown, the distribution for
x-bar is a “t” distribution with n-1
degrees of freedom.
• Use sample standard deviation “s” to
estimate σ.
• This test is more commonly used than
the z test.
Assumptions
• Random Sample.
• You must assume that the underlying
population, i.e. the underlying random
variable x is distributed normally.
• This test is very “robust” in that it
does not lose power for small
violations of the above assumption.
Methodology
You still need your 5 step Hypothesis Test
Methodology.
– The critical value approach is the same as
that for the z test.
– The p-value method does not work as
well when done by hand because of
limitations in the “t” table.
– One- and Two-tail tests are possible.
• You could add another step to check
assumptions.
EXAMPLE
• 9.54, page 323
9.5: z Test of Hypothesis for
the Proportion
• For the nominal variable—variable
values are categories and you tend to
describe the data set in terms of
proportions.
• Both one- and two-tail tests are
possible.
• Problem 9.72 on page 329 is a good
example.
Assumptions
• The number of observations of
interest (successes) and the number
of uninteresting observations
(failures) are both at least 5.