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CHAPTER fifteen
Learning Objectives
Statistical Testing
of Differences
Copyright © 2002
South-Western/Thomson Learning
Learning Objectives
Learning Objectives
1. To become aware of the nature of statistical
significance.
2. To understand the concept of hypothesis
development and how to test hypotheses.
3. To understand the differences between Type
I and Type II errors.
4. To be familiar with several of the more
common statistical tests of goodness of fit,
hypotheses about one mean, hypothesis
about two means, and hypotheses about
proportions.
5. To learn about analysis of variance.
Learning Objectives
Differences and Changes
To become aware of the nature
of statistical differences.
Are certain measures different from one another?
For example:
Did top-of-mind awareness really increase?
Did customer satisfaction really increase?
Learning Objectives
Statistical Significance
To become aware of the nature
of statistical differences.
It is possible for numbers to be different in a mathematical
sense but not statistically different in a statistical sense.
• Mathematical differences
• Statistical significance
• Managerially important differences
Hypothesis Testing
Learning Objectives
To understand the concept of
hypothesis development and how to
test hypotheses.
Hypothesis
An assumption that a researcher makes about some
characteristic of the population under study.
Steps in Hypothesis Testing
Step One: Stating the Hypothesis
Null hypothesis: Ho
Alternative hypothesis: Ha
Step Two: Choosing the Appropriate Test Statistic
Hypothesis Testing
Learning Objectives
To understand the concept of
hypothesis development and how to
test hypotheses.
Step Three: Developing a Decision Rule
Step Four: Calculating the Value of the Test Statistic
• Use the appropriate formula
• Compare calculated value to the critical value.
• State the result in terms of:
• rejecting the null hypothesis
• failing to reject the null hypothesis
Step Five: Stating the Conclusion
Other issues
Learning Objectives
To understand the differences
between Type I and Type II errors.
Types of Errors in Hypothesis Testing
Type I Error
Rejection of the null hypothesis when, in fact, it is true.
Type II Error
Acceptance of the null hypothesis when, in fact, it is
false.
Accepting Ho or Failing to Reject Ho?
One-Tailed Test or Two-Tailed Test?
Table 15.2
Table 15.2
Learning Objectives
Type I and Type II Errors
Actual State of the
Null Hypothesis
Fail to Reject Ho
Reject Ho
Ho is true
Correct (1-)
no error
Type I error ()
Ho is false
Type II error ()
Correct (1- )
no error
Commonly Used
Statistical Hypothesis Tests
Learning Objectives
To understand the concept
of hypothesis development
and testing a hypothesis.
Independent Versus Related Samples
Independent samples
Measurement of a variable in one population has
no effect on the measurement of the other variable
Related Samples
Measurement of a variable in one population may
influence the measurement of the other variable.
Degrees of Freedom
The number of observations minus the number of
constraints.
Learning Objectives
Goodness of Fit
To describe statistical tests
of goodness of fit.
Chi-Square
To determine whether an observed pattern of
frequencies corresponds to an “expected” pattern.
Chi-Square Test of a Single Sample
1. Specify the null and alternative hypotheses.
2. Determine the number of visitors that would be expected
in each category if the null hypotheses were correct (Ei).
Learning Objectives
Goodness of Fit
To describe statistical tests
of goodness of fit.
3. Calculate the X2 value using the
formula:
X2 =
k
(Oi - Ei ) 2
I=1
Ei

Oi = observed number in i th category
Ei = expected number in i th category
k = number of categories
Learning Objectives
Goodness of Fit
To describe statistical tests
of goodness of fit.
4. Select the level of significance .
5. Compare the calculated X2 value with the table value.
State the result.
Chi-Square Test of Two Independent Samples
Association between two or more variables.
1. State the null and alternative hypotheses.
Learning Objectives
Goodness of Fit
To describe statistical tests
of goodness of fit.
2. Place the observed (sample) frequencies in a k x r table
using the k columns for the sample groups and the r
rows for the conditions or treatments.
Calculate the sum of each row and each column.
Record marginal totals.
Calculate total for entire table.
3. Determine the expected frequency for each cell in the
contingency table by calculating the product of the two
marginal totals common to that cell and dividing that
value by N.
Learning Objectives
Goodness of Fit
To describe statistical tests
of goodness of fit.
4. Calculate the value of X2 using:
X2 =
r
k
I=1
j=1
 
(Oij - E ij) 2
Eij
Oij = observed number in i th row of the jth column
Eij = expected number in i th row of the jth column
5. Compare the calculated X2 with the tabular X2. State the
result.
Learning Objectives
Goodness of Fit
To describe statistical tests
of goodness of fit.
Kolmogorov-Smirnov Test
1. Specify the null and alternative hypotheses.
2. Establish the cumulative frequency distribution
expected under the null hypothesis.
3. Calculate the observed cumulative frequency
distribution from the sample.
4. Select the level of significance .
5. Determine the K-S statistic.
6. State the result.
Learning Objectives
About One Mean
To describe statistical tests
about one mean.
Z - Test
Hypothesis tests about a single mean if the sample
mean if the sample is large enough and drawn from a
normal population.
1. Specify the null and alternative hypotheses.
2. Specify the level of sampling error () allowed.
3. Determine the sample standard deviation.
4. Calculate the estimated standard error of the mean:
Learning Objectives
About One Mean
Sx =
To describe statistical tests
about one mean.
S
√n
where
Sx = the estimated standard error of the
mean.
Learning Objectives
About One Mean
To describe statistical tests
about one mean.
5. Calculate the test statistic:
Z=
(sample mean)
population mean under the
null hypothesis
estimated standard error of the mean
6. State the result.
Learning Objectives
About One Mean
To describe statistical tests
about one mean.
t - Test
Hypothesis tests about a single mean if the sample mean if
the sample is too small to use the Z - test
1. Specify the null and alternative hypotheses.
2. Specify the level of sampling error () allowed.
3. Determine the sample standard deviation.
Learning Objectives
To describe statistical tests
about one mean.
About One Mean
S =
√
n

(Xi - X) 2
I=1
n-1
where
Xi = observed sales per week in the ith store
X = average sales per week
n = the number of stores
Learning Objectives
About One Mean
To describe statistical tests
about one mean.
4. Calculate the estimated standard error
of the mean using:
Sx =
S
√n
Learning Objectives
About One Mean
To describe statistical tests
about one mean.
5. Calculate the t - statistic:
(sample mean)
t=
population mean under the
null hypothesis
estimated standard error of the mean
6. State the result.
Learning Objectives
About Two Means
To describe statistical tests
about two means.
Marketers are frequently interested in testing differences
between groups.
1. The null and alternative hypotheses
2. Set the level of sampling error ()
3. Estimate the standard error of the differences
between two means:
Learning Objectives
To describe statistical tests
about two means.
About Two Means
Sx m-ƒ
=
√
S2m
nm
+
S2 ƒ
nƒ
Sm = estimated standard deviation of population m (men)
Sf = estimated standard deviation of population f (women)
nm = sample size for sample m
nf = sample size for sample f
Learning Objectives
About Two Means
To describe statistical tests
about two means.
4. Calculate the test statistic.
Z is calculated as:
difference between
means of first sample
and second sample
Z=
difference between means
under the null hypothesis
standard error or the differences between the means
5. Compare the calculated value of Z with the critical
value. State the result.
Learning Objectives
About Proportions
To describe statistical tests
about proportions.
Test of a Proportion, One Sample
Phenomena expressed as percentages
1. Specify the null and alternative hypotheses
2. Set the level of sampling error ()
3. Estimate the standard error using the P-value
specified in the null hypothesis:
Learning Objectives
About Proportions
Sp
=
To describe statistical tests
about proportions.
√
P(1- P)
n-1
where:
P = proportion specified in the null hypothesis
n = sample size
Learning Objectives
About Proportions
To describe statistical tests
about proportions.
4. Calculate the test statistic as follows:
Z=
(observed proportion - proportion under null hypothesis)
estimated standard error (Sp)
Compare the calculated Z-value with the critical Z-value.
Learning Objectives
About Proportions
To describe statistical tests
about proportions.
Tests of Differences Between Two Proportions,
Independent Samples
Management may be interested in the difference
between the proportions of people in two different
groups that engage in a certain activity or have a
certain characteristic.
1. Specify the null and alternative hypotheses
2. Set the level of sampling error ()
3. Estimate the standard error of the differences between
the two proportions:
Learning Objectives
About Proportions
Spm-ƒ
where
P=
=
To describe statistical tests
about proportions.
√
P(1- P)
1
nm
nmPm + nfPf
nm + nf
Pm = proportion in sample m (men)
Pf = proportion in sample f (women)
nm = sample of size m
nf = sample of size f
+
1
nƒ
Learning Objectives
About Proportions
To describe statistical tests
about proportions.
4. Calculate the test statistic:
difference between
observed proportions
Z=
difference between
proportions under the null
hypothesis
estimated standard error or the differences between
the two means
5. Compare the calculated Z-value with the critical Z-value.
State the result.
Learning Objectives
To learn about analysis of
variance.
Analysis of Variance
Analysis of Variance (ANOVA)
Test for the differences among the means of two or more
variables.
1. Specify the null and alternative hypotheses
2. Calculate sum of squares among groups (SSA)
SSA =

j= 1
nj (X - X ) 2
i
t
Learning Objectives
Analysis of Variance
To learn about analysis of
variance.
3. Calculate the variation among the group means as
measured by the mean sum of squares among
groups (MSA):
MSA =
sum of squares among groups (SSA)
degrees of freedom (d.f.)
where Degrees of freedom = number of groups (C) -1
Learning Objectives
To learn about analysis of
variance.
Analysis of Variance
4. Calculate the sum of squares within groups or sum of
squared error (SSE):
nj
c
SSE =

j=1
I=1
(Xij- Xj) 2
5. Calculate the mean square error (MSE):
MSE =
sum of squares within groups (SSE)
degrees of freedom (d.f.)
Learning Objectives
Analysis of Variance
To learn about analysis of
variance.
6. Calculate the F - statistic as follows:
F=
MSA
MSE
7. State the result.
p - VALUES AND SIGNIFICANCE TESTING
P- Value
The 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.
Learning Objectives
SUMMARY
• Differences and Changes
• Statistical Significance
• Hypothesis Testing
• Other issues
• Hypothesis Tests
• Goodness of Fit
• About One Mean
• About Two Means
• About Proportions
• Analysis of Variance
Learning Objectives
The End
Copyright © 2002 South-Western/Thomson Learning