Chapter 8
Statistical inference: Significance
Tests About Hypotheses
 Learn
….
To use an inferential method called
a Significance Test
To analyze evidence that data provide
To make decisions based on data
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Two Major Methods for Making
Statistical Inferences about a
Population

Confidence Interval

Significance Test
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Questions that Significance
Tests Attempt to Answer

Does a proposed diet truly result in
weight loss, on the average?

Is there evidence of discrimination
against women in promotion decisions?

Does one advertising method result in
better sales, on the average, than another
advertising method?
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Section 8.1
What Are the Steps For
Performing a Significance Test?
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Hypothesis


A hypothesis is a statement about a
population, usually of the form that a
certain parameter takes a particular
numerical value or falls in a certain
range of values
The main goal in many research
studies is to check whether the data
support certain hypotheses
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Significance Test

A significance test is a method of
using data to summarize the evidence
about a hypothesis

A significance test about a hypothesis
has five steps
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Step 1: Assumptions

A (significance) test assumes that the
data production used randomization

Other assumptions may include:
• Assumptions about the sample size
• Assumptions about the shape of the
population distribution
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Step 2: Hypotheses

Each significance test has two
hypotheses:
• The null hypothesis is a statement that the
parameter takes a particular value
• The alternative hypothesis states that the
parameter falls in some alternative range
of values
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Null and Alternative Hypotheses

The value in the null hypothesis usually
represents no effect
• The symbol Ho denotes null hypothesis

The value in the alternative hypothesis
usually represents an effect of some type
• The symbol Ha denotes alternative
hypothesis
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Null and Alternative Hypotheses

A null hypothesis has a single
parameter value, such as Ho: p = 1/3

An alternative hypothesis has a range
of values that are alternatives to the
one in Ho such as
• Ha: p ≠ 1/3 or
• Ha: p > 1/3 or
• Ha: p < 1/3
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Step 3: Test Statistic


The parameter to which the
hypotheses refer has a point
estimate: the sample statistic
A test statistic describes how far that
estimate (the sample statistic) falls
from the parameter value given in the
null hypothesis
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Step 4: P-value

To interpret a test statistic value, we use a
probability summary of the evidence
against the null hypothesis, Ho
• First, we presume that Ho is true
• Next, we consider the sampling
•
distribution from which the test statistic
comes
We summarize how far out in the tail of
this sampling distribution the test statistic
falls
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Step 4: P-value

We summarize how far out in the tail
the test statistic falls by the tail
probability of that value and values
even more extreme
• This probability is called a P-value
• The smaller the P-value, the stronger
the evidence is against Ho
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Step 4: P-value
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Step 4: P-value


The P-value is the probability that the
test statistic equals the observed
value or a value even more extreme
It is calculated by presuming that the
null hypothesis H is true
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Step 5: Conclusion

The conclusion of a significance test
reports the P-value and interprets
what it says about the question that
motivated the test
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Summary: The Five Steps of a
Significance Test
1.
2.
3.
4.
5.
Assumptions
Hypotheses
Test Statistic
P-value
Conclusion
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Is the Statement a Null Hypothesis
or an Alternative Hypothesis?
In Canada, the proportion of adults who
favor legalize gambling is 0.50.
a. Null Hypothesis
b. Alternative Hypothesis
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Is the Statement a Null Hypothesis
or an Alternative Hypothesis?
a.
b.
The proportion of all Canadian college
students who are regular smokers is
less than 0.24, the value it was ten years
ago.
Null Hypothesis
Alternative Hypothesis
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 Section 8.4
Decisions and Types of Errors in
Significance Tests
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Type I and Type II Errors

When H0 is true, a Type I Error occurs
when H0 is rejected

When H0 is false, a Type II Error
occurs when H0 is not rejected
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Significance Test Results
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An Analogy: Decision Errors in
a Legal Trial
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P(Type I Error) = Significance
Level α

Suppose H0 is true. The probability of
rejecting H0, thereby committing a
Type I error, equals the significance
level, α, for the test.
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P(Type I Error)


We can control the probability of a
Type I error by our choice of the
significance level
The more serious the consequences
of a Type I error, the smaller α should
be
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Type I and Type II Errors

As P(Type I Error) goes Down, P(Type
II Error) goes Up
• The two probabilities are inversely
related
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A significance test about a
proportion is conducted using a
significance level of 0.05.
The test statistic is 2.58. The P-value is 0.01.
If Ho is true, for what probability of a
Type I error was the test designed?
a.
.01
b. .05
c.
2.58
d. .02
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A significance test about a
proportion is conducted using a
significance level of 0.05.
The test statistic is 2.58. The P-value is 0.01.
If this test resulted in a decision error,
what type of error was it?
a.
Type I
b. Type II
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