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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).

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.

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)

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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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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

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

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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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

1

Data set available online

Video Solution available

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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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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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