2: Population Models for Continuous Numerical Variables
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7.2
Population Models for Continuous Numerical Variables
343
This means that the area of the rectangle above each interval is equal to the relative frequency of values that fall in the interval. Because the area of a rectangle in the
density histogram specifies the proportion of the population values that fall in the
corresponding interval, it can be interpreted as the long-run proportion of time that
a value in the interval would occur if babies were randomly selected from the
population.
For the interval 4.5 to ,5.5 in Figure 7.6(a), the approximate probability that
the weight of a randomly selected individual falls in the interval is
P 14.5 , x , 5.52 < area of rectangle above the interval from 4.5 to 5.5
5 1density2 112
5 1.052 112
5 .05
Similarly,
P 17.5 , x , 8.52 < .25
The probability of observing a value in an interval other than those used to construct the density histogram can be approximated. For example, to approximate the
probability of observing a birth weight between 7 and 8 pounds, we could add half
the area of the rectangle for the 6.5 to 7.5 interval and half the area for the 7.5 to 8.5
interval. Because the area of each rectangle in the density histogram is equal to the
proportion of the population falling in the corresponding interval,
1
1area of rectangle for 6.5 to 7.52
2
1
1 1area of rectangle for 7.5 to 8.52
2
1
1
5 1.372 112 1 1.252 112
2
2
5 .31
P 17 , x , 82 <
The approximation of probabilities can be improved by increasing the number of
intervals on which the density histogram is based. As Figure 7.6(a) shows, a density
histogram based on a small number of intervals can be quite jagged. Figures 7.6(b)–
(d) show density histograms based on 14, 28, and 56 intervals, respectively. As the
number of intervals increases, the rectangles in the density histogram become much
narrower and the histogram appears smoother.
There are two important ideas that you should remember from this discussion.
First, when summarizing a population distribution with a density histogram, the area
of any rectangle in the histogram can be interpreted as the probability of observing a
variable value in the corresponding interval when an individual is selected at random
from the population. The second important idea is that when a density histogram
based on a small number of intervals is used to summarize a population distribution
for a continuous numerical variable, the histogram can be quite jagged. However,
when the number of intervals is increased, the resulting histograms become much
smoother in appearance. (You can see this in the histograms of Figure 7.6.)
It is often useful to represent a population distribution for a continuous variable
by using a simple smooth curve that approximates the actual population distribution.
For example, Figure 7.7 shows a smooth curve superimposed over the density histogram of Figure 7.6(d). Such a curve is called a continuous probability distribution.
Because the total area of the rectangles in a density histogram is equal to 1, we consider only smooth curves for which the total area under the curve is equal to 1.
A continuous probability distribution is an abstract but simplified description of
the population distribution that preserves important population characteristics (genCopyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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344
Chapter 7 Population Distributions
0.5
Density
0.4
0.3
0.2
0.1
0
FIGURE 7.7
3
A smooth curve specifies a continuous
distribution for birth weight.
4
5
6 7 8 9
Birth weight
10 11
eral shape, center, spread, etc.). Thus, it can serve as a model for the distribution of
values in the population. Because the area under the curve approximates the areas of
rectangles in the density histogram, the area under the curve and above any particular
interval can be interpreted as the approximate probability of observing a value in that
interval when an individual is selected at random from the population.
A continuous probability distribution is a smooth curve, called a density curve, that
serves as a model for the population distribution of a continuous variable.
Properties of continuous probability distributions are:
1. The total area under the curve is equal to 1.
2. The area under the curve and above any particular interval is interpreted as the
(approximate) probability of observing a value in the corresponding interval when
an individual or object is selected at random from the population.
Examples 7.5 to 7.7 show how a continuous probability distribution can be used
to make probability statements about a variable.
EXAMPLE 7.5
Departure Delays
FIGURE 7.8
Graphs for Example 7.5: (a) density
histogram of elapsed time values;
(b) continuous probability distribution
for elapsed time.
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0
0.10
Density
Density
A morning commuter train never leaves before its scheduled departure time. The
length of time that elapses between the scheduled departure time and the actual departure time is recorded on 365 occasions. The resulting observations are summarized
in the density histogram shown in Figure 7.8(a). This histogram can serve as an approximation to the population distribution of the variable x 5 elapsed time (in
minutes).
0
0 1 2 3 4 5 6 7 8 9 10
Elapsed time
(a)
0
10
Elapsed time
(b)
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7.2
Population Models for Continuous Numerical Variables
345
Because the histogram in Figure 7.8(a) is fairly flat, a reasonable model (smooth
curve) for the population distribution is the probability distribution “curve” shown
in Figure 7.8(b). This model is sometimes referred to as a uniform distribution. The
height of the curve (density 5 0.1) is chosen so that the total area under the density
curve is equal to 1.
The model can be used to approximate probabilities involving the variable x. For
example, the probability that between 5 and 7 minutes elapse between scheduled and
actual departure time is the area under the density curve and above the interval from
5 to 7, as shown in the accompanying illustration:
Density
Area = 2(0.1) = 0.2
0.1
0
5
7
10
Total area
under density
curve is 1
So
P(elapsed time is between 5 and 7) 5 P(5 , x , 7) 5 (2)(.1) 5 .2
Other probabilities are determined in a similar fashion. For example,
P(elapsed time is less than 2.5)
5 P(x , 2.5)
5 area under curve and above interval from 0 to 2.5
5 (2.5)(.1) = .25
For continuous numerical variables, probabilities are represented by an area under a probability distribution curve and above an interval. The area above an interval
is not changed by including the interval endpoints, because there is no area above a
single point. In Example 7.5, we found that P(x , 2.5) 5 .25. It is also true that
P(x # 2.5) 5 .25.
For continuous numerical variables and any particular numbers a and b,
P 1x # a 2 5 P 1x , a 2
P 1x $ b2 5 P 1x . b2
P 1a , x , b2 5 P 1a # x # b2
EXAMPLE 7.6
Priority Mail Package Weights
Two hundred packages shipped using the Priority Mail rate for packages under
2 pounds were weighed, resulting in a sample of 200 observations of the variable
x 5 package weight (in pounds)
from the population of all Priority Mail packages under 2 pounds. A density histogram constructed from the 200 weights is shown in Figure 7.9(a). Because the histogram is based on a sample of 200 packages, it provides only an approximation to the
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346
Chapter 7
Population Distributions
FIGURE 7.9
Graphs for Example 7.6: (a) density
histogram of package weight values;
(b) continuous probability distribution
for package weight.
P(x > 1.5)
1.0
Density
Density
1.0
0.5
0.75
0
0
0
0.50
1.00
Weight
1.50
2.00
0
1.00
Weight
(a)
1.50
2.00
(b)
population histogram. However, the shape of the sample density histogram does suggest that a reasonable model for the population might be the triangular distribution
shown in Figure 7.9(b).
Note that the total area under the probability distribution curve (the density
curve) is equal to
total area of triangle 5
1
1
1base2 1height2 5 122 112 5 1
2
2
The probability model can be used to compute the proportion of packages over
1.5 pounds, P(x . 1.5). This corresponds to the area of the shaded trapezoid in Figure 7.9(b). In this case, it is easier to compute the area of the unshaded region (which
corresponds to P(x # 1.5)), because this is just the area of a triangle:
P 1x # 1.52 5
1
11.52 1.752 5 .5625
2
Because the total area under a probability density curve is 1,
P(x . 1.5) 5 1 2 .5625 5 .4375
It is also the case that
P(x $ 1.5) 5 .4375
and that
P(x 5 1.5) 5 0
The last probability is a consequence of the fact that there is 0 area under the
density curve above a single x value.
EXAMPLE 7.7
Service Times
An airline’s toll-free reservation number recorded the length of time required to provide service to each of 500 callers. This resulted in 500 observations of the continuous
numerical variable
x 5 service time
A density histogram is shown in Figure 7.10(a).
The population of interest is all callers to the reservation line. After studying the
density histogram, we might think that a model for the population distribution
would be flat over the interval from 0 to 3 and higher but also flat over the interval
from 3 to 10. This type of model was thought to be reasonable, because service requests were usually one of two types: (1) requests to make a flight reservation and (2)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
7.2
347
Population Models for Continuous Numerical Variables
Area = 7
8
0.15
FIGURE 7.10
Graphs for Example 7.7: (a) density
histogram of service times; (b) continuous distribution of service times.
Density
Density
1
8
0.10
Area = 1
8
0.05
1
24
0
0
0
1
2
3
4
5 6
Time
7
8
9 10
0
10
3
(a)
(b)
requests to cancel a reservation. Canceling a reservation, which accounted for about
one-eighth of the calls to the reservation line, could usually be accomplished fairly
quickly, whereas making a reservation (seven-eighths of the calls) required more time.
Figure 7.10(b) shows the probability distribution curve proposed as a model
for the variable x 5 service time. The height of the curve for each of the two segments
was chosen so that the total area under the curve would be 1 and so that P(x # 3) 5
1/8 (these were thought to be cancellation calls) and P(x . 3) 5 7/8.
Once the model has been developed, it can be used to compute probabilities. For
example,
P 1x . 82 5 area under curve and above interval from 8 to 10
1
2
1
5 2a b 5 5
8
8
4
In the long run, one-fourth of all service requests will require more than 8 minutes.
Similarly,
P 12 , x , 42 5 area under curve and above interval from 2 to 4
5 1area under curve and above interval from 2 to 32
1 1area under curve and above interval from 3 to 42
1
1
5 1a b 1 1a b
24
8
1
3
5
1
24
24
4
5
24
1
5
6
In each of the previous examples, the continuous probability distribution used as
a model for the population distribution was simple enough that we were able to calculate probabilities (evaluate areas under the curve) using simple geometry. Example
7.8 shows that this is not always the case.
E X A M P L E 7 . 8 Online Registration Times
Students at a university use an online registration system to register for courses. The
variable
x 5 length of time required for a student to register
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
348
Chapter 7
Population Distributions
was recorded for a large number of students using the system, and the resulting values
were used to construct the density histogram of Figure 7.11. The general form of the
density histogram can be described as bell-shaped and symmetric, and a smooth curve
has been superimposed. This smooth curve serves as a reasonable model for the population distribution represented by the density histogram. Although this is a common
population model (there are many variables whose distributions are described by
curves of this sort), it is not obvious how we could use such a model to calculate probabilities, because at this point it is not clear how to find areas under such a curve.
Density
0.2
FIGURE 7.11
0.1
0
Density histogram and continuous
probability distribution for time to
register for Example 7.8.
5
7
9
11
13
15
Length of time
17
19
The probability model of Example 7.8 is an example of a type of symmetric bellshaped distribution known as a normal probability distribution. Normal distributions
have many and varied applications, and they are investigated in more detail in the
next section.
EX E RC I S E S 7. 1 0 - 7 . 1 4
7.10 Consider the population of batteries made by a
7.11 A particular professor never dismisses class early.
particular manufacturer. The following density curve
represents the probability distribution for the variable
x 5 lifetime (in hours):
Let x denote the amount of time past the hour (in minutes) that elapses before the professor dismisses class.
Suppose that the density curve shown in the following
figure is an appropriate model for the probability distribution of x:
Density
0
25
50
Shade the region under the curve corresponding to each
of the following probabilities (draw a new curve for each
part):
a. P(10 , x , 25)
b. P(10 # x # 25)
c. P(x , 30)
d. The probability that the lifetime is at least 25 hours
e. The probability that the lifetime exceeds 30 hours
Bold exercises answered in back
Data set available online
1
10
x
0
10
Time
(minutes)
a. What is the probability that at most 5 minutes elapse
before dismissal?
Video Solution available
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
7.2
b. What is the probability that between 3 and 5 minutes elapse before dismissal?
c. What do you think the value of the mean is for this
distribution?
7.12 Consider the population that consists of all soft
contact lenses made by a particular manufacturer, and
define the variable x 5 thickness (in millimeters). Suppose that a reasonable model for the population distribution is the one shown in the following figure:
Density
Population Models for Continuous Numerical Variables
349
Use the fact that the area of a trapezoid 5 (base)(average
of two side lengths) to answer each of the following
questions.
a. What is the probability that a randomly selected
package of this type weighs at most 0.5 pound?
b. What is the probability that a randomly selected
package of this type weighs between 0.25 pound and
0.5 pound?
c. What is the probability that a randomly selected
package of this type weighs at least 0.75 pound?
7.14 Let x denote the time (in seconds) necessary for an
individual to react to a certain stimulus. The probability
distribution of x is specified by the following density
curve:
5
Density
Thickness
0
0.20
0.40
a. Verify that the total area under the density curve is
equal to 1. [Hint: The area of a triangle is equal to
0.5(base)(height).]
b. What is the probability that x is less than .20? less
than .10? more than .30?
c. What is the probability that x is between .10 and
.20? (Hint: First find the probability that x is not
between .10 and .20.)
d. Because the density curve is symmetric, the mean of
the distribution is .20. What is the probability that
thickness is within 0.05 of the mean thickness?
7.13 A delivery service charges a special rate for any
package that weighs less than 1 pound. Let x denote the
weight of a randomly selected parcel that qualifies for
this special rate. The probability distribution of x is
specified by the following density curve:
x
1
a. What is the height of the density curve above x 5 0?
(Hint: Total area under the curve 5 1.)
b. What is the probability that reaction time exceeds
0.5 second?
c. What is the probability that reaction time is at
most 0.25 second?
Density
Density = 0.5 + x
1.5
1.0
0.5
x
0
Bold exercises answered in back
1
Data set available online
Video Solution available
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
350
Chapter 7
7.3
Population Distributions
Normal Distributions
Normal distributions formalize the notion of mound-shaped histograms introduced
in Chapter 4. Normal distributions are widely used for two reasons. First, they provide a reasonable approximation to the distribution of many different variables. They
also play a central role in many of the inferential procedures that will be discussed in
Chapters 9 to 11. Normal distributions are continuous probability distributions that
are bell-shaped and symmetric, as shown in Figure 7.12. Normal distributions are
also referred to as normal curves.
FIGURE 7.12
A normal distribution.
There are many different normal distributions, and they are distinguished from one
another by their mean m and standard deviation s. The mean m of a normal distribution describes where the corresponding curve is centered, and the standard deviation s
describes how much the curve spreads out around that center. As with all continuous
probability distributions, the total area under any normal curve is equal to 1.
Three normal distributions are shown in Figure 7.13. Notice that the smaller the
standard deviation, the taller and narrower the corresponding curve. Remember that
areas under a continuous probability distribution curve represent probabilities; therefore, when the standard deviation is small, a larger area is concentrated near the center
of the curve, and the chance of observing a value near the mean is much greater (because m is at the center).
Density
μ = 40, σ = 2.5
0.15
0.10
μ = 10, σ = 5
μ = 70, σ = 10
0.05
FIGURE 7.13
Three normal distributions.
0
0
50
100
The value of m is the number on the measurement axis lying directly below the
top of the bell. The value of s can also be ascertained from a picture of the curve.
Consider the normal curve in Figure 7.14. Starting at the top of the bell (above
m 5 100) and moving to the right, the curve turns downward until it is above the
value 110. After that point, it continues to decrease in height but is turning upward
rather than downward. Similarly, to the left of m 5 100, the curve turns downward
until it reaches 90 and then begins to turn upward. The curve changes from turning
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
7.3
Normal Distributions
351
Curve turns downward
Curve turns upward
Curve turns upward
σ = 10
FIGURE 7.14
m and s for a normal curve.
80
90
σ = 10
μ = 100
110
120
downward to turning upward at a distance of 10 on either side of m 5 100. In general, s is the distance to either side of m at which a normal curve changes from turning downward to turning upward, so s 5 10 for the normal curve in Figure 7.14.
If a particular normal distribution is to be used as a population model, a mean and
a standard deviation must be specified. For example, a normal distribution with mean
7 and standard deviation 1 might be used as a model for the distribution of x 5 birth
weight from Section 7.2. If this model is a reasonable description of the probability
distribution, we could use areas under the normal curve with m 5 7 and s 5 1 to
approximate various probabilities related to birth weight. The probability that a birth
weight is over 8 pounds (expressed symbolically as P(x . 8)) corresponds to the shaded
area in Figure 7.15(a). The shaded area in Figure 7.15(b) is the (approximate) probability P(6.5 , x , 8) of a birth weight falling between 6.5 and 8 pounds.
P(6.5 < x < 8)
P(x > 8)
FIGURE 7.15
Normal distribution for birth weight:
(a) shaded area 5 P(x . 8);
(b) shaded area 5 P(6.5 , x , 8).
7
(a)
8
6
7
8
(b)
Unfortunately, direct computation of such probabilities (areas under a normal
curve) is not simple. To overcome this difficulty, we rely on technology or a table of
areas for a reference normal distribution called the standard normal distribution.
DEFINITION
The standard normal distribution is the normal distribution with m 5 0 and
s 5 1. The corresponding density curve is called the standard normal curve. It
is customary to use the letter z to represent a variable whose distribution is
described by the standard normal curve. The term z curve is often used in place
of standard normal curve.
Few naturally occurring variables have distributions that are well described by the
standard normal distribution, but this distribution is important because it is also used
in probability calculations for other normal distributions. When we are interested in
finding a probability based on some other normal curve, we either rely on technology
or we first translate the problem into an “equivalent” problem that involves finding
an area under the standard normal curve. A table for the standard normal distribution
is then used to find the desired area. To be able to do this, we must first learn to work
with the standard normal distribution.
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352
Chapter 7 Population Distributions
The Standard Normal Distribution
In working with normal distributions, we need two general skills:
1. We must be able to use the normal distribution to compute probabilities, which
are areas under a normal curve and above given intervals.
2. We must be able to characterize extreme values in the distribution, such as the
largest 5%, the smallest 1%, and the most extreme 5% (which would include the
largest 2.5% and the smallest 2.5%).
The standard normal or z curve is shown in Figure 7.16(a). It is centered at
m 5 0, and the standard deviation, s 5 1, is a measure of the extent to which it
spreads out about its mean (in this case, 0). Note that this picture is consistent with
the Empirical Rule of Chapter 4: About 95% of the area (probability) is associated
with values that are within 2 standard deviations of the mean (between 22 and 2)
and almost all of the area is associated with values that are within 3 standard deviations of the mean (between 23 and 3).
Appendix Table 2 tabulates cumulative z curve areas of the sort shown in Figure
7.16(b) for many different values of z. The smallest value for which the cumulative
area is given is 23.89, a value far out in the lower tail of the z curve. The next smallest
value for which the area appears is 23.88, then 23.87, then 23.86, and so on in
increments of 0.01, terminating with the cumulative area to the left of 3.89.
Cumulative area = area to the left of z value
z curve
z
–3
–2
–1
0
2
3
–3
–2
–1
0
1
2
3
A particular z value
FIGURE 7.16
(a) A standard normal (z) curve;
(b) a cumulative area.
1
(a)
(b)
Using the Table of Standard Normal Curve Areas
For any number z* between 23.89 and 3.89 and rounded to two decimal
places, Appendix Table 2 gives
(area under z curve to the left of z*) 5 P(z , z*) 5 P(z # z*)
where the letter z is used to represent a variable whose distribution is the standard normal distribution.
To find this probability using the table, locate the following:
1. The row labeled with the sign of z* and the digit to either side of the decimal point (for example, 21.7 or 0.5)
2. The column identified with the second digit to the right of the decimal
point in z* (for example, .06 if z* 5 21.76)
The number at the intersection of this row and column is the desired probability, P(z , z*).
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7.3
Normal Distributions
353
A portion of the table of standard normal curve areas appears in Figure 7.17. To
find the area under the z curve to the left of 1.42, look in the row labeled 1.4 and the
column labeled .02 (the highlighted row and column in Figure 7.17). From the table,
the corresponding cumulative area is .9222. So
z curve area to the left of 1.42 5 .9222
We can also use the table to find the area to the right of a particular value. Because
the total area under the z curve is 1, it follows that
z curve area to the right of 1.42 5 1 2 1z curve area to the left of 1.422
5 1 2 .9222
5 .0778
These probabilities can be interpreted to mean that in a long sequence of observations, roughly 92.22% of the observed z values will be smaller than 1.42 and 7.78%
will be larger than 1.42.
FIGURE 7.17
z*
.00
.01
.02
.03
.04
.05
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
.5000
.5398
.5793
.6179
.6554
.6915
.7257
.7580
.7881
.8159
.8413
.8643
.8849
.9032
.9192
.9332
.9452
.9554
.9641
.5040
.5438
.5832
.6217
.6591
.6950
.7291
.7611
.7910
.8186
.8438
.8665
.8869
.9049
.9207
.9345
.9463
.9564
.9649
.5080
.5478
.5871
.6255
.6628
.6985
.7324
.7642
.7939
.8212
.8461
.8686
.8888
.9066
.9222
.9357
.9474
.9573
.9656
.5120
.5517
.5910
.6293
.6664
.7019
.7357
.7673
.7967
.8238
.8485
.8708
.8907
.9082
.9236
.9370
.9484
.9582
.9664
.5160
.5557
.5948
.6331
.6700
.7054
.7389
.7704
.7995
.8264
.8508
.8729
.8925
.9099
.9251
.9382
.9495
.9591
.9671
.5199
.5596
.5987
.6368
.6736
.7088
.7422
.7734
.8023
.8289
.8531
.8749
.8944
.9115
.9265
.9394
.9505
.9599
.9678
Portion of Appendix Table 2 (standard
normal curve areas).
P(z < 1.42)
EXAMPLE 7.9
Finding Standard Normal Curve Areas
The probability P(z , Ϫ1.76) is found at the intersection of the Ϫ1.7 row and the
.06 column of the z table. The result is
P(z , 21.76) 5 .0392
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