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Chapter 6
Putting Statistics
to Work
© 2008 Pearson Addison-Wesley.
All rights reserved
Chapter
6
Putting Statistics to Work
6A
Characterizing Data
6B
Measures of Variation
6C
The Normal Distribution
6D
Statistical Inference
Copyright © 2008 Pearson Education, Inc.
Slide 6-2
Unit
6A
Characterizing Data
Copyright © 2008 Pearson Education, Inc.
Slide 6-3
6-A
Measures of Center in a
Distribution
The mean is what we most commonly call the
average value. It is defined as follows:
The median is the middle value in the sorted
data set (or halfway between the two middle
values if the number of values is even).
The mode is the most common value (or group
of values) in a distribution.
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Slide 6-4
6-A
Mean vs. Average
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Slide 6-5
6-A
Finding the Median for an Odd
Number of Values
6.72
3.46
3.46
3.60
3.60
6.44
6.44
6.72
26.70
26.70
(sorted list)
(odd number of values)
exact middle
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median is 6.44
Slide 6-6
6-A
Finding the Median for an
Even Number of Values
6.72
3.46
3.60
6.44
3.46
3.60
6.44
6.72
(sorted list)
(even number of values)
3.60 + 6.44
2
Copyright © 2008 Pearson Education, Inc.
median is 5.02
Slide 6-7
6-A
Mode Examples
a. 5 5 5 3 1 5 1 4 3 5

Mode is 5
b. 1 2 2 2 3 4 5 6 6 6 7 9

Bimodal (2 and 6)
c. 1 2 3 6 7 8 9 10

No Mode
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Slide 6-8
6-A
Shapes of Distributions
Two single-peaked (unimodal) distributions.
A double-peaked (bimodal) distribution.
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Slide 6-9
Symmetric and Skewed
Distributions
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6-A
Slide 6-10
6-A
Variation
From left to right, these three distributions have
increasing variation.
Variation describes how widely data values are
spread out about the center of a distribution.
Copyright © 2008 Pearson Education, Inc.
Slide 6-11
Unit
6B
Measures of Variation
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Slide 6-12
6-B
Why Variation Matters
Big Bank (three line wait times):
4.1
5.2
5.6
6.2
6.7
7.2
7.7
7.7
8.5
9.3
11.0
Best Bank (one line wait times):
6.6
6.7
6.7
6.9
7.1
7.2
7.3
7.4
7.7
7.8
7.8
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Slide 6-13
6-B
Five Number Summaries &
Box Plots
Big Bank
low value (min) =
lower quartile =
median =
upper quartile =
high value (max) =
4.1
5.6
7.2
8.5
11.0
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Best Bank
low value (min) =
lower quartile =
median =
upper quartile =
high value (max) =
6.6
6.7
7.2
7.7
7.8
Slide 6-14
6-B
Standard Deviation
Let A = {2, 8, 9, 12, 19} with a mean of 10. Use the data
set A above to find the sample standard deviation.
x (data value)
2
8
9
12
19
x – mean
(deviation)
2 – 10 = –8
8 – 10 = –2
9 – 10 = –1
12 – 10 = 2
19 – 10 = 9
Total
(deviation)2
(-8)2 = 64
(-2)2 = 4
(-1)2 = 1
(2)2 = 4
(9)2 = 81
154
2
standard deviaton = sum of (deviations from the mean)
total number of data values  1
Copyright © 2008 Pearson Education, Inc.
Slide 6-15
Range Rule of Thumb for
Standard Deviation
standard deviation »
6-B
range
4
Estimate standard deviation by taking an
approximate range, (usual high – usual low),
and dividing by 4.
If we know the standard deviation, we can
estimate:
low value  mean – 2 x standard deviation
high value  mean + 2 x standard deviation
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Slide 6-16
Unit
6C
The Normal Distribution
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Slide 6-17
The 68-95-99.7 Rule for
a Normal Distribution
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6-C
Slide 6-18
6-C
Z-Score Formula
standard score = z = data value  mean
standard deviation
Example:
If the nationwide ACT mean were 21 with a standard
deviation of 4.7, find the z-score for a 30.
What does this mean?
This means that an ACT score of 30 would be about 1.91
standard deviations above the mean of 21.
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Slide 6-19
6-C
Standard Scores and Percentiles
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Slide 6-20
Unit
6D
Statistical Inference
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Slide 6-21
6-D
Inferential Statistics Definitions
A set of measurements or observations in a
statistical study is said to be statistically significant if
it is unlikely to have occurred by chance.
0.05 (1 out of 20) or 0.01 (1 out of 100) are two very
common levels of significance that indicate the
probability that an observed difference is simply due
to chance.
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Slide 6-22
Margin of Error and Confidence
Interval
6-D
The margin of error for 95% confidence is
1
margin of error »
n
The 95% confidence interval is found by subtracting
and adding the margin of error from the sample
proportion.
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Slide 6-23
6-D
Null and Alternative
Hypotheses
The null hypothesis claims a specific value for a
population parameter. It takes the form
null hypothesis: population parameter = claimed value
The alternative hypothesis is the claim that is
accepted if the null hypothesis is rejected.
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Slide 6-24
6-D
Outcomes of a Hypothesis Test

Rejecting the null hypothesis, in which case we
have evidence that supports the alternative
hypothesis.

Not rejecting the null hypothesis, in which case
we lack sufficient evidence to support the
alternative hypothesis.
Copyright © 2008 Pearson Education, Inc.
Slide 6-25