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ZIKMUND BABIN
CARR GRIFFIN
BUSINESS
MARKET
RESEARCH
EIGHTH EDITION
© 2010 South-Western/Cengage Learning. All rights reserved. May not
be scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
Chapter 21
Univariate
Statistical Analysis
LEARNING OUTCOMES
After studying this chapter, you should be able to
1. Implement the hypothesis-testing procedure
2. Use p-values to assess statistical significance
3. Test a hypothesis about an observed mean compared
to some standard
4. Know the difference between Type I and Type II errors
5. Know when a univariate χ2 test is appropriate and how
to conduct one
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–2
Hypothesis Testing
• Types of Hypotheses
 Relational hypotheses

Examine how changes in one variable vary with changes in
another.
 Hypotheses about differences between groups

Examine how some variable varies from one group to
another.
 Hypotheses about differences from some standard

Examine how some variable differs from some preconceived
standard. These tests typify univariate statistical tests.
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–3
Types of Statistical Analysis
• Univariate Statistical Analysis
 Tests of hypotheses involving only one variable.
 Testing of statistical significance
• Bivariate Statistical Analysis
 Tests of hypotheses involving two variables.
• Multivariate Statistical Analysis
 Statistical analysis involving three or more variables
or sets of variables.
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–4
The Hypothesis-Testing Procedure
•
Process
1. The specifically stated hypothesis is derived from
the research objectives.
2. A sample is obtained and the relevant variable is
measured.
3. The measured sample value is compared to the
value either stated explicitly or implied in the
hypothesis.


If the value is consistent with the hypothesis, the hypothesis
is supported.
If the value is not consistent with the hypothesis, the
hypothesis is not supported.
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–5
Statistical Analysis: Key Terms
• Hypothesis
 Unproven proposition: a supposition that tentatively
explains certain facts or phenomena.
 An assumption about nature of the world.
• Null Hypothesis
 Statement about the status quo.
 No difference in sample and population.
• Alternative Hypothesis
 Statement that indicates the opposite of the null
hypothesis.
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–6
Significance Levels and p-values
• Significance Level
 A critical probability associated with a statistical
hypothesis test that indicates how likely an inference
supporting a difference between an observed value
and some statistical expectation is true.
 The acceptable level of Type I error.
• p-value
 Probability value, or the observed or computed
significance level.
p-values are compared to significance levels to test
hypotheses.
 Higher p-values equal more support for an hypothesis.

© 2010 South-Western/Cengage Learning. All rights reserved. May not be
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website, in whole or in part.
21–7
EXHIBIT 21.1
p-Values and Statistical Tests
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website, in whole or in part.
21–8
EXHIBIT 21.2
As the observed mean
gets further from the
standard (proposed
population mean), the
p-value decreases.
The lower the p-value,
the more confidence
you have that the
sample mean is
different.
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–9
An Example of Hypothesis Testing
The null hypothesis: the mean is equal to 3.0:
The alternative hypothesis: the mean does not equal to 3.0:
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–10
An Example of Hypothesis Testing
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scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–11
—
EXHIBIT 21.3
A Hypothesis Test Using the Sampling Distribution of X
under the Hypothesis µ = 3.0
Critical Values
Values that lie exactly
on the boundary of
the region of
rejection.
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website, in whole or in part.
21–12
Type I and Type II Errors
• Type I Error
 An error caused by rejecting the null hypothesis when
it is true.
 Has a probability of alpha (α).
 Practically, a Type I error occurs when the researcher
concludes that a relationship or difference exists in
the population when in reality it does not exist.
 “There really are no monsters under the bed.”
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–13
Type I and Type II Errors (cont’d)
• Type II Error
 An error caused by failing to reject the null hypothesis
when the alternative hypothesis is true.
 Has a probability of beta (β).
 Practically, a Type II error occurs when a researcher
concludes that no relationship or difference exists
when in fact one does exist.
 “There really are monsters under the bed.”
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–14
EXHIBIT 21.4
Type I and Type II Errors in Hypothesis Testing
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website, in whole or in part.
21–15
Choosing the Appropriate Statistical
Technique
• Choosing the correct statistical technique
requires considering:
 Type of question to be answered
 Number of variables involved
 Level of scale measurement
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scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–16
Parametric versus Nonparametric Tests
• Parametric Statistics
 Involve numbers with known, continuous distributions.
 Appropriate when:
Data are interval or ratio scaled.
 Sample size is large.

• Nonparametric Statistics
 Appropriate when the variables being analyzed do not
conform to any known or continuous distribution.
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–17
EXHIBIT 21.5
Univariate Statistical
Choice Made Easy
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website, in whole or in part.
21–18
The t-Distribution
• t-test
 A hypothesis test that
uses the t-distribution.
 A univariate t-test is
appropriate when the
variable being
analyzed is interval or
ratio.
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scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
• Degrees of freedom
(d.f.)
 The number of
observations minus the
number of constraints
or assumptions
needed to calculate a
statistical term.
21–19
EXHIBIT 21.6
The t-Distribution for Various Degrees of Freedom
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website, in whole or in part.
21–20
Calculating a Confidence Interval Estimate Using
the t-Distribution
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scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–21
Calculating a Confidence Interval Estimate Using
the t-Distribution (cont’d)
  X  t c .l . S X
X  3.89 S  2.81 n  18
2.81
 3.89  2.12(
)  2.49
18
2.81
 3.89  2.12(
)  5.28
18
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–22
One-Tailed Univariate t-Tests
• One-tailed Test
 Appropriate when a research hypothesis implies that
an observed mean can only be greater than or less
than a hypothesized value.

Only one of the “tails” of the bell-shaped normal curve is
relevant.

A one-tailed test can be determined from a two-tailed test
result by taking half of the observed p-value.

When there is any doubt about whether a one- or two-tailed
test is appropriate, opt for the less conservative two-tailed
test.
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–23
Two-Tailed Univariate t-Tests
• Two-tailed Test
 Tests for differences from the population mean that
are either greater or less.

Extreme values of the normal curve (or tails) on both the right
and the left are considered.

When a research question does not specify whether a
difference should be greater than or less than, a two-tailed
test is most appropriate.

When the researcher has any doubt about whether a one- or
two-tailed test is appropriate, he or she should opt for the less
conservative two-tailed test.
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–24
Univariate Hypothesis Test Utilizing the
t-Distribution
• Example:
 Suppose a Pizza Inn manager believes the
average number of returned pizzas each day
to be 20.
 The store records the number of defective
assemblies for each of the 25 days it was
opened in a given month.
 The mean was calculated to be 22, and the
standard deviation to be 5.
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scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–25
Univariate Hypothesis Test Utilizing the
t-Distribution: An Example
The sample mean is
equal to 20.
H 0 :   20
The sample mean is
equal not to 20.
H1 :   20
S X  S / n  5 / 25  1
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scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–26
Univariate Hypothesis Test Utilizing the
t-Distribution: An Example (cont’d)
• The researcher desired a 95 percent confidence;
the significance level becomes 0.05.
• The researcher must then find the upper and
lower limits of the confidence interval to
determine the region of rejection.
 Thus, the value of t is needed.
 For 24 degrees of freedom (n-1= 25-1),
the t-value is 2.064.
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scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–27
Univariate Hypothesis Test Utilizing the
t-Distribution: An Example (cont’d)
Lower limit =   tc .l . S X
 5 
 20  2.064
  17.936
 25 
Upper limit =   tc .l . S X
 5 
 20  2.064
  22.064
 25 
© 2010 South-Western/Cengage Learning. All rights reserved. May not be
scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–28
Univariate Hypothesis Test Utilizing the
t-Distribution: An Example (cont’d)
Univariate Hypothesis Test t-Test
tobs
22  20 2
X 



1
SX
1
2
This is less than the critical t-value of 2.064 at the 0.05
level with 24 degrees of freedom  hypothesis is not
supported.
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scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–29
The Chi-Square Test for Goodness of Fit
• Chi-square (χ2) test
 Tests for statistical significance.
 Is particularly appropriate for testing hypotheses
about frequencies arranged in a frequency or
contingency table.
• Goodness-of-Fit (GOF)
 A general term representing how well some computed
table or matrix of values matches some population or
predetermined table or matrix of the same size.
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scanned, copied or duplicated, or posted to a publically accessible
website, in whole or in part.
21–30
The Chi-Square Test for Goodness of Fit:
An Example
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website, in whole or in part.
21–31
The Chi-Square Test for Goodness of Fit:
An Example (cont’d)
(Oi  Ei )²
²  
Ei
χ² = chi-square statistics
Oi = observed frequency in the ith cell
Ei = expected frequency on the ith cell
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website, in whole or in part.
21–32
Chi-Square Test: Estimation for Expected
Number for Each Cell
E ij 
R iC
j
n
Ri = total observed frequency in the ith row
Cj = total observed frequency in the jth column
n = sample size
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21–33
Hypothesis Test of a Proportion
• Hypothesis Test of a Proportion
 Is conceptually similar to the one used when the
mean is the characteristic of interest but that differs in
the mathematical formulation of the standard error of
the proportion.
Z obs
p 

Sp
π is the population proportion
p is the sample proportion
π is estimated with p
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website, in whole or in part.
21–34