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CHAPTER 8
Hypothesis Testing
© Copyright McGraw-Hill 2004
8-1
Objectives

Understand the definitions used in hypothesis
testing.

State the null and alternative hypotheses.

Find critical values for the z test.

State the five steps used in hypothesis testing.

Test means for large samples using the z test.

Test means for small samples using the t test.
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Objectives (cont’d.)

Test proportions using the z test.

Test variances or standard deviations using
the chi square test.

Test hypotheses using confidence intervals.

Explain the relationship between type I and
type II errors and the power of a test.
© Copyright McGraw-Hill 2004
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Introduction

Statistical hypothesis testing is a decisionmaking process for evaluating claims about a
population.

In hypothesis testing, the researcher must
define the population under study, state the
particular hypotheses that will be
investigated, give the significance level, select
a sample from the population, collect the
data, perform the calculations required for the
statistical test, and reach a conclusion.
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Hypothesis Testing

Hypotheses concerning parameters such as
means and proportions can be investigated.

The z test and the t test are used for
hypothesis testing concerning means.
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Methods to Test Hypotheses

The three methods used to test hypotheses
are:
1. The traditional method.
2. The P-value method.
3. The confidence interval method.
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Statement of a Hypothesis

A statistical hypothesis is a conjecture about a
population parameter which may or may not
be true.

There are two types of statistical hypotheses
for each situation: the null hypothesis and the
alternative hypothesis.
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Hypotheses

The null hypothesis, symbolized by H0, is a
statistical hypothesis that states that there is
no difference between a parameter and a
specific value, or that there is no difference
between two parameters.

The alternative hypothesis, symbolized by H1,
is a statistical hypothesis that states the
existence of a difference between a parameter
and a specific value, or states that there is a
difference between two parameters.
© Copyright McGraw-Hill 2004
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Hypothesis-Testing Common Phrases


Is greater than
Is less than
Is increased
Is decreased or
reduced from

Is greater than or
equal to
Is at least


Is less than or
equal to
Is at most

Is equal to
Is not equal to
Has not changed
from
Has changed from
© Copyright McGraw-Hill 2004
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Design of the Study

After stating the hypotheses, the researcher’s
next step is to design the study. The
researcher selects the correct statistical test,
chooses an appropriate level of significance,
and formulates a plan for conducting the
study.
© Copyright McGraw-Hill 2004
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Statistical Test

A statistical test uses the data obtained from a
sample to make a decision about whether or
not the null hypothesis should be rejected.

The numerical value obtained from a
statistical test is called the test value.
© Copyright McGraw-Hill 2004
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Possible Outcomes of a Hypothesis Test
H0 True
H0 False
Reject
H0
Error
Type I
Correct
Decision
Do
not
reject
H0
Correct
Decision
Error
Type II
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Summary of Possible Outcomes

A type I error occurs if one rejects the null
hypothesis when it is true.

A type II error occurs if one does not reject the
null hypothesis when it is false.
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Error Probabilities

The level of significance is the maximum
probability of committing a type I error. This
probability is symbolized by ; that is ,
P( type I error )   .

The probability of a type II error is symbolized
by . That is, P( type II error )   .
© Copyright McGraw-Hill 2004
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 and  Probabilities

In most hypothesis testing situations, 
cannot easily be computed; however,  and 
are related in that decreasing one increases
the other.
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Hypothesis Testing

In a hypothesis testing situation, the
researcher decides what level of significance
to use.

After a significance level is chosen, a critical
value is selected from a table for the
appropriate test.
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Critical Values

The critical value(s) separates the critical
region from the noncritical region. The symbol
for critical value is C.V.
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Critical Values (cont’d)

The critical or rejection region is the range of
values of the test value that indicates that
there is a significant difference and that the
null hypothesis should be rejected.

The noncritical or nonrejection region is the
range of values of the test value that indicates
that the difference was probably due to
chance and that the null hypothesis should
not be rejected.
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One-Tailed Test

A one-tailed test indicates that the null
hypothesis should be rejected when the test
value is in the critical region on one side of
the mean.

A one-tailed test is either right-tailed or lefttailed, depending on the direction of the
inequality of the alternative hypothesis.
© Copyright McGraw-Hill 2004
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Left-Tailed Test
H0 :   k
H1 :   k
  010
. , C. V.  128
.
  0.05, C. V.  165
.
  0.01, C. V.  2.33
Noncritical
region
Critical
region
-z
0
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Right-Tailed Test
H0 :   k
H1 :   k
  010
. , C. V.  128
.
  0.05, C. V.  165
.
  0.01, C. V.  2.33
Noncritical
region
0
Critical
region
+z
© Copyright McGraw-Hill 2004
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Two-Tailed Test

In a two-tailed test, the null hypothesis
should be rejected when the test value is in
either of the two critical regions.
© Copyright McGraw-Hill 2004
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Two-Tailed Test (cont’d.)
H0 :  k
H1 :   k
  0.10,C.V.  1.65
  0.05,C.V.  1.96
  0.01,C.V.  2.58
Noncritical
Noncritical
region
region
Critical
region
-z
00
Critical
Critical
region
region
+z
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Hypothesis-Testing (Traditional Method)

Step 1
State the hypothesis, and identify the
claim.

Step 2
Find the critical value from the
appropriate table.

Step 3
Compute the test value.

Step 4
Make the decision to reject or not
reject the null hypothesis.

Step 5
Summarize the results.
© Copyright McGraw-Hill 2004
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The z Test

The z test is a statistical test for the mean of a
population. It can be used when n  30, or
when the population is normally distributed
and  is known.
© Copyright McGraw-Hill 2004
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The z Test Formula

The formula for the z test is:
X 
z 
 n
where
X = sample mean
 = hypothesized population mean
 = population standard deviation
n = sample size
© Copyright McGraw-Hill 2004
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The z Test When  is Unknown

The central limit theorem states that when
the population standard deviation  is
unknown, the sample standard deviation s
can be used in the formula as long as the
sample size is 30 or more.
z
X 
s n
© Copyright McGraw-Hill 2004
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The P-value

The P-value (or probability value) is the
probability of getting a sample statistic (such
as the mean) or a more extreme sample
statistic in the direction of the alternative
hypothesis when the null hypothesis is true.
© Copyright McGraw-Hill 2004
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The P-value (cont’d.)

The P-value is the actual area under the
standard normal distribution curve (or other
curve depending on what statistical test is
being used) representing the probability of a
particular sample statistic or a more extreme
sample statistic occurring if the null
hypothesis is true.
© Copyright McGraw-Hill 2004
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Decision Rule When Using a P-Value
© Copyright McGraw-Hill 2004
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Statistical vs. Practical Significance

The researcher should distinguish between
statistical significance and practical
significance.

When the null hypothesis is rejected at a
specific significance level, it can be concluded
that the difference is probably not due to
chance and thus is statistically significant.
However, the results may or may not have any
practical significance.
© Copyright McGraw-Hill 2004
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The t Test

The t test is a statistical test of the mean of a
population and is used when the population
is normally or approximately normally
distributed,  is unknown, and the sample
size is less than 30.

The formula for the t test is:
X 
t
s n

The degrees of freedom are d.f. = n–1.
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z Test for a Proportion

A hypothesis test involving a population
proportion can be considered as a binomial
experiment when there are only two outcomes
and the probability of a success does not
change from trial to trial.
© Copyright McGraw-Hill 2004
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Formula for the z Test for Proportions

pp
z
pq / n
where

X
p 
n
p  population proportion
n = sample size
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Chi-Square Test for Single Variance

The formula is:
 
2
(n  1)s
2
2
with d.f.=n-1 where
n  sample size
2
s = sample variance
 2 =population variance
© Copyright McGraw-Hill 2004
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Assumptions for Chi-Square Test for Single Variance

The sample must be randomly selected from
the population.

The population must be normally distributed
for the variable under study.

The observations must be independent of
each other.
© Copyright McGraw-Hill 2004
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Confidence Intervals & Hypothesis Testing

There is a relationship between confidence
intervals and hypothesis testing.

When the null hypothesis is rejected in a
hypothesis testing situation, the confidence
interval for the mean using the same level of
significance will not contain the hypothesized
mean.

Likewise, when the null hypothesis is not
rejected, the confidence interval computed
will contain the hypothesized mean.
© Copyright McGraw-Hill 2004
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Power of a Statistical Test

The power of a test measures the sensitivity of
the test to detect a real difference in
parameters if one actually exists.

The higher the power, the more sensitive the
test is to detecting a real difference between
parameters if there is a difference.
© Copyright McGraw-Hill 2004
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Summary

A statistical hypothesis is a conjecture about
a population.

There are two types of statistical hypotheses:
the null hypothesis states that there is no
difference, and the alternative hypothesis
specifies a difference.
© Copyright McGraw-Hill 2004
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Summary (cont’d.)

The z test is used when the population
standard deviation is known and the variable
is normally distributed or when  is not
known and the sample size is greater than or
equal to 30.

When the population standard deviation is
not known and the variable is normally
distributed, the sample standard deviation is
used, but a t test should be conducted if the
sample size is less than 30.
© Copyright McGraw-Hill 2004
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Summary (cont’d.)

Researchers compute a test value from the
sample data in order to decide whether the
null hypothesis should or should not be
rejected.

Statistical tests can be one-tailed or twotailed, depending on the hypotheses.
© Copyright McGraw-Hill 2004
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Summary (cont’d.)

The null hypothesis is rejected when the
difference between the population parameter
and the sample statistic is said to be
significant.

The difference is significant when the test
value falls in the critical region of the
distribution.

The critical region is determined by , the
level of significance of the test.
© Copyright McGraw-Hill 2004
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Summary (cont’d.)

The significance level of a test is the
probability of committing a type I error.

A type I error occurs when the null hypothesis
is rejected when it is true.

The type II error can occur when the null
hypothesis is not rejected when it is false.

One can test a single variance by using a chisquare test.
© Copyright McGraw-Hill 2004
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Conclusions

Researchers are interested in answering many
types of questions. For example:
“Will a new drug lower blood pressure?”
“Will seat belts reduce the severity of injuries
caused by accidents?”

These types of questions can be addressed
through statistical hypothesis testing, which
is a decision-making process for evaluating
claims about a population.
© Copyright McGraw-Hill 2004
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