Identifying Appropriate Test Statistics Involving Population Mean

Steps for Hypothesis Testing
Fig. 15.3
Draw Marketing Research Conclusion
Formulate H0 and H1
Select Appropriate Test
Choose Level of Significance
Determine Prob
Assoc with Test Stat
Determine Critical
Value of Test Stat
TSCR
Determine if TSCR
falls into (Non)
Rejection Region
Compare with Level
of Significance, 
Reject/Do not Reject H0
Calculate Test Statistic TSCAL

Step 1: Formulate the Hypothesis
• A null hypothesis is a statement of the status
quo, one of no difference or no effect. If the null
hypothesis is not rejected, no changes will be
made.
• An alternative hypothesis is one in which some
difference or effect is expected.
• The null hypothesis refers to a specified value of
the population parameter (e.g., ), not a
sample statistic (e.g., ).
m, s, p
X

For the data in Table 15.1, suppose we wanted to test
the hypothesis that the mean familiarity rating exceeds
4.0, the neutral value on a 7 point scale. A significance
level of = 0.05 is selected. The hypotheses may be
formulated as:
Example of a Hypothesis Test

tCAL = (4.724-4.0)/0.293 = 2.471
< 4.0
H0:

m > 4.0
t = (X - m)/sX
sX = 0.293

m
H1:

•The df for the t stat is n - 1. In this case, n - 1 = 28.
•The probability assoc with 2.471 is less than 0.05. So the null hypothesis is
rejected
• Alternatively, the critical tα value for a significance level of 0.05 is 1.7011
•Since, 1.7011 <2.471, the null hypothesis is rejected.
•The familiarity level does exceed 4.0.
•Note that if the population standard deviation was known to be 1.5, rather
than estimated from the sample, a z test would be appropriate.
One Sample : t Test

• A null hypothesis may be rejected, but it can
never be accepted based on a single test.
• In marketing research, the null hypothesis is
formulated in such a way that its rejection
leads to the acceptance of the desired
conclusion.
• A new Internet Shopping Service will be
introduced if more than 40% people use it:
H0: p  0.40
H1: p > 0.40

• In eg on previous slide, the null hyp is a
one-tailed test, because the alternative
hypothesis is expressed directionally.
• If not, then a two-tailed test would be
required as foll:
H0: p = 0.40
H1: p  0.40

• The test statistic measures how close the sample
has come to the null hypothesis.
• The test statistic often follows a well-known
distribution (eg, normal, t, or chi-square).
• In our example, the z statistic, which follows the
standard normal distribution, would be appropriate.
Step 2: Select an Appropriate Test
z =
p - p
sp
Where σp is standard deviation

Type I Error
• Occurs if the null hypothesis is rejected when it is in fact true.
• The probability of type I error ( α ) is also called the level of
significance.
Type II Error
• Occurs if the null hypothesis is not rejected when it is in fact
false.
• The probability of type II error is denoted by β .
• Unlike α, which is specified by the researcher, the magnitude
of β depends on the actual value of the population
parameter (proportion).
It is necessary to balance the two types of errors.
Step 3: Choose Level of Significance

Power of a Test
• The power of a test is the probability
(1 - β) of rejecting the null hypothesis
when it is false and should be rejected.
• Although β is unknown, it is related to
α. An extremely low value of α (e.g.,
= 0.001) will result in intolerably high β
errors.
Step 3: Choose Level of Significance

Probability of z with a One-
Tailed Test
Unshaded Area
= 0.0301
Fig. 15.5
Shaded Area
= 0.9699
zCAL = 1.88
0

• The required data are collected and the value
of the test statistic computed.
• In our example, 30 people were surveyed and
17 shopped on the internet. The value of the
sample proportion is
= 17/30 = 0.567.
• The value of is:
Step 4: Collect Data and Calculate
Test Statistic
p
sp
sp =0.089

The test statistic z can be calculated as follows:
s
p
p
p
zCAL
-
=
ˆ
= 0.567-0.40
0.089
= 1.88
Step 4: Collect Data and Calculate
Test Statistic

• Using standard normal tables (Table 2 of the Statistical
Appendix), the area to the right of zCAL is .0301 (zCAL =1.88)
• The shaded area between 0 and 1.88 is 0.4699. Therefore,
the area to the right of 1.88 is 0.5 - 0.4699 = 0.0301.
• Thus, the p-value is .0301
• Alternatively, the critical value of z, called zα, which will give
an area to the right side of the critical value of α=0.05, is
between 1.64 and 1.65. Thus zα =1.645.
• Note, in determining the critical value of the test statistic,
the area to the right of the critical value is either α or α/2. It
is α for a one-tail test and α/2 for a two-tail test.
Step 5: Determine Probability Value/
Critical Value

• If the prob associated with the calculated value
of the test statistic ( zCAL) is less than the level of
significance (α), the null hypothesis is rejected.
• In our case, the p-value is 0.0301.This is less
than the level of significance of α =0.05. Hence,
the null hypothesis is rejected.
• Alternatively, if the calculated value of the test
statistic is greater than the critical value of the
test statistic ( zα), the null hypothesis is rejected.
Steps 6 & 7: Compare Prob and
Make the Decision

• The calculated value of the test statistic zCAL= 1.88
lies in the rejection region, beyond the value of
zα=1.645. Again, the same conclusion to reject the
null hypothesis is reached.
• Note that the two ways of testing the null hypothesis
are equivalent but mathematically opposite in the
direction of comparison.
• Writing Test-Statistic as TS:
If the probability of TSCAL < significance level ( α )
then reject H0 but if TSCAL > TSCR then reject H0.
Steps 6 & 7: Compare Prob and
Make the Decision

• The conclusion reached by hypothesis testing must be
expressed in terms of the marketing research problem.
• In our example, we conclude that there is evidence that
the proportion of Internet users who shop via the
Internet is significantly greater than 0.40. Hence, the
department store should introduce the new Internet
shopping service.
Step 8: Mkt Research Conclusion

Using a t-Test
• Assume that the random variable X is normally dist, with
unknown pop variance estimated by the sample variance s 2.
• Then a t test is appropriate.
• The t-statistic, is t distributed with n - 1 df.
• The t dist is similar to the normal distribution: bell-shaped and
symmetric. As the number of df increases, the t dist
approaches the normal dist.
t = (X - m)/sX

Broad Classification of Hyp Tests
Means Proportions
Tests of
Association
Tests of
Differences
Hypothesis Tests
Means Proportions

Hypothesis Testing for Differences
Independent
Samples
* Two-Group t
test
* Z test
* Paired
t test
Hypothesis Tests
One Sample Two or More
Samples
* t test
* Z test
Parametric Tests
(Metric)
Non-parametric Tests
(Nonmetric)

Two Independent Samples: Means
• In the case of means for two independent samples, the
hypotheses take the following form.
• The two populations are sampled and the means and
variances computed based on samples of sizes n1 and n2.
• The idea behind the test is similar to the test for a single
mean, though the formula for standard error is different
• Suppose we want to determine if internet usage is different
for males than for females, using data in Table 15.1
m
m 2
1
0
: =
H
m
m 2
1
1
: 
H

Identifying Appropriate Test Statistics Involving Population Mean

Two Independent-Samples: t Tests
Table 15.14 Summary Statistics
Number Standard
of Cases Mean Deviation
Male 15 9.333 1.137
Female 15 3.867 0.435
F Test for Equality of Variances
F 2-tail
value probability
15.507 0.000
t Test
Equal Variances Assumed Equal Variances Not Assumed
t Degrees of 2-tail t Degrees of 2-tail
value freedom probability value freedom probability
4.492 28 0.000 -4.492 18.014 0.000
-
Table
15.14

•Consider data of Table 15.1
•Is the proportion of respondents using the Internet
for shopping the same for males and females?
The null and alternative hypotheses are:
•The test statistic is similar to the one for difference
of means, with a different formula for standard error.
Two Independent Samples: Proportions

H0: p1 = p2
H1: p1  p2

Summary of Hypothesis Tests
for Differences
Sample Application Level of Scaling Test/Comments
One Sample
One Sample Means Metric t test, if variance is unknown
z test, if variance is known
Proportion Metric Z test

Summary of Hypothesis Tests
for Differences
Two Indep Samples
Two indep samples Means Metric Two-groupt test
F test for equality of
variances
Two indep samples Proportions Metric z test
Nonmetric Chi -square test
Application Scaling Test/Comments

Identifying Appropriate Test Statistics Involving Population Mean

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