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Chapter FifteenChapter Fifteen
Copyright © 2006McGraw-Hill/Irwin
Data Analysis: Testing for Significant Differences
McGraw-Hill/Irwin 2
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.
Learning Objectives
McGraw-Hill/Irwin 3
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.
Learning Objectives
McGraw-Hill/Irwin 4
• 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
Value of Testing for Differences in Data
McGraw-Hill/Irwin 16
• 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
Calculate the mean, median, and mode as measures of
central tendency
Measures of Central Tendency
McGraw-Hill/Irwin 17
Calculate the mean, median, and mode as measures of
central tendencyExhibit 15.8
McGraw-Hill/Irwin 18
Calculate the mean, median, and mode as measures of
central tendencyExhibit 15.9
McGraw-Hill/Irwin 19
• 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
Explain the range and standard deviation of a frequency distribution as measures
of dispersion
Measures of Dispersion
McGraw-Hill/Irwin 20
• 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
Explain the range and standard deviation of a frequency distribution as measures
of dispersion
Measures of Dispersion
McGraw-Hill/Irwin 21
Explain the range and standard deviation of a frequency distribution as measures
of dispersionExhibit 15.10
McGraw-Hill/Irwin 22
• 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.
Explain hypothesis testing and assess potential error in its useHypothesis Testing
McGraw-Hill/Irwin 23
• 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
Understand the difference between independent and related samplesHypothesis Testing
McGraw-Hill/Irwin 24
• 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
Explain hypothesis testing and assess potential error in its useHypothesis Testing
McGraw-Hill/Irwin 25
• 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
Explain hypothesis testing and assess potential error in its useHypothesis Testing
McGraw-Hill/Irwin 26
• 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
Explain hypothesis testing and assess potential error in its useHypothesis Testing
McGraw-Hill/Irwin 27
• 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)
Explain hypothesis testing and assess potential error in its useHypothesis Testing
McGraw-Hill/Irwin 28
• 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
Analyzing Relationships of
Sample Data
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• 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
Analyzing Relationships of
Sample Data
Understand univariate and bivariate statistical tests
McGraw-Hill/Irwin 30
Explain the range and standard deviation of a frequency distribution as measures
of dispersionExhibit 15.11
McGraw-Hill/Irwin 31
• 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
Analyzing Relationships of
Sample Data
Understand univariate and bivariate statistical tests
McGraw-Hill/Irwin 32
• 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
Analyzing Relationships of
Sample Data
Understand univariate and bivariate statistical tests
McGraw-Hill/Irwin 33
• 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.
Analyzing Relationships of
Sample Data
Understand univariate and bivariate statistical tests
McGraw-Hill/Irwin 34
• The formula for calculating the t value is _ _
Z = x1 – x2 S x1 – x2
Analyzing Relationships of
Sample Data
Understand univariate and bivariate statistical tests
McGraw-Hill/Irwin 35
Explain the range and standard deviation of a frequency distribution as measures
of dispersionExhibit 15.12
McGraw-Hill/Irwin 36
Explain the range and standard deviation of a frequency distribution as measures
of dispersionExhibit 15.14
McGraw-Hill/Irwin 37
• 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
Analyzing Relationships of
Sample Data
Apply and interpret the results of the ANOVA and n-way
ANOVA statistical methods
McGraw-Hill/Irwin 38
• 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
Analyzing Relationships of
Sample Data
Apply and interpret the results of the ANOVA and n-way
ANOVA statistical methods
McGraw-Hill/Irwin 39
• Determining Statistical Significance in ANOVA
– F-test–used to statistically evaluate the differences between the group means in ANOVA
– Total variance–separated into between-group and within-group variance
Analyzing Relationships of
Sample Data
Apply and interpret the results of the ANOVA and n-way
ANOVA statistical methods
McGraw-Hill/Irwin 40
• 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
Analyzing Relationships of
Sample Data
Apply and interpret the results of the ANOVA and n-way
ANOVA statistical methods
McGraw-Hill/Irwin 41
• 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
Analyzing Relationships of
Sample Data
Apply and interpret the results of the ANOVA and n-way
ANOVA statistical methods
McGraw-Hill/Irwin 42
• 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
Analyzing Relationships of
Sample Data
Apply and interpret the results of the ANOVA and n-way
ANOVA statistical methods
McGraw-Hill/Irwin 43
Explain the range and standard deviation of a frequency distribution as measures
of dispersionExhibit 15.15
McGraw-Hill/Irwin 44
Explain the range and standard deviation of a frequency distribution as measures
of dispersionExhibit 15.16
McGraw-Hill/Irwin 45
• 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
Analyzing Relationships of
Sample Data
Apply and interpret the results of the ANOVA and n-way
ANOVA statistical methods
McGraw-Hill/Irwin 46
• 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
Perceptual MappingUtilize perceptual mapping to
simplify presentation of research findings
McGraw-Hill/Irwin 47
Explain the range and standard deviation of a frequency distribution as measures
of dispersionExhibit 15.17
McGraw-Hill/Irwin 48
Explain the range and standard deviation of a frequency distribution as measures
of dispersionExhibit 15.18
McGraw-Hill/Irwin 49
• Applications in Marketing Research
1. New-product development
2. Image measurements
3. Advertising
4. Distribution
Perceptual MappingUtilize perceptual mapping to
simplify presentation of research findings
McGraw-Hill/Irwin 50
• 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
Summary