Histogram of Speed
Speed
Density
0
20
40
60
80
100
0.000
0.005
0.010
0.015
0.020
Figure 1: Densityscale histogram of speed for Question 1.
Statistics 21: Homework 1
1. The table below depicts data collected in a (hypothetical) survey studying the distribution of traffic speed on the
Bay Bridge. Each group includes the left endpoint but not the right. (So, for instance, the first group includes
cars driving 0 mph (not moving) but does not include cars driving exactly 10 mph.) Use this information to
answer the following questions:
Speed (mph)
Percentage of Total Cars
010
20
1020
10
2040
40
4080
20
8095
10
(a) Draw a histogram of speed on the density scale.
See Figure 1.
(b) Approximately how many cars are going less than 25 miles per hour?
For all histogram problems, we will define the function
P
(
a
≤
X < b
) to be the percentage of data in [
a, b
),
which is given by the area of the histogram in this interval.
We can divide the cars going less than 25 miles per hour into three groups: those going between 0 and 10,
those between 10 and 20, and those between 20 and 25. From the table, we know that 20% of the cars are
going between 0 and 10, and 10% are going between 10 and 20. To determine the number going between
20 and 25 miles per hour, we need to rely upon the uniformity assumption within the 2040 class interval
of speed. That is, we think every car within this class is equally likely to be going any speed between 20
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and 40 miles per hour, and so the cars are spread evenly throughout the class. (This is what ensures that
the histogram is drawn as a rectangle.) Under this assumption, the number of cars going 2025 miles per
hour is approximately equal to the number going 2530, 3035, or 3540 miles per hour. That is, we can
divide the 2040 class into 4 equal blocks, each of which has a width of 5 miles per hour and a height of
Area
Total length
=
40
20
= 2%. We now have enough information to approximate the number of cars going
less than 25 miles per hour:
Percent less than 25 mph = Percent in [0
,
10) + Percent in [10
,
20) + Percent in [20
,
25)
=
P
(0
≤
X <
10) +
P
(10
≤
X <
20) +
P
(20
≤
X <
25)
= 20% + 10% + [
width of
[20
,
25)]
*
[
Density in
[20
,
25)] = 20% + 10% + 5
*
40
20
= 40%.
(c) Approximately how many cars are going between 18 and 72 miles per hour?
P
(18
≤
X <
72) =
P
(18
≤
X <
20) +
P
(20
≤
X <
40) +
P
(40
≤
X <
72)
= [
width of
[18
,
20)]
*
[
Density in
[18
,
20)] +
P
(20
≤
X <
40) + [
width of
[40
,
72)]
*
[
Density in
[40
,
72)]
= (20

18)
*
10%
10

0
+ 40% + (72

40)
*
20%
80

40
= 58%.
(d) Approximate the 32nd percentile of speed.
We know that
P
(0
≤
X <
20) =
P
(0
≤
X <
10) +
P
(10
≤
X <
20) = 20% + 10% = 30%, so 20 miles per
hour is the 30th percentile of speed. Likewise, we also know that
P
(0
≤
X <
40) =
P
(0
≤
X <
20) +
P
(20
≤
X <
40) = 30% + 40% = 70%,
so 40 miles per hour is the 70th percentile of speed. Therefore, the 32nd percentile must be between 20
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 Spring '08
 anderes
 Statistics, Approximation, Fitness approximation

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