Application of Statistical Techniques in Industry

7/29/2019 Application of Statistical Techniques in Industry

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Akhilesh Menon

MMS Batch-2,Roll No.90

Application of Statistical Techniques in Industry

1:Chi-Square Test:

A chi-squared test, also referred to as chi-square test or test, is any

statistical hypothesis test in which the sampling distribution of the test

statistic is a chi-squared distribution when the null hypothesis is true, or

any in which this is asymptotically true, meaning that the sampling

distribution (if the null hypothesis is true) can be made to approximate

a chi-squared distribution as closely as desired by making the sample

size large enough. Its properties were first investigated by Karl Pearson

in 1900. In contexts where it is important to make a distinction

between the test statistic and its distribution, names similar to Pearson

X-squared test or statistic are used. It tests a null hypothesis stating

that the frequency distribution of certain events observed in a sample

is consistent with a particular theoretical distribution. The events

considered must be mutually exclusive and have total probability 1. A

common case for this is where the events each cover an outcome of a

categorical variable. A simple example is the hypothesis that an

ordinary six-sided die is "fair", i. e., all six outcomes are equally likely to

occur

Pearson's chi-squared test is used to assess two types of comparison:

tests of goodness of fit and tests of independence.


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A test of goodness of fit establishes whether or not an observedfrequency distribution differs from a theoretical distribution.

A test of independence assesses whether paired observations ontwo variables, expressed in a contingency table, are independentof each other (e.g. polling responses from people of different

nationalities to see if one's nationality affects the response).

Calculating the test-statistic:

The value of the test-statistic is

where

= Pearson's cumulative test statistic, which asymptotically

approaches a distribution.

= an observed frequency;

= an expected (theoretical) frequency, asserted by the null

hypothesis;

n= the number of cells in the table.

Application of Chi-Square Test in Marketing:

A number of marketing problems involve decision situations in which it

is important for a marketing manager to know whether the pattern of

frequencies that are observed fit well with the expected ones. The
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appropriate test is the c test of goodness of fit. The illustration given

below will clarify the role of c in which only one categorical variable is

involved.

Example: In consumer marketing, a common problem that anymarketing manager faces is the selection of appropriate colors for

package design. Assume that a marketing manager wishes to compare

five different colors of package design. He is interested in knowing

which of the five is the most preferred one so that it can be introduced in

the market. A random sample of 400 consumers reveals the following:

Package Color

Preference by

Consumers

Red 70

Blue 106

Green 80

Pink 70

Orange 74

Total 400

Do the consumer preferences for package colors show any significant

difference?

Solution: If you look at the data, you may be tempted to infer that Blue

is the most preferred color. Statistically, you have to find out whether

this preference could have arisen due to chance. The appropriate test

statistic is the test of goodness of fit.

Null Hypothesis: All colors are equally preferred.

Alternative Hypothesis: They are not equally preferred


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Please note that under the null hypothesis of equal preference for all

colors being true, the expected frequencies for all the colors will be

equal to 80. Applying the formula

we get the computed value of chi-square () = 11.400

The critical value of at 5% level of significance for 4 degrees of

freedom is 9.488. So, the null hypothesis is rejected. The inference is

that all colors are not equally preferred by the consumers. In particular,

Blue is the most preferred one. The marketing manager can introduce

blue color package in the market.


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2 : Normal Distribution

In many natural processes, random variation conforms to a particular

probability distribution known as the normal distribution, which is the

most commonly observed probability distribution. The shape of thenormal distribution resembles that of a bell, so it sometimes is referred

to as the "bell curve".

We can quickly estimate the spread of the data given the mean and

standard deviation of a data set that follows the normal distribution.

For a normal distribution:

68% of the data will fall within 1 standard deviation of the mean 95% of the data will fall within 2 standard deviations of the mean Almost all (99.7%) of the data will fall within 3 standard deviations

of the mean

Characteristics

The bells curve has the following characteristics:

Symmetric Unimodal Extends to +/- infinity Area under the curve = 1

Applications of Normal Distribution

Operations Management

In the field of operations management, results of many processes fall

along the Normal Distribution Curve.


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Human Resources

The Normal Probability Distribution governs many aspects of human

performance. Human resource professionals often use the Normal

Distribution to describe employee performance.

Investment Portfolio

Modern portfolio theory commonly assumes that the returns of a

diversified asset portfolio follow a normal distribution.

Physics

Certain quantities in physics are distributed normally. Examples of suchquantities are:

Velocities of the molecules in the ideal gas. More generally,velocities of the particles in any system in thermodynamic

equilibrium will have normal distribution, due to the maximum

entropy principle.

Probability density function of a ground state in a quantumharmonic oscillator.

Marketing Research

Normal distribution is used in marketing research when conducting a

sample survey. The results obtained are supposed to be normally

distributed. This can be used to determine the success of new product

launch.

Finance

The normal distribution helps financial analysts and investors make


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better financial decisions based on the statistical information provided

by the normal distribution.

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Application of Statistical Techniques in Industry