(13) normal distribution

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Applied Statistics and Computing Lab

NORMAL DISTRIBUTION


Indian School of Business


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Learning Goals

To understand the concept of Normal

Distribution

Useful properties of Normal Distribution

Finding Normal Probabilities

Applications of Normal Distribution

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Normal Distribution

One of the most important continuous distributions

A number of real life examples If a random variable is affected by many independent causes, and the

effect of each cause is not overwhelmingly large compared to othereffects, then the random variable will closely follow a normal distribution.The lengths of pins made by an automatic machine, the times taken by anassembly worker to complete the assigned task repeatedly, the weights of

baseballs, the tensile strengths of a batch of bolts, and the volumes ofsoup in a particular brand of canned soup are good examples of normallydistributed random variables. - Aczel- Sounderpandian

All of these are affected by a number of independent causes where theeffect of each cause is small

For eg length of a pin is affected by many independent causes such asvibrations, temperature, wear and tear on the machine, and raw materialproperties.

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Bell shaped Curve Data can be spread out in a number of ways- The following histograms

(relative frequency on y-axis) illustrates a few different shapes-

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Skewed to the left Skewed to the right All jumbled up

The bell shaped or

normal curve

The normal curve is

symmetric- that is,

neither right-skewed

nor left skewed.


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Normal Curve and real life data

We look at how many real life data like weights of new born babies, heights of

men and women resemble the bell shaped normal curve-

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All the above diagrams show the results of fitting normal density curves to

real life data


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What is Normal Density?

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The density function of a normal distribution is given as:

where is the mean and is the variance of the normal distribution.

These are the parameters of the distribution.

To check that its is a pdf, if we integrate f(x) over the entire range of x we get a valueof 1

Normal distributions are symmetric around their mean.

The mean, median, and mode of a normal distribution are equal.

68% of the area of a normal distribution is within one standard deviation of the mean.

Approximately 95% of the area of a normal distribution is within two standard

deviations of the mean.


7/16Applied Statistics and Computing Lab

Normal Distribution(S) Though normal distribution refers to bell shaped curves, the mean and variance

of the normal distribution will, in general, differ from one normal distribution toanother resulting in different shapes of the bell- the mean and variance are thus

the parameters of the normal distribution

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In the diagram on the

left, all the shapes are

that for bell shapednormal curves, but note

how the shapes differ

with different means

and variances



Standard Normal Distribution: Need for

standardization How to compare normal distributions with different and ?

We define the standard normal variable Z= (X- )/ , where and are respectively the mean andstandard deviation of the normal variable X

Z follows normal distribution with mean= 0 and standard deviation=1

Why Standardize? By standardizing a normally distributed variable, we can find the area under its normal curve using a

table. This is because the percentage of observations of the original normally distributed variablethat lie between a and b is the same as the percentage of observations of the standard normalvariable, z, that lie between (a)/ and (b)/

Also, it facilitates comparison and helps you make decision about your data. Eg: Prof Snape has giventhe following marks in an exam ( out of 60, 30 is the qualifying marks)- 20, 15, 26, 32, 18, 28, 35, 14,26, 22, 17

So, only one student has passed!

The mean marks= 23 and the standard deviation= 6.6. Prof. Snape decides to set a new qualifyingmarks- only those students who would score less than 1 standard deviation from the mean will notqualify.

These are the standard scores: -0.45, -1.21, 0.45, 1.36, -0.76, 0.76, 1.82, -1.36, 0.45, -0.15, -0.91. So,now only two students fail

This is the importance of standardization

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Reading the standard normal table

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There is the standard normal table available which shows

the area of the normal curve to the left of the standard

normal variable

A snapshot of the table:

Source: Wikipedia

For eg, from the table what is

P(z



Properties of Standard Normal Distribution

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Most useful properties of the normal distribution are based on the symmetry

property of normal distribution.

P(z=a)= 1-p(z



An example: A survey indicates that for each trip to the supermarket, a shopper spends an

average =45 minutes with a standard deviation of =12 minutes. The length oftime spent in the store is normally distributed and is represented by the variable x.

A shopper enters the store. (a) Find the probability that the shopper will be in the

store for each interval of time listed below. (b) If 200 shoppers enter the store,

how many shoppers would you expect to be in the store for each interval of time

listed below?

1) Between 24 and 54 minutes 2) More than 39 minutes

Solution: The graph at the left shows a normal curve with =45 minutes and =12

minutes. The area forxbetween 24 and 54 minutes is shaded .

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a) The z-scores corresponding to x=24 and x=54 are:

Z1= (24-45)/12= -1.75, Z2 = (54-45)/12= .75

So, the probability that a shopper will be in the storebetween 24 and 54 minutes is

P(-1.75



Solution Continued

b) Another way of interpreting this probability is to say that 73.33% of shoppers willbe in the store between 24 and 54 minutes after entering. So if 200 shoppers

enter the stop, we expect (200*.7333)=146.66 or 147 shoppers to stay between 24

and 54 minutes.

The graph below shows the normal curve with =45 minutes and =12 minutes

and the area greater than 39 minutes is shaded-

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The z-score corresponding to 39 mins is Z= (39-45)/12= -.5

P(Z> (-.5))= 1- P(Z



Example: Given probability, finding the Z

ordinate

The amount of fuel consumed by the engines of a jetliner on a flightbetween two cities is a normally distributed random variableX withmean of 5.7 tons and standarddeviation of 0.5. Carrying too much fuel isinefficient as it slows the plane. If, however, too little fuel is loaded on theplane, an emergency landing may be necessary. The airline would like todetermine the amount of fuel to load so that there will be a 0.99

probability that the plane will arrive at its destination. Solution: We first find the value of Z such that P(Z



Snapshot of Standard normal Table

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Further Applications

If the weekly wage of 20,00 workers in a factory follow normal distribution

with mean 5,000 and standard deviation 500 respectively, find the expectednumber of workers whose weekly wages are a) between Rs 4000-4500

b) Less than Rs 4,000 c) More than Rs 5,000

The marks obtained by a group of students for Mathematics are assumed tobe normally distributed with mean 60 and standard deviation 8. If 5 studentsare taken at random from this set, what is the probability that exactly one of

them will have marks above 70?( Hint: First find the probability that the marks is above 70 by using normaldistribution. Then letting Y denote the number of students who have marksabove 70 out of the 5 students, find the binomial probability for Y taking thevalue 1)

Normal Distribution has many applications in business:

For eg, modern portfolio theory assumes that the return of a diversified assetportfolio follows a normal distribution

In HR management, the performance of employees is often assumed to benormally distributed

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(13) normal distribution

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