In this model we take the difference of each pair and create a new population of differences, so if effect,
the hypothesis test is a one population test of mean that we already covered in the prior section.
Example  rental cars
An
independent
testing
agency
is
comparing the daily rental cost for renting
a compact car from Hertz and Avis.
A
random sample of 15 cities is obtained
and
the
following
rental
information
obtained.
At the .05 significance level can the
testing agency conclude that there is a
difference in the rental charged?
Notice in this example that cities are the single population being sampled and that two measurements
(Hertz and Avis) are being taken from each city. Using the matched pair design, we can eliminate the
variability due to cities being differently priced (Honolulu is cheap because you can’t drive very far on
Oahu!)
Design
Research Hypotheses:
H
o
:
µ
1
=
µ
2
(Hertz and Avis have the same mean price for compact cars.)
H
a
:
µ
1
≠
µ
2
(Hertz and Avis do not have the same mean price for compact cars.)
Model will be matched pair ttest and these hypotheses can be restated as:
H
o
:
µ
d
=
0
H
a
:
µ
d
≠
0
The test will be run at a level of significance (
α
) of 5%.
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Chapter 8 / Exercise 2
Finite Mathematics for the Managerial, Life, and Social Sciences: An Applied Approach
Tan
Expert Verified
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170
Model is twotailed matched pairs ttest with 14 degrees of freedom. Reject Ho if t < 2.145 or t >2.145.
Data/Results
We take the difference for each pair and find the sample mean and
standard deviation.
𝑡
=
1.80
−
0
2.513
√
15
⁄
= 2.77
Reject Ho under either the critical value or pvalue method.
Conclusion
There is a difference in mean price for compact cars between Hertz and Avis. Avis has lower mean
prices.
The advantage of the matched pair design is clear in this example. The sample standard deviation for the
Hertz prices is $5.23 and for Avis it is $5.62. Much of this variability is due to the cities, and the matched
pairs design dramatically reduces the standard deviation to $2.51, meaning the matched pairs ttest has
significantly more power in this example.
10.4
Independent sampling – comparing two population variances or standard deviations
Sometimes we want to test if two populations have the same spread or variation, as measured by
variance or standard deviation.
This may be a test on its own or a way of checking assumptions when
deciding between two different models (e.g.: pooled variance ttest vs. unequal variance ttest). We will
now explore testing for a difference in variance between two independent samples.
15
513
.
2
80
.
1
=
=
=
n
s
X
d
d
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171
Characteristics of F Distribution
•
It is positively skewed
•
It is nonnegative
•
There are 2 different degrees of freedom (df
num
, df
den
)
•
When the degrees of freedom change,
a new distribution is created
•
The expected value is 1.
10.4.1 F distribution
The F distribution is a family of distributions related to the Normal Distribution. There are two different
degrees of freedom, usually represented as numerator (df
num
)
and denominator (df
den
).
Also, since the F
represents squared data, the inference will be about the variance rather than the about the standard
deviation.