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Chapter 11: Testing a Claim
Section11.1
Significance Tests: The Basics
The Practice of Statistics, 3rd edition – For AP*
STARNES, YATES, MOORE
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Chapter 11
Testing a Claim
 11.1
Significance Tests: The Basics
 11.2
Tests about a Population Proportion
 11.3
Tests about a Population Mean
+ Section 11.1
Significance Tests: The Basics
Learning Objectives
After this section, you should be able to…

STATE correct hypotheses for a significance test about a population
proportion or mean.

INTERPRET P-values in context.

INTERPRET a Type I error and a Type II error in context, and give
the consequences of each.

DESCRIBE the relationship between the significance level of a test,
P(Type II error), and power.
Introduction…
In addition to estimation, hypothesis testing is a procedure
for making inferences about a population.
Population
Sample
Inference
Statistic
Parameter
Hypothesis testing allows us to determine whether enough
statistical evidence exists to conclude that a belief (i.e.
hypothesis) about a parameter is supported by the data.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.4
A significance test is a formal procedure for comparing observed
data with a claim (also called a hypothesis) whose truth we want
to assess. The claim is a statement about a parameter, like the
population proportion p or the population mean µ. We express the
results of a significance test in terms of a probability that
measures how well the data and the claim agree.
In this chapter, we’ll learn the underlying logic of statistical tests,
how to perform tests about population proportions and population
means, and how tests are connected to confidence intervals.
Significance Tests: The Basics
Confidence intervals are one of the two most common types of
statistical inference. Use a confidence interval when your goal is
to estimate a population parameter. The second common type of
inference, called significance tests, has a different goal: to
assess the evidence provided by data about some claim
concerning a population.
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 Introduction
Reasoning of Significance Tests
We can use software to simulate 400 sets of 50 shots
assuming that the player is really an 80% shooter.
You can say how strong the evidence
against the player’s claim is by giving the
probability that he would make as few as
32 out of 50 free throws if he really makes
80% in the long run.
The observed statistic is so unlikely if the
actual parameter value is p = 0.80 that it
gives convincing evidence that the player’s
claim is not true.
Significance Tests: The Basics
Statistical
deal with
claims
abouttoa be
population.
Tests ask ifshooter.
sample
Suppose atests
basketball
player
claimed
an 80% free-throw
good evidence
against
a claim.
A test might He
say,makes
“If we 32
took
Todata
test give
this claim,
we have him
attempt
50 free-throws.
of
manyHis
random
samples
andof
the
claimshots
wereistrue,
we=would
them.
sample
proportion
made
32/50
0.64. rarely get a
result like this.” To get a numerical measure of how strong the sample
What
can weis,conclude
about
the term
claim“rarely”
based on
sample data?
evidence
replace the
vague
by this
a probability.
+
 The
Reasoning of Significance Tests
In reality, there are two possible explanations for the fact that he
made only 64% of his free throws.
1) The player’s claim is correct (p = 0.8), and by bad luck, a
very unlikely outcome occurred.
2) The population proportion is actually less than 0.8, so the
sample result is not an unlikely outcome.
Basic Idea
An outcome that would
rarely happen if a claim
were true is good evidence
that the claim is not true.
Significance Tests: The Basics
Based on the evidence, we might conclude the player’s claim is
incorrect.
+
 The
Hypotheses
Definition:
The claim tested by a statistical test is called the null hypothesis (H0).
The test is designed to assess the strength of the evidence against the
null hypothesis. Often the null hypothesis is a statement of “no
difference.”
The claim about the population that we are trying to find evidence for is
the alternative hypothesis (Ha).
In the free-throw shooter example, our hypotheses are
H0 : p = 0.80
Ha : p < 0.80
where p is the long-run proportion of made free throws.
Significance Tests: The Basics
A significance test starts with a careful statement of the claims we want to
compare. The first claim is called the null hypothesis. Usually, the null
hypothesis is a statement of “no difference.” The claim we hope or
suspect to be true instead of the null hypothesis is called the alternative
hypothesis.
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 Stating
Hypotheses
Definition:
The alternative hypothesis is one-sided if it states that a parameter is
larger than the null hypothesis value or if it states that the parameter is
smaller than the null value.
It is two-sided if it states that the parameter is different from the null
hypothesis value (it could be either larger or smaller).
 Hypotheses always refer to a population, not to a sample. Be sure
to state H0 and Ha in terms of population parameters.
 It is never correct to write a hypothesis about a sample statistic,
ˆ  0.64 or x  85.
such as p
Significance Tests: The Basics
In any significance test, the null hypothesis has the form
H0 : parameter = value
The alternative hypothesis has one of the forms
Ha : parameter < value
Ha : parameter > value
Ha : parameter ≠ value
To determine the correct form of Ha, read the problem carefully.
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 Stating
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Activity: For each pair of
Must use indicate
parameterwhich
(population)
x
hypotheses,
are not
Must be(sample)
NOT equal!
is
a
statistic
legitimate & explain why
a) H 0 :   15 ; H a :   15
p is the population
b) H 0 : x  123 ;proportion!
H a : x  123
Must use same
is parameter
for
population
number
as
H
!
c) H
:


.
1
;
H
0
0
a :   .1
correlation coefficient – but H0
d) H 0 : MUST
 .4;beH“=“
!  .6
a :
e) H 0 :   0 ; H a :   0
Concepts of Hypothesis Testing…
Once the null and alternative hypotheses are stated, the next
step is to randomly sample the population and calculate a test
statistic.
If the test statistic’s value is inconsistent with the null
hypothesis we reject the null hypothesis and infer that the
alternative hypothesis is true. However, if the test statistic’s
value is consistent with the null hypothesis, we cannot reject
it.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.11
Concepts of Hypothesis Testing …
So there are two possible decisions that can be made:
Conclude that there is enough evidence to support the
alternative hypothesis
(also stated as: rejecting the null hypothesis in favor of the
alternative)
Conclude that there is not enough evidence to support the
alternative hypothesis
(also stated as: not rejecting the null hypothesis in favor of
the alternative)
NOTE: we do not say that we accept the null hypothesis…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
11.12
Studying Job Satisfaction
a) Describe the parameter of interest in this setting.
The parameter of interest is the mean µ of the differences (self-paced
minus machine-paced) in job satisfaction scores in the population of all
assembly-line workers at this company.
b) State appropriate hypotheses for performing a significance test.
Because the initial question asked whether job satisfaction differs, the
alternative hypothesis is two-sided; that is, either µ < 0 or µ > 0. For
simplicity, we write this as µ ≠ 0. That is,
H0: µ = 0
Ha: µ ≠ 0
Significance Tests: The Basics
Does the job satisfaction of assembly-line workers differ when their work is machinepaced rather than self-paced? One study chose 18 subjects at random from a
company with over 200 workers who assembled electronic devices. Half of the
workers were assigned at random to each of two groups. Both groups did similar
assembly work, but one group was allowed to pace themselves while the other
group used an assembly line that moved at a fixed pace. After two weeks, all the
workers took a test of job satisfaction. Then they switched work setups and took
the test again after two more weeks. The response variable is the difference in
satisfaction scores, self-paced minus machine-paced.
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 Example:
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HW #24 Pg 690 #1,3,5,6
P-Values
Definition:
The probability, computed assuming H0 is true, that the statistic would
take a value as extreme as or more extreme than the one actually
observed is called the P-value of the test. The smaller the P-value, the
stronger the evidence against H0 provided by the data.
 Small P-values are evidence against H0 because they say that the
observed result is unlikely to occur when H0 is true.
 Large P-values fail to give convincing evidence against H0 because
they say that the observed result is likely to occur by chance when H0
is true.
Significance Tests: The Basics
The null hypothesis H0 states the claim that we are seeking evidence
against. The probability that measures the strength of the evidence
against a null hypothesis is called a P-value.
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 Interpreting
Studying Job Satisfaction pg 697
self - paced environment, on average, 17 points higher. Researchers performed a
significance test using the sample data and found a p - value of 0.2302.
a) Explain what it means for the null hypothesis to be true in this setting.
In this setting, H0: µ = 0 says that the mean difference in satisfaction
scores (self-paced - machine-paced) for the entire population of
assembly-line workers at the company is 0. If H0 is true, then the workers
don’t favor one work environment over the other, on average.
b) Interpret the P-value in context.
Significance Tests: The Basics
For the job satisfaction study, the hypotheses are
H0: µ = 0
Ha: µ ≠ 0
Data from the 18 workers gave x  17 and sx  60. That is, these workers rated the
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 Example:
 The P-value is the probability of observing a sample result as extreme or more
extreme in the direction specified by Ha just by chance when H0 is actually true.
An outcome that would occur so often just by chance
Because the alternative hypothesis is two - sided, the P - value is the probability of
(almost
in every
samples
18observed
workers)
when
 getting 1
a value
of x as4farrandom
from 0 in either
directionof
as the
x  17
when
H0 H
is 0true.
average
difference of evidence
17 or more points
betweenHthe
is That
trueis,isannot
convincing
against
0. two
work environments would
happen
of theHtime
just by chance in random
We fail
to23%
reject
0: µ = 0.
samples of 18 assembly - line workers when the true population mean is  = 0.
Significance

If our sample result is too unlikely to have happened by chance
assuming H0 is true, then we’ll reject H0.

Otherwise, we will fail to reject H0.
Note: A fail-to-reject H0 decision in a significance test doesn’t mean
that H0 is true. For that reason, you should never “accept H0” or use
language implying that you believe H0 is true.
In a nutshell, our conclusion in a significance test comes down to
P-value small → reject H0 → conclude Ha (in context)
P-value large → fail to reject H0 → cannot conclude Ha (in context)
Significance Tests: The Basics
The final step in performing a significance test is to draw a conclusion
about the competing claims you were testing. We will make one of two
decisions based on the strength of the evidence against the null
hypothesis (and in favor of the alternative hypothesis) -- reject H0 or fail
to reject H0.
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 Statistical
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HW #25 Pg 698 #7-9,11,12
Hypotheses: Null H0 & Alternative Ha
• Think of the null hypothesis as the status quo
• Think of the alternative hypothesis as something has
changed or is different than expected
• We can not prove the null hypothesis! We only can find
enough evidence to reject the null hypothesis or not.
Example 4
Several cities have begun to monitor paramedic response
times. In one such city, the mean response time to all
accidents involving life-threatening injuries last year was
μ=6.7 minutes with σ=2 minutes. The city manager
shares this info with the emergency personnel and
encourages them to “do better” next year. At the end of
the following year, the city manager selects a SRS of 400
calls involving life-threatening injuries and examines
response times. For this sample the mean response time
was x-bar = 6.48 minutes. Do these data provide good
evidence that the response times have decreased since
last year?
List parameter, hypotheses and conditions check
Example 4 cont
Parameter:
H0: μ = 6.7 minutes (unchanged)
Ha: μ < 6.7 minutes (they got “better”)
Conditions Check:
1) SRS : stated in problem statement
2) Normality : n = 400 suggest CLT would apply to x-bar
3) Independence:
n = 400 means we must assume over 4000 calls
each year that involve life-threatening injuries
P-value
• P-value is the probability of getting a more
extreme value if H0 is true (measures the tails)
• Small P-values are evidence against H0
– observed value is unlikely to occur if H0 is true
• Large P-values fail to give evidence against H0
P-Value Approach
z0
-|z0|
|z0|
z0
P-Value is the
area highlighted
Test Statistic:
x – μ0
z0 = ------------σ/√n
Reject null hypothesis, if
P-Value < α
• Probability(getting a result further away from the point
estimate) = p-value
• P-value is the area in the tails!!
Significance
Definition:
If the P-value is smaller than alpha, we say that the data are
statistically significant at level α. In that case, we reject the null
hypothesis H0 and conclude that there is convincing evidence in favor
of the alternative hypothesis Ha.
When we use a fixed level of significance to draw a conclusion in a
significance test,
P-value < α → reject H0 → conclude Ha (in context)
P-value ≥ α → fail to reject H0 → cannot conclude Ha (in context)
Significance Tests: The Basics
There is no rule for how small a P-value we should require in order to reject
H0 — it’s a matter of judgment and depends on the specific
circumstances. But we can compare the P-value with a fixed value that
we regard as decisive, called the significance level. We write it as α,
the Greek letter alpha. When our P-value is less than the chosen α, we
say that the result is statistically significant.
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 Statistical
Example 5: P-Values
For each α and observed significance level (p-value)
pair, indicate whether the null hypothesis would be
rejected.
a) α = . 05, p = .10
α < P  fail to reject Ho
b) α = .10, p = .05
P < α  reject Ho
c) α = .01 , p = .001
P < α  reject Ho
d) α = .025 , p = .05
α < P  fail to reject Ho
e) α = .10, p = .45
α < P  fail to reject Ho
Example 4 cont
• What is the P-value associated with the data in
example 4?
x – μ0
6.48 – 6.7
Z0 = ----------- = -------------σ/√n
0.10
= -2.2
P(z < Z0) = P(z < -2.2)
= 0.0139 (unusual !)
• What if the sample mean was 6.61?
x – μ0
6.61 – 6.7
Z0 = ----------- = -------------- = - 0.9
σ/√n
0.10
P(z < Z0) = P(z < -0.9)
= 0.1841 (not unusual !)
Two-sided Test P-value
• P-value is the sum of both tail areas in the two sided
test case
Statistical Significance Definition
• Statistically significant means simply that it is
not likely to happen just by chance
• Significant in the statistical sense does not
mean important
• Very large samples can make very small
differences statistically significant, but not
practically important
Statistical Significance – P-value
When using a P-value, we compare it with a level
of significance, α, decided at the start of the test.
• Not significant when α < P
• Significant when α ≥ P
Fail to Reject H0
Reject H0
Statistical Significance Interpretation
Remember the three C’s:
Conclusion, connection, context
• Conclusion: Either we have evidence to reject
H0 in favor of Ha or we fail to reject
• Connection: connect your calculated values
to your conclusion
• Context: Always put it in terms of the
problem (don’t use generalized statements)
Statistical Significance Warnings
• If you are going to draw a conclusion base on
statistical significance, then the significance
level α should be stated before the data are
produced
– Deceptive users of statistics might set an α level
after the data have been analyzed to manipulate
the conclusion
– P-values give a better sense of how strong the
evidence against H0 is
• This is just as inappropriate as choosing an
alternative hypothesis to be one-sided in a
particular direction after looking at the data
Summary
• Summary
– Significance test assesses evidence provided by
data against H0 in favor of Ha
– Ha can be two-sided (different, ≠) or one-sided
(specific direction, < or >)
– Same three conditions as with confidence intervals
– Test statistic is usually a standardized value
– P-value, the probability of getting a more extreme
value given that H0 is true  is small we reject H0
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HW #26 Pg 701 #13-14, 18, 20-24
Better Batteries
a) What conclusion can you make for the significance level α = 0.05?
Since the P-value, 0.0276, is less than α = 0.05, the sample result is
statistically significant at the 5% level. We have sufficient evidence to
reject H0 and conclude that the company’s deluxe AAA batteries last
longer than 30 hours, on average.
b) What conclusion can you make for the significance level α = 0.01?
Since the P-value, 0.0276, is greater than α = 0.01, the sample result is
not statistically significant at the 1% level. We do not have enough
evidence to reject H0 in this case. therefore, we cannot conclude that the
deluxe AAA batteries last longer than 30 hours, on average.
Significance Tests: The Basics
A company has developed a new deluxe AAA battery that is supposed to last longer
than its regular AAA battery. However, these new batteries are more expensive to
produce, so the company would like to be convinced that they really do last longer.
Based on years of experience, the company knows that its regular AAA batteries last
for 30 hours of continuous use, on average. The company selects an SRS of 15 new
batteries and uses them continuously until they are completely drained. A significance
test is performed using the hypotheses
H0 : µ = 30 hours
Ha : µ > 30 hours
where µ is the true mean lifetime of the new deluxe AAA batteries. The resulting Pvalue is 0.0276.
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 Example:
+ Section 11.1
Significance Tests: The Basics
Summary
In this section, we learned that…

A significance test assesses the evidence provided by data against a null
hypothesis H0 in favor of an alternative hypothesis Ha.

The hypotheses are stated in terms of population parameters. Often, H0 is a
statement of no change or no difference. Ha says that a parameter differs
from its null hypothesis value in a specific direction (one-sided alternative)
or in either direction (two-sided alternative).

The reasoning of a significance test is as follows. Suppose that the null
hypothesis is true. If we repeated our data production many times, would we
often get data as inconsistent with H0 as the data we actually have? If the
data are unlikely when H0 is true, they provide evidence against H0 .

The P-value of a test is the probability, computed supposing H0 to be true,
that the statistic will take a value at least as extreme as that actually
observed in the direction specified by Ha .
+ Section 11.1
Significance Tests: The Basics
Summary

Small P-values indicate strong evidence against H0 . To calculate a P-value,
we must know the sampling distribution of the test statistic when H0 is true.
There is no universal rule for how small a P-value in a significance test
provides convincing evidence against the null hypothesis.

If the P-value is smaller than a specified value α (called the significance
level), the data are statistically significant at level α. In that case, we can
reject H0 . If the P-value is greater than or equal to α, we fail to reject H0 .