Download Hypothesis Testing

Chapter
Fifteen
Data Analysis:
Testing for
Significant
Differences
Copyright © 2006
McGraw-Hill/Irwin
Learning Objectives
1. Understand how to prepare graphical
presentations of data.
2. Calculate the mean, median, and mode as
measures of central tendency.
3. Explain the range and standard deviation of a
frequency distribution as measures of
dispersion.
4. Understand the difference between independent
and related samples.
McGraw-Hill/Irwin
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Learning Objectives
5.Explain hypothesis testing and assess
potential error in its use.
6.Understand univariate and bivariate
statistical tests.
7.Apply and interpret the results of the
ANOVA and n-way ANOVA statistical
methods.
McGraw-Hill/Irwin
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Value of Testing for Differences in Data
• Basic statistics and descriptive analysis
– common to all marketing research projects
– Central tendency and dispersion
– t-distribution and associated confidence interval
estimation
– Hypothesis testing
– Analysis of variance
McGraw-Hill/Irwin
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Measures of Central
Tendency
•
Calculate the mean, median,
and mode as measures of
central tendency
Three Measures of Central Tendency–
strengths and weaknesses
1. Nominal Data–mode is the best measure
2. Median–ordinal data
3. Mean–interval or ratio data
McGraw-Hill/Irwin
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Exhibit 15.8
McGraw-Hill/Irwin
Calculate the mean, median,
and mode as measures of
central tendency
17
Exhibit 15.9
McGraw-Hill/Irwin
Calculate the mean, median,
and mode as measures of
central tendency
18
Measures of
Dispersion
Explain the range and standard deviation
of a frequency distribution as measures
of dispersion
• Measures of Central Tendency–cannot tell the
whole story about a distribution of responses
• Measures of Dispersion–how close to the mean or
other measure of central tendency the rest of the
values in the distribution fall
• Range–the distance between the smallest and
largest value in a set of responses
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Measures of
Dispersion
Explain the range and standard deviation
of a frequency distribution as measures
of dispersion
• Standard Deviation–average distance of the
dispersion values from the mean
– Deviation–difference between a particular
response and the distribution mean
– Average squared deviation–used as a
measure of dispersion for a distribution
• Variance–average squared deviations about the
mean of a distribution of values
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Exhibit 15.10
McGraw-Hill/Irwin
Explain the range and standard deviation
of a frequency distribution as measures
of dispersion
21
Hypothesis Testing
Explain hypothesis testing and assess
potential error in its use
• Hypothesis–empirically testable though yet
unproven statement developed in order to
explain phenomena
– Preconceived notion of the relationships that
the captured data should present–a
hypothesis.
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Hypothesis Testing
Understand the difference between
independent and related samples
• Independent Samples–two or more groups of
responses that are tested as though they may come
from different populations
• Related Samples–two or more groups of responses
that originated from the sample population
• Paired sample–questions are independent–the
respondents are the same
– Paired samples t-test--used for differences in related
samples
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Hypothesis Testing
Explain hypothesis testing and assess
potential error in its use
• First Step–to develop the hypotheses that is to be tested
– Developed prior to the collection of data
– Developed as part of a research plan
– Make comparisons between two groups of respondents to
determine if there are important differences between the groups
– Important considerations in hypothesis testing are:
• Magnitude of the difference between the means
• Size of the sample used to calculate the means
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Hypothesis Testing
•
Explain hypothesis testing and assess
potential error in its use
Null Hypothesis (Ho)–a statement that asserts the status quo
– Alternative Hypothesis (H1)
• a statement that is the opposite of the null hypothesis, that the difference
exists in reality not simply due to random error
• Represents the condition desired
– Null hypothesis is accepted–there is no change to the status quo
– Null hypothesis is rejected–the alternative hypothesis is accepted
and the conclusion is that there has been a change in opinions or
actions
– Null hypothesis refers to a population parameter–not a sample
statistic
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Hypothesis Testing
Explain hypothesis testing and assess
potential error in its use
• Statistical Significance
– Inference Regarding a Population
– Type I Error–made by rejecting the null
hypothesis when it is true; the probability of
alpha (α)
• Level of Significance--.10, .05, or .01
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Hypothesis Testing
Explain hypothesis testing and assess
potential error in its use
• Type II Error–failing to reject the null hypothesis
when the alternative hypothesis is true; the
probability of beta (β).
– Unlike alpha (α), which is specified by the researcher,
beta (β) depends on the actual population parameter.
– Type I and Type II errors–sample size can help
control these errors
• Can select an alpha (α) and the sample size in order to
increase the power of the test and beta (β)
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Analyzing
Relationships of
Sample Data
• Purpose of Inferential Statistics
– Sample
– Sample Statistics
– Population Parameter
• The actual population parameters are unknown
since the cost to perform a census of the
population is prohibitive
• Frequency Distribution
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Analyzing
Relationships of
Sample Data
Understand univariate and
bivariate statistical tests
• Univariate Tests of Significance
– involve hypothesis testing using one variable
at a time
• z-test
– sample size >30 and the standard deviation is
unknown
• t-test–
– sample size <30 and the standard deviation is
unknown, assumption of a normal distribution
is not valid
McGraw-Hill/Irwin
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Exhibit 15.11
McGraw-Hill/Irwin
Explain the range and standard deviation
of a frequency distribution as measures
of dispersion
30
Analyzing
Relationships of
Sample Data
Understand univariate and
bivariate statistical tests
• Bivariate Hypotheses–where more than one
group is involved
• Null hypotheses–that there is no difference
between the group means
µ1 = µ2 or µ1 - µ2 = 0
McGraw-Hill/Irwin
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Analyzing
Relationships of
Sample Data
Understand univariate and
bivariate statistical tests
• Using the t-Test to Compare Two Means
– Univariate t-test and the Bivariate t-test–require
interval or ratio data
• t-test –useful when the sample size is < 30 and the
population standard deviation is unknown
• Bivariate test—assumption is that the samples are
drawn from populations with normal distributions
and that the variances of the populations are equal
McGraw-Hill/Irwin
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Analyzing
Relationships of
Sample Data
Understand univariate and
bivariate statistical tests
• t-test for differences between group
means–as the difference between the means
divided by the variability of random means
– t-value–ratio of the difference between two
sample means and the standard error
– t-test–provides a rational way of determining
if the difference between the two sample
means occurred by chance.
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Analyzing
Relationships of
Sample Data
Understand univariate and
bivariate statistical tests
• The formula for calculating the t value is
_ _
Z = x1 – x2
S x1 – x2
McGraw-Hill/Irwin
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Exhibit 15.12
McGraw-Hill/Irwin
Explain the range and standard deviation
of a frequency distribution as measures
of dispersion
35
Exhibit 15.14
McGraw-Hill/Irwin
Explain the range and standard deviation
of a frequency distribution as measures
of dispersion
36
Analyzing
Relationships of
Sample Data
Apply and interpret the results
of the ANOVA and n-way
ANOVA statistical methods
• Analysis of Variance (ANOVA)–statistical
technique that determines if three or more
means are statistically different from each
other
• Multivariate Analysis of Variance
(MANOVA)–multiple dependent variables can
be analyzed together
McGraw-Hill/Irwin
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Analyzing
Relationships of
Sample Data
•
Apply and interpret the results
of the ANOVA and n-way
ANOVA statistical methods
Requirements for the ANOVA
– The dependent variable be either interval or ratio scaled
– The independent variable be categorical
•
Null hypothesis for ANOVA–states that there is no difference between
the groups–the null hypothesis would be
µ1 = µ2 = µ3
•
ANOVA technique–focuses on the behavior of the variance with a set
of data
•
ANOVA–if the calculated variance between the groups is compared to
the variance within the groups, a rational determination can be made as
to whether the means are significantly different
McGraw-Hill/Irwin
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Analyzing
Relationships of
Sample Data
Apply and interpret the results
of the ANOVA and n-way
ANOVA statistical methods
• Determining Statistical Significance in
ANOVA
– F-test–used to statistically evaluate the
differences between the group means in
ANOVA
– Total variance–separated into betweengroup and within-group variance
McGraw-Hill/Irwin
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Analyzing
Relationships of
Sample Data
Apply and interpret the results
of the ANOVA and n-way
ANOVA statistical methods
• F distribution–ratio of these two components of total variance
and can be calculated as follows
– F ratio = Variance between groups
Variance within groups
• The larger the F ratio
– The larger the difference in the variance between groups
– Implies significant differences between the groups
– the more likely that the null hypothesis will be rejected
McGraw-Hill/Irwin
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Analyzing
Relationships of
Sample Data
Apply and interpret the results
of the ANOVA and n-way
ANOVA statistical methods
• ANOVA–cannot identify which pairs of means
are significantly different from each other
– Follow-up Tests—test that flag the means that
are statistically different from each other
• Sheffé
• Tukey, Duncan and Dunn
McGraw-Hill/Irwin
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Analyzing
Relationships of
Sample Data
Apply and interpret the results
of the ANOVA and n-way
ANOVA statistical methods
• n-Way ANOVA
– In a one-way ANOVA–only one independent
variable
– For several independent variables–a n-way
ANOVA would be used
– Use of experimental designs–provides different
groups in a sample with different information to
see how their responses change
McGraw-Hill/Irwin
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Exhibit 15.15
McGraw-Hill/Irwin
Explain the range and standard deviation
of a frequency distribution as measures
of dispersion
43
Exhibit 15.16
McGraw-Hill/Irwin
Explain the range and standard deviation
of a frequency distribution as measures
of dispersion
44
Analyzing
Relationships of
Sample Data
Apply and interpret the results
of the ANOVA and n-way
ANOVA statistical methods
• MANOVA–designed to examine multiple
dependent variables across single or multiple
independent variables
– Statistical calculations for MANOVA–similar
to n-way ANOVA and are in the statistical
software packages such as SPSS and SAS
McGraw-Hill/Irwin
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Perceptual Mapping
Utilize perceptual mapping to
simplify presentation of
research findings
• Perceptual Mapping–process that is used to
develop maps showing the perceptions of
respondents. The maps are visual representations
of respondents’ perceptions of a company, product,
service, brand, or any other object in two
dimensions
– Has a vertical and a horizontal axis that are
labeled with descriptive adjectives
– Development of the perceptual map–rankings,
mean ratings, and multivariate techniques
McGraw-Hill/Irwin
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Exhibit 15.17
McGraw-Hill/Irwin
Explain the range and standard deviation
of a frequency distribution as measures
of dispersion
47
Exhibit 15.18
McGraw-Hill/Irwin
Explain the range and standard deviation
of a frequency distribution as measures
of dispersion
48
Perceptual Mapping
•
Utilize perceptual mapping to
simplify presentation of
research findings
Applications in Marketing Research
1. New-product development
2. Image measurements
3. Advertising
4. Distribution
McGraw-Hill/Irwin
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Summary
•
•
•
•
•
•
•
Value of Testing for Differences in Data
Guidelines for Graphics
Measures of Central Tendency
Measures of Dispersion
Hypothesis Testing
Analyzing Relationships of Sample Data
Perceptual Mapping
McGraw-Hill/Irwin
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