Section I

Introduction

Our business world is uncertain. It is numerically oriented, dynamic, and risky and is affected by multiplicity of causes. Here while we take some decision, we undertake some risks.

Since success in business relies on making correct decisions in proper time and executing them properly, it is vital for any business to make sure that the decisions it takes avoids unnecessary risks and at the same time provides proper return for the risks it has undertaken. In addition, it has to ensure that the decisions taken are timely and relevant for the purpose. Finally, the business has to make sure that the decisions taken are executed properly so that its goals are achieved properly. Failure in any of these matters may prove crucial for the business.

For dealing with these matters, a business needs proper information in time, make accurate & timely decisions, measure and mange risks, and confirms the effective application of decisions. For these, our business world heavily depends on statistics as statistics provides qualitative and quantitative information for a specific purpose, process, analyze and interpret collected information, provides important features of the data and thus aids in decision making.

Normal distribution is one of the very important tools used in statistics. It helps to determine certain characteristics of the data and also provides as a base for using other certain statistical tools for decision making. For these reason, here we study Normal distribution and its application in business in our report.

The rationale of the study

For the following reasons, we choose Normal distribution as our study-

1. In real world, many of the variables studied tend to be normally distributed. Variables that are not normally distributed can be brought into Normal distribution by simple transformation of the variable. It also helps to study many discrete variables, as the sample size gets larger.
2. It helps quality control, cost management, and business operations by helping to determine the most sensitive part of the variable.
3. Sampling, test of hypothesis and other important statistical tools applied for decision-making are based on the assumption that samples have been drawn from a normally distributed population. Thus, Normal distribution serves as the base of other statistical tools

Objectives

The objectives of this report is to-

• Identify Normal distribution and normal curve
• Determine the several characteristics of Normal distribution
• Use of Normal distribution in decision making
• Show how it helps business
• Test the use of Normal distribution through a business application

Methodology

Data collection: For preparing this report, we have collected data from mainly form secondary sources. And the sources were our textbooks, several study materials, internet and other sources.

Variables covered: since Normal distribution is a form of continuous probability distribution, the variables we used for our study are assumed to be continuous variable.

Techniques applied: the techniques we applied to prepare this report are-

• The normal curve table
• Graphical methods

Section II: Normal Distribution

Normal Distribution & Normal Curve

Normal distribution is a distribution of a continuous random variable with a single- peaked, bell- shaped curve. Normal distribution is a useful continuous probability distribution. The normal curve is a theoretical mathematical curve. Normal curve is used for normal distribution. For practical purposes, normal curve should be converted into a standard normal curve and a given variable needs to be converted into a standard normal variate.

The following formula is used for Normal distribution-

i. y = the computed height of an ordinate at a distance of X from the mean.

ii. s = Standard deviation of the given normal distribution.

iii. p = constant = 3.1416

iv. e = constant = 2.7183

v. m = Mean of the random variable X.

History of Normal distribution:

The normal distribution was discovered by De Moivre as the limiting case of Binomial model in 1733. Through a historical error it has been credited to Gauss who first made reference to it in 1809. Through the 18^th and 19^th centuries, various efforts were made to establish the normal model as the underlying law ruling all continuous random variables—thus the name Normal. These efforts failed because of the false premises. The normal model has, nevertheless, become the most important probability model in statistical analysis. It is also known as Gaussian distribution.

Properties Of The Normal Curve:

The normal curve has the following properties-

• The curve has a single peak. It has the bell shape.

• The mean of a normally distributed population lies at the center of its normal curve.
• For a normal curve, the mean, median and mode are the same value.
• The two tails of the normal probability distribution extend indefinitely and never touch the horizontal axis.
• Since there is only one maximum point, therefore, the normal curve is unimodal,
  • The points inflection occur at

  

x = m ± 6 , y = f(x)

• As distinguished from binomial and Poisson distributions where variable is discrete, the variable distributed according to the normal curve is a continuous one.
• First and third quartiles are equidistant from the median.
• Linear combination of independent normal varieties is also a normal variate.
• Mean deviation about mean is =4/5 or, more precisely, 0.7979 times of standard deviation.
• All odd moments of the normal distribution are zero.

• Since b1=0 the normal distribution is perfectly symmetrical and b2=3 implies that normal curve is neither leptokurtic nor platykurtic.
• Mean ± 6, Mean ±26, and mean ±36 covers 68.27%, 95.45% and 99.73% area respectively.

Importance Of Normal Distribution In Business

The normal distribution has great significance in statistical work because of the following reason:

• The normal distribution has the remarkable property stated in the so-called central limit theorem.
• Even if a variable is not normally distributed, it can sometimes be brought to normal form by simple transformation of variable.
• Many of the sampling distributions like student’s t, Snadeeos’s F, etc. Also tend to normal distribution.
• The sampling theory and tests of significance are based upon the assumption that samples have been drawn from a normal population with mean and variance.
• Normal distribution finds large applications in Statistical Quality Control.
• As n becomes large, the normal distribution serves as a good approximation for many discrete distributions (such as Binomial, Poisson, etc.).
• In theoretical statistics many problems can be solved only under the assumption of a normal population. In applied work we often find that methods developed under the normal probability law yield satisfactory results, even when the assumption of a normal population is not fully met, despite the fact that the problem can have a formal solution only if such a premise is hypothesized.
• The normal distribution has numerous mathematical properties which make it popular and comparatively easy to manipulate.

The Standard Normal Curve & Normal Curve Table

The Standard Normal Curve: There may be numerous normal curves for different variables. So, to use normal distribution, we have to convert a normal distribution into standard normal curve.

The green line is the standard normal curve

The Standard Deviation And Normal Curve

In a normal curve, an exact percentage of observations in the distribution fall within ranges established by the standard deviation in conjunction with the mean. The mean plus 1 standard deviation and minus 1 standard deviation, i.e., µ+ 1 covers 68.27% of the observations in the distribution and 34.135% of observation will fall on either side of mean. Similarly, µ+ 2covers 95.45% of observations and µ+ 3covers 99.73% of observations.

It can be converted in a standard normal distribution where standardized normal

variate = z

Normal Curve Table:

The table shows the area of the normal curve between mean ordinate and ordinates at various sigma distances from the mean as percentage of the total area

AREA RELATIONSHIP

Using Normal Curve For Decision Making

Normally distributed random variable x with parameter m and s can be transformed to the standardized normally distributed random variate z, therefore the table heading of area under the normal curve may be used.

a) Area between mean and some point lying above it.

b) Area between mean and some point lying below it.

c) Area to the left of a value above the mean.

d) Area to the right of a value below the mean.

(e) Area under portion overlapping the mean

i) Area between two values lying below the mean.

Relation between Binomial, Poisson and Normal Distribution

The Binomial, Poisson and Normal distributions are very closely related to each other. The relations are explained below:

There is a relation between Poisson and normal distribution. When n is large and the probability p of occurrence of an event is close to zero so that np remains a finite constant, then the binomial distribution tends to Poisson distribution.

Similarly, there is a relation between binomial and normal distributions. Normal distribution is a limiting form of binomial distribution under the following conditions:

1. n, the number of trials is very large, i.e., nà ∞; and
2. Neither p nor q is very small.

In fact it can be proved that the binomial distribution approaches a normal distribution with standardized variable, i.e.,

Or  will follow the normal distribution with mean zero and variance one.

Similarly, Poisson distribution  also approaches a normal distribution with standardized variable, i.e.

In other words,  will follow the normal distribution with mean zero and variance one.

Business Application

Example 1

The salary of 10,000 workers in Renata Ltd was approximately normally distributed with mean salary Tk. 12,000 and standard deviation salary Tk. 3,000.

1. Find the number of workers receiving salary Tk 12,000 and Tk 18,000.
2. Find the number of workers receiving salary between Tk. 7,000 and Tk. 10,000.
3. Find the number of workers receiving salary of Tk. 16,000 or less
4. Find the number of workers receiving salary of Tk. 8,500 and more
5. Find the number of workers receiving salary between tk.7,000 and Tk. 14,000.
6. Find the % of workers receiving salary of Tk. 20,000 or more.
7. Find the % of workers receiving salary of Tk. 5,000 or less
8. Find the % of workers receiving salary between Tk. 13,500 and 17,250.

Solution:

Given that,

N= total number of workers = 10,000

= mean salary = 12,000

=standard deviation of salary = 3,000

And standard normal variate-

Here, the corresponding area (CA) under normal curve = 0.4772

So, number of workers = N 0.4772

= 10000 0.4772

= 4772

Here, the corresponding area (CA) under normal curve = 0.4525

So, number of workers = N 0.4525

= 10000 0.4525

= 4525

Here, the corresponding area (CA)  between mean and z-value under normal curve = 0.4080

So, the desired area = 0.50 + 0.4080 = 0.9080

So, number of workers = N 0.9080

= 10000 0.9080

= 9080

Here, the corresponding area (CA) under normal curve = 0.2967

So, desired area = 0.4525 + 0.2967 = 0.7492

So, number of workers = N 0.7492

= 10000 0.7492

= 7492

Here, the corresponding area (CA) under normal curve = 0.4901

So, desired area = 0.5-0.4901 = 0.0099

So, the percentage of workers = 100 x 0.0099

= 0.99%

Here, the corresponding area (CA₁) under normal curve = 0.1915

= 1.75

Here, the corresponding area (CA₁) under normal curve = 0.4599

So, desired area = 0.4599 – 0.1915 = 0.2684

So, the percentage of workers = 100 x 0.2684

= 26.84%

Example 2

The world famous cell phone company NOKIA wishes to set a minimum hours guarantee on its new model NOK-333 a high quality cell phone. Tests reveal the mean hours are 67,900 with a standard deviation of 2,050 hours and that the distribution of hours follows the normal distribution. They want to set the minimum guaranteed hours so that no more than 4% of the cell phones will have to be replaced. What minimum guaranteed hours should NOKIA announce?

Solution:

Given that,

Here there are two unknown values, z and x. to find x, we first find z, and then solve for x. we know, the area under the normal curve to the left of is 0.50 and the area between x and  is 0.4600 ( 0.50 – 0.04).

From the table value of z, we find that z = -1.75 when CA = 0.4600 (since the value is to the left of the mean, z-value should be negative)

Now,

 Our Over View

The normal distribution is a convenient model of quantitative phenomena in the natural and behavioral sciences. A variety of psychological test scores and physical phenomena like photon counts have been found to approximately follow a normal distribution. While the underlying causes of these phenomena are often unknown, the use of the normal distribution can be theoretically justified in situations where many small effects are added together into a score or variable that can be observed. The normal distribution also arises in many areas of statistics: for example, the sampling distribution of the mean is approximately normal, even if the distribution of the population the sample is taken from is not normal. In addition, the normal distribution maximizes information entropy among all distributions with known mean and variance, which makes it the natural choice of underlying distribution for data summarized in terms of sample mean and variance. The normal distribution is the most widely used family of distributions in statistics and many statistical tests are based on the assumption of normality. In probability theory, normal distributions arise as the limiting distributions of several continuous and discrete families of distributions.

Bibliography

1. Gupta and Gupta, “Business Statistics”, 11^th revised Edition.
2. Lind, Marchal and Wathen, “Statistical Techniques in Business & Economics”.