Chapter 15 Data Analysis: Testing for Significant Differences.
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Transcript of Chapter 15 Data Analysis: Testing for Significant Differences.
Chapter 15Data Analysis: Testing for
Significant Differences
Chapter 15Data Analysis: Testing for
Significant Differences
Value of Testing for Data
Differences
Analysis of variance (ANOVA)Analysis of variance (ANOVA)
Hypothesis testingHypothesis testing
t-distribution and associated confidence interval estimationt-distribution and associated
confidence interval estimation
Central tendency and dispersionCentral tendency and dispersion
Common to all marketing research projects
Common to all marketing research projects
Basic Statistics and Descriptive
Analysis
15-2
• Mode – most common value in a set of responses; i.e., the question response most often given. (Nominal data)
• Median – middle value of a rank ordered distribution; half of the responses are above and half below the median value. (Ordinal data)
• Mean – average of the sample. (Interval and Ratio data)
• Range – the distance between the smallest and largest values of the variable.
• Standard deviation – the average distance of the dispersion of the values from the mean.
• Variance – the squared standard deviation.
Statistical Measures
15-3
• Inferential statistics – to make a determination about a population on the basis of a sample.
oSample – a subset of the population.
oSample statistics – measures obtained directly from sample data.
oPopulation parameter – a measured characteristic of the population.• Actual population parameters are unknown since the
cost to perform a census of the population is prohibitive.
Analyzing Relationships of Sample Data
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Type I:Rejection of the null hypothesis when, in fact, it is true.(Convict an innocent man)
Type II:Acceptance of the null hypothesis when, in fact, it is false.(Set a guilty man free)
Tests are either one or two-tailed. This approach depends on the nature of the situation and what the researcher is demonstrating.
One-Tailed Tests (provide some direction)“If you take the medicine, you will get better”
Two-Tailed Tests (there is a difference)“If you take the medicine, you will get either better or worse.”
Chapter 15
Hypothesis Testing – Error Types
Actual State of theNull Hypothesis
Fail to Reject Ho Reject Ho
Ho is true
Ho is false
Correct (1-) no error
Type II error ()
Type I error ()
Correct (1- ) no error
Chapter 15
Significance Testing Error Issues
Type I and Type II Errors
Alpha and Beta (chances of making a Type I or Type II error) are related. If we set one very small (Alpha is .0001%), then we make the other very large. We are most concerned with minimizing Alpha, therefore a common percentage for a typical research study is 5% so we are 95% confident in the results.
Chapter 15
Significance Testing Error Issues
When do we do a means test?
When do we do a proportions test?
When do we do a chi-square (frequency distribution) test?
In making these decisions, we consider the level of data. We cannot calculate a mean on nominal or ordinal data, therefore we must do a proportion of frequency distribution test. If we have interval or ratio level data, we typically conduct a means test. If we have nominal or ordinal data, and we have one group or two groups, we usually do a proportion’s z-test. If we have more than two groups, we usually do a chi-square test.
Chapter 15
Types of Hypothesis Tests
One Mean
Used to test whether a sample mean is significantly different from an expected or pre-determined mean.Z-Test - usually for samples of about 30 and above.
t-Test - usually for samples below 30.
Two MeansZ-TestTests the difference between means for two.
More than Two MeansANOVA (Analysis of Variance)Tests the difference between means for more than two groups.
Chapter 15
Types of Mean Hypothesis Tests
Chapter 15
Types of Proportion Hypothesis TestsOne Sample
Z-TestTest to determine whether the difference between proportions is greater than would be expected because of sampling error.
Two Proportions in Independent Samples
Z-TestTest to determine the proportional differences between two or more groups.
More than Two Groups
Chi-square χ2
Test to determine whether the difference between three or more groups is greater than would be expected.
p-valueThe exact probability of getting a computed test statistic that was largely due to chance. The smaller the p-value, the smaller the probability that the observed result occurred by chance. The smaller the p-value, the more likely the test is significant.
For example a p-value of .045 is equivalent to a statistically significant test at a 95.5% level of
confidence (4.5% alpha level).
Chapter 15
Hypothesis Testing Term
• Independent samples – two or more groups of respondents that are tested as though they may come from different populations (independent samples t-test).
• Related samples – two or more groups of respondents that originated from the sample population (paired samples t-test).
• Paired samples – questions are independent but respondents are the same.
Hypothesis Testing
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Analyzing Relationships of
Sample Data
BivariateHypothesisBivariate
Hypothesis
NullHypothesis
NullHypothesis
. . . more than one group is involved.. . . more than one group is involved.
. . . there is no difference between
the group means.
• µ1 = µ2 or that µ1 - µ2 = 0
. . . there is no difference between
the group means.
• µ1 = µ2 or that µ1 - µ2 = 0
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Bivariate Statistical Tests
Cross-tabulation – is useful for examining relationships and reporting
the findings for two variables. The purpose of cross-tabulation is to
determine if differences exist between subgroups of the total sample.
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Chi-Square (X2)
Analysis
. . . test for significance between the frequency
distributions of two or more nominally scaled variables in a
cross-tabulation table to determine if there is any
association.
15-15
Assesses how closely the observed frequencies fit the pattern of the expected frequencies and is referred to as a ”goodness-of-fit” test.
Assesses how closely the observed frequencies fit the pattern of the expected frequencies and is referred to as a ”goodness-of-fit” test.
Used to analyze nominal data which cannot be analyzed with other types of statistical analysis, such as ANOVA or t-tests.
Used to analyze nominal data which cannot be analyzed with other types of statistical analysis, such as ANOVA or t-tests.
Results will be distorted if more than 20 percent of the cells have an expected count of less than 5.
Results will be distorted if more than 20 percent of the cells have an expected count of less than 5.
Chi-Square (X2) Analysis
15-16
Analyzing Data
Relationships• Requirements for ANOVA odependent variable can be either
interval or ratio scaled.oindependent variable is categorical.
• Null hypothesis for ANOVA – states there is no difference between the groups – the null hypothesis is . . . oµ1 = µ2 = µ3
15-17
ANOVAANOVA
Total variance – separated into
between-group and within-group variance.
Total variance – separated into
between-group and within-group variance.
F-test – used to statistically evaluate the differences between the
group means.
F-test – used to statistically evaluate the differences between the
group means.DeterminingStatistical
Significance
DeterminingStatistical
Significance
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ANOVA – Testing Statistical
Significance
The larger the F ratio . . .
The larger the F ratio . . .
. . . the larger the difference in the variance between groups.
. . . the larger the difference in the variance between groups.
. . . the more likely the null hypothesis will be rejected.. . . the more likely the null hypothesis will be rejected.
Based on the F-distribution . . .
Based on the F-distribution . . .
. . . Examines the ratio of two components of total variance and is calculated as shown below . . .
. . . Examines the ratio of two components of total variance and is calculated as shown below . . .
F ratio = Variance between groups
Variance within groups
F ratio = Variance between groups
Variance within groups
. . . implies significant differences between the groups.
. . . implies significant differences between the groups.
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Analyzing Relationships of
Sample DataANOVA – cannot identify which pairs of means are significantly different from each other.
Must perform follow-up tests to identify the means that are statistically different from each other. Including:
•Sheffé
•Tukey, Duncan and Dunn
15-20
Analyzing Data
Relationships
ANOVA(analysis of variance)
ANOVA(analysis of variance)
. . . determines if three or more means
are statistically different from each
other (single dependent variable)
. . . determines if three or more means
are statistically different from each
other (single dependent variable)
. . . same as ANOVA but multiple
dependent variables can be analyzed
together.
. . . same as ANOVA but multiple
dependent variables can be analyzed
together.
MANOVA(multivariate analysis
of variance)
MANOVA(multivariate analysis
of variance)
15-21
Perceptual Maps
. . . have a vertical and a horizontal axis that are labeled
with descriptive adjectives.
. . . have a vertical and a horizontal axis that are labeled
with descriptive adjectives.
To develop perceptual maps – can use rankings, mean ratings, and
multivariate methods.
To develop perceptual maps – can use rankings, mean ratings, and
multivariate methods.
15-22
15-23
Perceptual Perceptual
MappingMapping
DistributionDistribution
AdvertisingAdvertising
Image developmentImage development
New product developmentNew product development
Applicationsin Marketing
Research
Applicationsin Marketing
Research
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