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CHAPTERS 16-17
HYPOTHESIS TESTING, AND
DETERMINING AND
INTERPRETING BETWEEN TWO
VARIABLES
Important Topics of These
Chapters
Hypothesis and testing.
Steps involved in hypotheses testing.
Type I and Type II errors.
Independent and related samples.
Degrees of freedom.
Hypotheses about single mean.
Cross-tabulations.
Goodness of fit and chi-square tests.
How to interpret a chi-square result.
Hypotheses and Hypothesis
Testing
Hypotheses:
Assumptions, intuition, prior knowledge or
theories that a researcher or manager makes
statements about population parameter under
study. Most commonly takes the form of exact
specification as what the population parameter
value is.
Hypothesis Testing:
Statistical procedure used to ‘accept’ or ‘reject’ the
hypothesis based on sample information. For
hypothesis testing, sample is the most current
information about population.
Hypothesis Testing (cont.)
•
Statement of hypotheses:
• Null Hypothesis(Ho:):
• Stated hypothesis:
• Mean Income of population is equal to $36,000.
• Alternative Hypothesis (Ha:):
• Alternative that tested against the ‘Null hypothesis’.
• Mean income of population is not equal to $36,000.
(Two tails test).,or
• Directional Hypothesis:
• Mean income of population is less than
$36,000.(one tail test), or
• Mean income of population is higher than
$36,000. (one tail test).
Hypothesis testing (cont.)
• Statistical technique to test the Null and Alternative
Hypotheses:
• Cross tabulation, or other available statistical techniques.
• Decision rule as the basis for determining whether to reject or fail to
reject the null hypothesis:
• Computed and table value of test statistic (for cross-tabulation, it is the table
value of Chi-Square statistics at certain degree of freedom (d.f .= c-1 X r-1),
against to computed value of Chi-Square statistic.
• Significance level:
• At .01 level (99% confidence), or at .05 level (95% confidence), or at .10 level
(90% confidence).
• Reject, or fail to reject the Null Hypothesis by basing upon decision
rule.
• State the conclusion from the perspective of the original research
problem or question.
Steps Involved in Hypothesis Testing
Formulate H0 and H1
Select Appropriate Test
Choose Level of Significance, 
Collect Data and Calculate Test Statistic
Determine Probability
Associated with Test
Statistic
Determine Critical
Value of Test Statistic
TSCR
Compare with Level of
Significance,
Determine if TSCR falls
into (Non) Rejection
Region
Reject, or Fail to Reject H0
Draw Marketing Research Conclusion
Other Issues in Hypothesis
Testing
Types of errors in hypothesis testing.
Type I error
Type II error
Rejection of a null
hypothesis when, in
fact, it is true
Fail to reject the
null hypothesis
when, in fact, it is
false
Other Issues in Hypothesis
Testing (cont.)
Type I and Type II Errors
Actual State of
the Null Hypothesis
Ho is true
Ho is false
Fail to Reject Ho
Reject Ho
Correct (1 - ) no error
Type II error (b)
Type I Error ()
Correct (1 - b)
no error
Probabilities of Type I & Type II Error
95% of
Total Area
 = 0.05
= 15
Z  = 1.645
Critical Value
99% of
of Z
Total Area
Z
b = 0.01
 = 17
Z b = -2.33
Z
Other Issues in Hypothesis
Testing (cont.)
Accepting Ho or Failing to Reject (FTR) Ho:
Researchers often fail to make a distinction between
accepting and failing to reject (FTR) Ho.
One-Tailed Test or Two-Tailed Test:
The decision of whether to use a one-tailed test or a
two-tailed test depends on the nature of the
situation and what you are trying to demonstrate
when you stated the Null Hypothesis.
Hypothesis Tests
Independent versus Related Samples
Independent Samples:
Samples in which measurement of a variable in
one population has no effect on the
measurement of the variable in another.
Related Samples:
Samples in which the measurement of a
variable in one population may influence the
measurement of the variable in another.
Hypothesis Tests (cont.)
Degrees of Freedom:
Degrees of freedom are the number of
observations in a statistical problem that
are not restricted or are free to vary.
The number of degrees of freedom (d.f.) is
equal to the number of observations minus
the number of assumptions or constraints
necessary to calculate a statistic.
Hypotheses About One Mean
Z-test:
Hypothesis test about a single mean if the
sample is large enough (n > 30) and
drawn from a normal population.
Calculation of the Test Statistic:
Z=
population mean under the
(
)
(sample mean) null hypothesis
estimated standard error of the mean
Hypotheses About One
Mean(cont.)
T-test:
Hypothesis test about a single mean if the
sample is too small (n < 30) to use the Ztest.
Calculation of the Test Statistic:
t=
population mean specified
(sample mean) - under the null hypothesis
estimated standard error of the mean
(
)
Probability of z with a One-Tailed Test
Shaded Area
= 0.9664
Unshaped Area
= 0.0336
0
z = 1.83
A Broad Classification of Hypothesis Tests
Hypothesis Tests
Tests of
Differences
Tests of
Association
Distributions
Means
Proportions
Median/
Rankings
Cross-Tabulations
Monotonic relationships:
Researcher can assign only a general direction
(increase or decrease) between two variables.
Non-monotonic relationships:
The presence(or absence) of one variable is
systematically associated with the presence(or
absence) of another variable.
Cross tabulation and associated chi-square:
It used to assess whether a non-monotonic
relationship exits between two nominal-scaled
variables.
Gender and Internet Usage
Sex
Internet Usage
Male
Female
Row
Total
Light (1)
5
10
15
Heavy (2)
10
5
15
15
15
Column Total
Internet Usage by Sex
Sex
Internet Usage
Male
Female
Light
33.3%
66.7%
Heavy
66.7%
33.3%
Column total
100%
100%
Sex by Internet Usage
Internet Usage
Sex
Light
Heavy
Total
Male
33.3%
66.7%
100.0%
Female
66.7%
33.3%
100.0%
Purchase of Fashion Clothing by
Marital Status
Purchase of
Fashion
Clothing
Current Marital Status
Married
Unmarried
High
31%
52%
Low
69%
48%
Column
100%
100%
700
300
Number of
respondents
Purchase of Fashion Clothing by
Marital Status and Gender
Pur chase of
Fashion
Clothing
Sex
Male
Marr ied
Female
High
35%
Unmarried
Not
Mar r ied
40%
Mar r ied
Low
65%
60%
75%
40%
Column
totals
Number of
cases
100%
100%
100%
100%
400
120
300
180
25%
Unmarried
Not
Mar r ied
60%
Ownership of Expensive
Automobiles by Education Level
Own Expensive
Automobile
Education
College Degr ee
No College Degr ee
Yes
32%
21%
No
68%
79%
Column totals
100%
100%
250
750
Number of cases
Desire to Travel Abroad by Age
Desir e to Tr avel Abr oad
Age
Less than 45
45 or Mor e
Yes
50%
50%
No
50%
50%
Column totals
100%
100%
500
500
Number of respondents
Eating Frequently in Fast Food
Restaurants by Family Size
Eat Fr equently in Fast
Food Restaurants
Family Size
Small
Lar ge
Yes
65%
65%
No
35%
35%
Column totals
100%
100%
500
500
Number of cases
Ownership of Expensive Automobiles
by Education Level and Income Levels
Own
Expensive
Automobile
Low Income
College
Degr ee
Income
High Income
Yes
20%
No
College
Degr ee
20%
College
Degr ee
40%
No
College
Degr ee
40%
No
80%
80%
60%
60%
Column
totals
Number of
r espondents
100%
100%
100%
100%
100
700
150
50
Eating Frequently in Fast Food
Restaurants by Family Size & Income
Eat
Fr equently
in Fast Food
Restaurants
Low
Family size
Income
High
Family size
Small
Large
Small
Lar ge
Yes
65%
65%
65%
65%
No
35%
35%
35%
35%
Column
totals
Number of
Respondents
100%
100%
100%
100%
250
250
250
250
Desire to Travel Abroad
by Age and Gender
Desir e to
Tr avel
Abr oad
Sex
Male
Age
Female
Age
< 45
>=45
<45
>=45
Yes
60%
40%
35%
65%
No
40%
60%
65%
35%
Column
totals
Number of
Cases
100%
100%
100%
100%
300
300
200
200
Goodness of Fit
Chi-Square Test:
Test of the goodness of fit between the
observed distribution and the expected
distribution of a variable.
Statement of Hypotheses:
Ho: There is not an association (relationship)
between variable ‘X’ and variable ‘Y’.
Ha: There is an association (relationship)
between variable ‘X’ and variable ‘Y’.
Chi-Square Analysis
The computed Chi-Square value:
n
(observed - Expected)
X2 =
-------------------------------i-1
Expected
Where:
Observed: Observed frequency of cell i.
Expected: Expected frequency of cell I.
n: number of cells.
Chi-Square Analysis (cont.)
The Chi-Square Distribution:
Table value or critical value of Chi-Square
at certain degree of freedom (d.f).
Degrees of freedom (d.f.) for Chi-Square
statistics:
(r-1) (c-1)
Chi-Square Distribution: One Tail Test
Fail to Reject
H0
Reject H0
Critical
Value, or table
value of X2
2
How to Interpret a Chi-Square
Result
It yields the probability that researcher find
evidence in support of the null hypothesis.
It should be pointed out that whether or not
a non-monotonic relationship exits between
variable ‘X’ and variable “Y’.
The chi-square test does not indicate the
nature or direction of association between the
two variables.
The chi-square test indicated the strength of
association that exits between two variables.
A Classification of Hypothesis Testing
Procedures for Examining Differences
Hypothesis Tests
Non-parametric Tests
(Nonmetric Tests)
Parametric Tests
(Metric Tests)
One Sample
* t test
* Z test
Two or More
Samples
Independent
Samples
* Two-Group
t test
* Z test
Paired
Samples
* Paired
t test
One Sample
* Chi-Square
* K-S
* Runs
* Binomial
Two or More
Samples
Independent
Samples
* Chi-Square
* Mann-Whitney
* Median
* K-S
Paired
Samples
* Sign
* Wilcoxon
* McNemar
* Chi-Square