Chapter 14
Statistical Inference:
Review of Chapters 12 & 13
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14.1
Which technique to use?
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14.2
Identifying the Correct Technique…
The two most important factors in determining the correct
statistical technique to use are:
 the problem objective,
(i.e. describe one population or compare two populations)
 and the data type.
(i.e. interval data or nominal data)
Once these factors are determined, our analysis extends to
other factors (e.g. type of descriptive measure [central
location? variability?], etc.)
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.3
Figure 14.1…
The flowchart in your textbook (Figure 14.1) describes the
logical process that allows us to identify the appropriate
method to use for the problem.
Start at the top and work
your way down the chart…
The following slides are
an interactive version of
this flowchart…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.4
Figure 14.1: Flowchart of Techniques…
Problem objective?
Describe a population
Compare two populations
Click on the mouse icon to follow the
branch of the flowchart to the next level…
Skip flowchart,
go to examples…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.5
Figure 14.1: Flowchart of Techniques…
Problem objective?
Describe a population
Data type?
Interval
Nominal
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14.6
Figure 14.1: Flowchart of Techniques…
Problem objective?
Describe a population
Data type?
Type of descriptive
measurement?
Interval
Central location
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Variability
14.7
Slide 12.19 : Identifying Factors…
Factors that identify the t-test and estimator of
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:
Top of
Flowchart
14.8
Slide 12.29 : Identifying Factors…
Factors that identify the chi-squared test and
estimator of :
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Top of
Flowchart
14.9
Figure 14.1: Flowchart of Techniques…
Problem objective?
Describe a population
Data type?
Nominal
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14.10
Slide 12.42 : Identifying Factors…
Factors that identify the z-test and
interval estimator of p:
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Top of
Flowchart
14.11
Figure 14.1: Flowchart of Techniques…
Problem objective?
Compare two populations
Data type?
Interval
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
Nominal
14.12
Figure 14.1: Flowchart of Techniques…
Problem objective?
Compare two populations
Data type?
Nominal
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.13
Slide 13.64 : Identifying Factors…
Factors that identify the z-test
and estimator for p1–p2
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
Top of
Flowchart
14.14
Figure 14.1: Flowchart of Techniques…
Problem objective?
Compare two populations
Data type?
Interval
Type of descriptive
measurement?
Central location
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
Variability
14.15
Figure 14.1: Flowchart of Techniques…
Problem objective?
Compare two populations
Data type?
Interval
Type of descriptive
measurement?
Variability
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.16
Slide 13.49 : Identifying Factors…
Factors that identify the F-test and
estimator of
:
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
Top of
Flowchart
14.17
Figure 14.1: Flowchart of Techniques…
Problem objective?
Central Location
Experimental design?
Compare two populations
Data type?
Type of descriptive
measurement?
Interval
Matched pairs
Independent samples
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.18
Figure 14.1: Flowchart of Techniques…
Problem objective?
Central Location
Experimental design?
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
Compare two populations
Data type?
Type of descriptive
measurement?
Interval
Matched pairs
14.19
Slide 13.40 : Identifying Factors…
Factors that identify the t-test and estimator of
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
:
Top of
Flowchart
14.20
Figure 14.1: Flowchart of Techniques…
Problem objective?
Central Location
Compare two populations
Data type?
Type of descriptive
measurement?
Interval
Experimental design?
Independent samples
Population variances?
Equal
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
Unequal
14.21
Figure 14.1: Flowchart of Techniques…
Problem objective?
Central Location
Compare two populations
Data type?
Type of descriptive
measurement?
Interval
Experimental design?
Independent samples
Population variances?
Equal
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.22
Slide 13.30 : Identifying Factors…
Factors that identify the equal-variances t-test
and estimator of
:
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
Top of
Flowchart
14.23
Figure 14.1: Flowchart of Techniques…
Problem objective?
Central Location
Compare two populations
Data type?
Type of descriptive
measurement?
Interval
Experimental design?
Independent samples
Population variances?
Unequal
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.24
Slide 13.31 : Identifying Factors…
Factors that identify the unequal-variances t-test and
estimator of
:
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
Top of
Flowchart
14.25
Example 14.1…
Is anti-lock braking system (ABS) in cars really effective?
We would expect if it were effective that:
 The number of accidents would decrease, and
 The cost of accident repairs would be less.
Data were collected on 500 cars with ABS and 500 cars
without. The number of cars involved in accidents was
recorded, as was the cost of repairs. What can we conclude?
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14.26
Example 14.1 (a)…
IDENTIFY
Is there sufficient evidence to infer that the accident rate is
lower in ABS-equipped cars than in cars without ABS?
(If ABS is effective, we would expect a lower accident rate in ABS-equipped cars.)
Accident rate = number of cars in accidents
total number of cars
This is nominal (i.e. categorical) data; either a car had an
accident or it didn’t. The accident rate is a proportion. We
want to compare cars with ABS against cars without.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.27
Example 14.1 (a)…
IDENTIFY
Identify the correct technique…
Problem objective?
Describing a single population
Compare two populations
Data type?
Interval
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
Nominal
14.28
Example 14.1 (a)…
IDENTIFY
The correct technique:
has been identified. The next step is to translate the rest of
the problem into the symbols and language of statistics:
p1 = proportion of cars without ABS involved in an accident
p2 = proportion of cars with ABS involved in an accident
We want to test if ABS is effective, that is, we want to
research if: p1 > p2 , that is if H1: (p1 – p2 ) > 0
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14.29
Example 14.1 (a)…
COMPUTE
Since H1: (p1 – p2 ) > 0
we have our null hypothesis: H0: (p1 – p2 ) = 0
Hence this is a Case 1 type problem.
Upon calculating our sample proportions…
…we can use Excel to complete our analysis…
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14.30
Example 14.1 (a)…
INTERPRET
Noting that z = .4663 is not greater than zCritical = 1.6449 (or
alternatively, by looking at the p-value of .3205), we cannot
reject H0 in favor of H1, that is, there is not enough evidence
to infer that ABS equipped cars have fewer accidents than
non-ABS equipped cars…
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14.31
Example 14.1 (b)…
IDENTIFY
When accidents do occur, we would expect the severity of
accidents to be lower in ABS-equipped cars (assuming that
ABS is effective), thus we are interested in this question:
Is there sufficient evidence to infer that the cost of repairing
accident damage in ABS-equipped cars is less than that of
cars without ABS?
The cost of repairs is interval data. We need a measure to
compare the two populations of cars in a meaningful way…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.32
Example 14.1 (b)…
IDENTIFY
Identify the correct technique…
Problem objective?
Describing a single population
Compare two populations
Data type?
Interval
Nominal
Type of descriptive
measurements?
Central location
Variability
…continues…
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14.33
Example 14.1 (b)…
IDENTIFY
Identify the correct technique (continued)…
Central location
Experimental design?
Independent samples
Matched pairs
Population variances
equal?
Equal??
Unequal??
e.g. we are not comparing a
head-on collision of an ABS
equipped car with a head-on
collision of a non-ABS car…
Which one is it?
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14.34
Example 14.1 (b)…
IDENTIFY
Identify the correct technique (continued)…
Population variances
equal?
Equal??
Apply the
F-test of
Unequal??
there is not enough
evidence to infer that
the variances differ…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.35
Example 14.1 (b)…
IDENTIFY
Identify the correct technique (continued)…
Population variances
equal?
Equal
Unequal
…we have the right technique! Let’s proceed with our
hypotheses set-up…
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.
14.36
Example 14.1 (b)…
IDENTIFY
We want to research whether or not the mean cost of
repairing cars without ABS brakes (population 1) is greater
than the mean cost of repair of cars equipped with ABS
brakes (population 2), i.e.:
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14.37
Example 14.1 (b)…
INTERPRET
Applying the Data Analysis tools in Excel to our data…
Indeed, there is sufficient evidence to support the belief that
non-ABS equipped cars do indeed have higher accident
repair costs than ABS equipped cars.
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14.38
Example 14.1 (c)…
IDENTIFY
In part (b) we’ve shown that ABS-equipped cars suffer less
damage in accidents (as measured by repair costs); can we
estimate how much cheaper they are to repair on average
compared to cars without ABS brakes.
Our path through the flowchart is the same, that is, we are
comparing the measure of central location of independent
samples of interval data from two populations who’s
variances are equal…
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14.39
Example 14.1 (c)…
CALCULATE
The estimator of interest is:
Assuming a 95% confidence interval…
…we estimate the cost of repair for a non-ABS equipped car
of between $71 and $651 over an ABS equipped car.
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14.40