Quantitative Methods for Business chapter 13 pdf

Smooth running – continuous probability distributions and basicqueuing theory 13 Chapter objectives This chapter will help you to: ■ make use of the normal distribution and appreciate it

Smooth running – continuous probability distributions and basic

queuing theory

13

Chapter objectives

This chapter will help you to:

■ make use of the normal distribution and appreciate its ance

import-■ employ the Standard Normal Distribution to investigate mal distribution problems

nor-■ apply the exponential distribution and be aware of its ness in analysing queues

useful-■ analyse a simple queuing system

■ use the technology: continuous probability distribution

■ become acquainted with business uses of the normal distribution

In the previous chapter we looked at how two different types of oretical probability distributions, the binomial and Poisson distribu-tions, can be used to model or simulate the behaviour of discrete

the-random variables These types of variable can only have a limited orfinite number of values, typically only whole numbers like the number

of defective products in a pack or the number of telephone callsreceived over a period of time

Discrete random variables are not the only type of random variable.You will also come across continuous random variables, variableswhose possible values are not confined to a limited range In the sameway as we used discrete probability distributions to help us investigatethe behaviour of discrete random variables we use continuous prob-ability distributions to help us investigate the behaviour of continuousrandom variables Continuous probability distributions can also beused to investigate the behaviour of discrete variables that can havemany different values

The most important continuous probability distribution in Statistics

is the normal distribution As the name suggests, this distribution resents the pattern of many ‘typical’ or ‘normal’ variables You mayfind the distribution referred to as the Gaussian distribution after the German mathematician Carl Friedrich Gauss (1777–1855) whodeveloped it to model observation errors that arose in surveying andastronomy measurements

rep-The normal distribution has a very special place in Statistics because

as well as helping us to model variables that behave in the way that thenormal distribution portrays, it is used to model the way in whichresults from samples vary This is of great importance when we want touse sample results to make predictions about entire populations

13.1 The normal distribution

Just as we saw that there are different versions of the binomial tion that describe the patterns of values of binomial variables, and dif-ferent versions of the Poisson distribution that describe the behaviour

distribu-of Poisson variables, there are many different versions distribu-of the normal tributions that display the patterns of values of normal variables

dis-Each version of the binomial distribution is defined by n, the ber of trials, and p, the probability of success in any one trial Each ver-

num-sion of the Poisson distribution is defined by its mean In the same way,each version of the normal distribution is identified by two definingcharacteristics or parameters: its mean and its standard deviation.The normal distribution has three distinguishing features:

■ It is unimodal, in other words there is a single peak

■ It is symmetrical, one side is the mirror image of the other

■ It is asymptotic, that is, it tails off very gradually on each sidebut the line representing the distribution never quite meetsthe horizontal axis.

Because the normal distribution is a symmetrical distribution with asingle peak, the mean, median and mode all coincide at the middle ofthe distribution For this reason we only need to use the mean as themeasure of location for the normal distribution Since the average weuse is the mean, the measure of spread that we use for the normal distribution is the standard deviation

The normal distribution is sometimes described as bell-shaped Figure13.1 illustrates the shape of the normal distribution It takes the form of

a smooth curve This is because it represents the probabilities that a tinuous variable takes values right across the range of the distribution

con-If you look back at the diagrams we used to represent discrete ability distributions in Figures 12.1 and 12.2 in Chapter 12 you will seethat they are bar charts that consist of separate blocks Each distinctblock represents the probability that the discrete random variable inquestion takes one of its distinct values Because the variable can onlytake discrete, or distinct, values we can represent its behaviour with adiagram consisting of discrete, or distinct, sections

If we want to use a diagram like Figure 12.1 or 12.2 to find the ability that the discrete variable it describes takes a specific value, wecan simply measure the height of the block against the vertical axis Incontrast, using the smooth or continuous curve in Figure 13.1 to findthe probability that the continuous variable it describes takes a particu-lar value is not so easy

prob-To start with we need to specify a range rather than a single valuebecause we are dealing with continuous values For instance, referring to

the probability of a variable, X being 4 is inadequate as in a continuous

0.0 0.1 0.2 0.3 0.4

distribution it implies the probability that X is precisely 4.000 Instead we would have to specify the probability that X is between 3.500 and 4.499.

This probability would be represented in a diagram by the area belowthe curve between the points 3.500 and 4.499 on the horizontal axis as aproportion of the total area below the curve The probability that a con-tinuous variable takes a precise value is, in effect zero This means that

in practice there is no difference between, say, the probability that X is less than 4, P(X 4.000) and the probability that X is less than or equal

to 4, P(X 4.000) Similarly the probability that X is more than 4, P(X

X is more than or equal to 4, P(X 4.000) For convenience the ities are left out of the probability statements in what follows

equal-When we looked at the binomial and Poisson distributions inChapter 12 we saw how it was possible to calculate probabilities in thesedistributions using the appropriate formulae In fact, in the days beforethe sort of software we now have became available, if you needed to use

a binomial or a Poisson distribution you had to start by consulting lished tables However, because of the sheer number of distributions,the one that you wanted may not have appeared in the tables In such

pub-a situpub-ation you hpub-ad to cpub-alculpub-ate the probpub-abilities yourself

Calculating the probabilities that make up discrete distributions istedious but not impossible, especially if the number of outcomes involved

is quite small The nature of the variables concerned, the fact that theycan only take a limited number of values, restricts the number of cal-culations involved

In contrast, calculating the probabilities in continuous distributionscan be daunting The variables, being continuous, can have an infinitenumber of different values and the distribution consists of a smooth curverather than a collection of detached blocks This makes the mathematicsinvolved very much more difficult and puts the task beyond many people.Because it was so difficult to calculate normal distribution probabil-ities, tables were the only viable means of using the normal distribution.However, the number of versions of the normal distribution is literallyinfinite, so it was impossible to publish tables of all the versions of thenormal distribution

The solution to this problem was the production of tables describing

a benchmark normal distribution known as the Standard Normal tribution The advantage of this was that you could analyse any version

Dis-of the normal distribution by comparing points in it with equivalentpoints in the Standard Normal Distribution Once you had these equiva-lent points you could use published Standard Normal Distributiontables to assist you with your analysis

Although modern software means that the Standard NormalDistribution is not as indispensable as it once was, it is important thatyou know something about it Not only is it useful in case you do nothave access to appropriate software, but more importantly, there aremany aspects of further statistical work you will meet that are easier tounderstand if you are aware of the Standard Normal Distribution.

13.2 The Standard Normal Distribution

The Standard Normal Distribution describes the behaviour of the

vari-able Z, which is normally distributed with a mean of zero and a ard deviation of one Z is sometimes known as the Standard Normal Variable and the Standard Normal Distribution is known as the Z Distribution The distribution is shown in Figure 13.2.

stand-If you look carefully at Figure 13.2 you will see that the bulk of thedistribution is quite close to the mean, 0 The tails on either side getcloser to the horizontal axis as we get further away from the mean, butthey never meet the horizontal axis They are what are called asymptotic

As you can see from Figure 13.2, half of the Standard NormalDistribution is to the left of zero, and half to the right This means that

half of the z values that make up the distribution are negative and half

are positive

Table 5 on pages 621–622, Appendix 1 provides a detailed breakdown

of the Standard Normal Distribution You can use it to find the

probabil-ity that Z, the Standard Normal Variable, is more than a certain value, z,

0.0 0.1 0.2 0.3 0.4

or less than z In order to show you how this can be done, a section of

Table 5 is printed below:

Suppose you need to find the probability that the Standard Normal

Variable, Z, is greater than 0.62, P(Z value of z, 0.62, to just one decimal place, i.e 0.6, amongst the values listed in the column headed z on the left hand side Once you have found 0.6 under z, look along the row to the right until you reach the

figure in the column headed 0.02 The figure in the 0.6 row and the0.02 column is the proportion of the distribution that lies to the right

of 0.62, 0.2676 This area represents the probability that Z is greater than 0.62, so P(Z

If you want the probability that Z is less than 1.04, P(Z 1.04), look

first for 1.0 in the z column and then proceed to the right until you

reach the figure in the column headed 0.04, 0.1492 This is the area to

the right of 1.04 and represents the probability that Z is more than 1.04 To get the probability that Z is less than 1.04, subtract 0.1492 from 1: P(Z

In Example 13.1 you will find a further demonstration of the use ofTable 5

Use Table 5 to find the following:

(a) The probability that Z is greater than 0.6, P(Z

(b) The probability that Z is less than 0.6, P(Z 0.6)

(c) The probability that Z is greater than 2.24, P(Z

(d) The probability that Z is greater than

(e) The probability that Z is less than 1.37, P(Z 1.37).

(f ) The probability that Z is greater than 0.38 and less than 2.71, P(0.38

z value(s) of interest and the area that represents the probability you want.

(a) The probability that Z is greater than 0.6, P(Z

The value of Z in this case is not specified to two places of decimals, so we take the

fig-ure to the immediate right of 0.6 in Table 5, in the column headed 0.00, which is

0.2743 This is the probability that Z is greater than 0.6 We could also say that 27.43%

of z values are greater than 0.6.

This is represented by the shaded area in Figure 13.3

(b) The probability that Z is less than 0.6, P(Z 0.6)

In part (a) we found that 27.43% of z values are bigger than 0.6 This implies that 72.57% of z values are less than 0.6, so the answer is 1  0.2743 which is 0.7257.

This is represented by the unshaded area in Figure 13.3

0.0 0.1 0.2 0.3 0.4

(c) The probability that Z is greater than 2.24, P(Z

The figure in the row to the right of 2.2 and in the column headed 0.04 in Table 5 is

0.0125 This means that 1.25% of z values are bigger than 2.24 The probability that Z is

bigger than 2.24 is therefore 0.0125

This is represented by the shaded area in Figure 13.4

(d) The probability that Z is greater than

The figure in the row to the right of 1.3 and the column headed 0.07 in Table 5 is0.9147 This is the area of the distribution to the right of 1.37 and represents the

probability that Z is greater than 1.37.

This is shown as the shaded area in Figure 13.5

(e) The probability that Z is less than 1.37, P(Z 1.37).

From part (d) we know that the probability that Z is greater than 1.37 is 0.9147, so the probability that Z is less than 1.37 (by which we mean 1.4, 1.5 and so on) is

1 0.9147, which is 0.0853

This is represented by the unshaded area in Figure 13.5

(f ) The probability that Z is greater than 0.38 and less than 2.71, P(0.38 Z 2.71) The probability that Z is greater than 0.38, P(Z

for 0.3 and the column headed 0.08, 0.3520 You will find the probability that Z is

0.0 0.1 0.2 0.3 0.4

greater than 2.71 in the row for 2.7 and the column headed 0.01, 0.0034 We can obtain

the probability that Z is more than 0.38 and less than 2.71 by taking the probability that

Z is more than 2.71 away from the probability that Z is more than 0.38:

P(0.38

This is represented by the shaded area in Figure 13.6

0.0 0.1 0.2 0.3 0.4

Another way of approaching this is to say that if 35.20% of the area is to the right of 0.38and 0.34% of the area is to the right of 2.71, then the difference between these two per-centages, 34.86%, is the area between 0.38 and 2.71.

(g) The probability that Z is greater than 1.93 and less than 0.88, P(1.93

Z 0.88)

In Table 5 the figure in the 1.9 row and the 0.03 column, 0.9732, is the probability

that Z is more than

(h) The probability that Z is greater than 0.76 and less than 1.59, P(0.76

headed 0.09, 0.0559 The probability that Z is between 0.76 and 1.59 is the probability that Z is greater than 0.76 minus the probability that Z is greater than 1.59:

P(

This is represented by the shaded area in Figure 13.8

0.0 0.1 0.2 0.3 0.4

Sometimes we need to use the Standard Normal Distribution in a

rather different way Instead of starting with a value of Z and finding a probability, we may have a probability and need to know the value of Z

associated with it

Example 13.2

Use Table 5 on pages 621–622 to find the specific value of Z, which we will call z ␣, so that

the area to the right of z ␣ , the probability that Z is greater than z ␣ , P(Z ␣) is:

(a) 0.4207

(b) 0.0505

(c) 0.0250

(a) If you look carefully down the list of probabilities in Table 5, you will see that

0.4207 appears to the right of the z value 0.2, so in this case the value of zis 0.2

The probability that Z is greater than 0.2, P(Z

This is represented by the shaded area in Figure 13.9.

(b) To find the value of Z that has an area of 0.0505 to the right of it you will have to

look further down Table 5 The figure 0.0505 appears in the row for 1.6 and the

column headed 0.04, 0.0505 is the probability that Z is more than 1.64, P(Z

This is represented by the shaded area in Figure 13.10

3 2 1

1 0

The symbol we used in Example 13.2 to represent the value of Z we wanted to find, z ␣, is a symbol that you will come across in later work

 represents the area of the distribution beyond z ␣, in other words, the

probability that z is beyond z ␣

times refer to such as area as a tail area.

Sometimes it is convenient to represent a particular value of Z by the letter z followed by the tail area beyond it in the distribution in the form of a suffix For instance, the z value 1.96 could be written as z0.0250

because there is a tail of 0.0250 of the distribution beyond 1.96 We

might say that the z value 1.96 ‘cuts off’ a tail area of 0.0250 from the

rest of the distribution

(c) The value of Z that has an area of 0.0250, or 2.5% of the distribution, to the right

of it, is 1.96 because the figure 0.0250 is in the row for 1.9 and the column

headed 0.06 in Table 5 So 0.0250 is the probability that Z is more than 1.96, P(Z

This is represented by the shaded area in Figure 13.11

3 2

1 0

In later work you will find that particular z values are often referred

to in this style because it is the area of the tail that leads us to use a

par-ticular z value and we may want to emphasize the fact Values of Z that

cut off tails of 5%, 2.5%, 1% and 1⁄2% crop up in the topics we will look

at in Chapter 16 The z values that cut off these tail areas, 1.64, 1.96, 2.33 and 2.58, are frequently referred to as z0.05, z0.025, z0.01 and z0.005

respectively

At this point you may find it useful to try Review Question 13.1 at the

end of the chapter

13.2.1 Using the Standard Normal Distribution

To use the Standard Normal Distribution to analyse other versions ofthe normal distribution we need to be able to express any value of the

normal distribution that we want to investigate as a value of Z This is sometimes known as finding its Z-equivalent or Z score.

The Z-equivalent of a particular value, x, of a normal variable, X, is the difference between x and the mean of X, ␮, divided by the standard deviation of X, ␴.

Because we are dividing the difference between the value, x, and the

mean of the distribution it belongs to, ␮, by the standard deviation of

the distribution, ␴, to get it, the Z-equivalent of a value is really just the

number of standard deviations the value is away from the mean

Once we have found the Z-equivalent of a value of a normal

distri-bution we can refer to the Standard Normal Distridistri-bution table, Table 5

on pages 621–622, to assess probabilities associated with it

(a) a steak that has an uncooked weight of more than 10 oz?

(b) a steak that has an uncooked weight of more than 9.5 oz?

(c) a steak that has an uncooked weight of less than 10.5 oz?

If X represents the uncooked weights of the steaks, we want to find P(X

represented by the shaded area in Figure 13.12

The Z-equivalent of x 10 is

So the probability that X is more than 10 is equivalent to the probability that Z is more

than 0.4 From Table 5 on pages 621–622:

So the probability that X is more than 9.5 is equivalent to the probability that Z is more

than0.6 From Table 5:

The probability that X is less than 10.5 is the same as the probability that Z is less than 1.4 According to Table 5 the probability that Z is more than 1.4 is 0.0808, so the prob- ability that Z is less than 1.4 is 1 0.0808 which is 0.9192, or 91.92%

3 1

1 0