Assignment Qt Col Mba Semester 1

Assignment No. 1

QUANTITATIVE TECHNIQUES (5564) Executive MBA/MPA (Col)

ZAHID NAZIR Roll.No. AB523655 Semester:Autumn 2008

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Question 1 a). Explain the use of Quantitative Techniques in Business and Management? Marks: 10 b). What are limitations of Statistics? Marks: 10

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a). QUATITATIVE TECHNIQUES Quantitative techniques refers to the group of statistical and operations research techniques. All these techniques require preliminary knowledge of certain topics in mathematics. Quantitative Techniques

Statistical Techniques Techniques

Operations Research

USE OF QUANTITATIVE TECHNIQUES IN BUSINESS AND MANAGMENT Due to increasing complexity in business and industry, decision making based on intuition has become highly questionable especially when the decision involves the choice among several courses of action each of which can achieve several management actions. So there is need for training the people who can manage a system efficiently and creatively. Quantitative Techniques now have a major role in effective decision making in various functional areas of management i.e. marketing, finance, production and personnel. These techniques are also widely used in planning, transportation, public health, communication, military, agriculture etc. Quantitative techniques are also used extensively as an aid in business decision making. Some of the areas where quantitative techniques can be used are:

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MANAGEMENT i).

Marketing • Analysis of marketing research information • Statistical records for building and maintaining an extensive market • Sales forecasting

ii). • • • •

Production Production planning, control and analysis Evaluation of machine performance Quality control requirement (to analyze the data/trends) Inventory control measures

• • • • •

Finance, Accounting and Investment Financial forecast, budget preparation Financial investment decisions Selection of securities Auditing function Credit policies, credit risk and delinquent accounts

• • • •

Personnel Labour turn over rate Employment trends Performance appraisal Wage rates and incentive plans

iii).

iv).

ECONOMICS • Measurement of gross national product and input-output analysis • Determination of business cycle, long term growth and seasonal fluctuations • Comparison of market prices, cost and profits of individual firms • Analysis of population, land economies and economic geography Page 4 of 29

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• Operational studies of public utilities • Formulation of appropriate economic policies and evaluation of their effect RESEARCH & DEVELOPMENT • Development of new product lines • Optimal use of resources • Evaluation of existing products NATURAL SCIENCE • Diagnosis of disease based on data like temperature, pulse rate, blood pressure etc. • Judging the efficiency of a particular drug for curing a certain disease • Study of plant life In the competitive and dynamic business world, firms/companies who likes to succeed and survive are those which are capable of maximizing the use of tools of management like quantitative techniques.

b) LIMITATIONS OF STATISTICS Statistics with all its wide application in every sphere of human activity has its own limitations. Some of them are given below. 1. STATISTICS IS NOT SUITABLE TO THE STUDY OF QUALITATIVE PHENOMENON: Since statistics is basically a science and deals with a set of numerical data, it is applicable to the study of only these subjects of enquiry, which can be expressed in terms of quantitative measurements. As a matter of fact, qualitative phenomenon like honesty, poverty, beauty, intelligence etc, cannot be expressed numerically and any Page 5 of 29

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statistical analysis cannot be directly applied on these qualitative phenomenons. Nevertheless, statistical techniques may be applied indirectly by first reducing the qualitative expressions to accurate quantitative terms. For example, the intelligence of a group of students can be studied on the basis of their marks in a particular examination. 2. STATISTICS DOES NOT STUDY INDIVIDUALS: Statistics does not give any specific importance to the individual items; in fact it deals with an aggregate of objects. Individual items, when they are taken individually do not constitute any statistical data and do not serve any purpose for any statistical enquiry. 3. STATISTICAL LAWS ARE NOT EXACT: It is well known that mathematical and physical sciences are exact. But statistical laws are not exact and statistical laws are only approximations. Statistical conclusions are not universally true. They are true only on an average. 4. STATISTICS TABLE MAY BE MISUSED: Statistics must be used only by experts; otherwise, statistical methods are the most dangerous tools on the hands of the inexpert. The use of statistical tools by the inexperienced and untraced persons might lead to wrong conclusions. Statistics can be easily misused by quoting wrong figures of data. 5. STATISTICS IS ONLY, ONE OF THE METHODS OF STUDYING A PROBLEM: Statistical method do not provide complete solution of the problems because problems are to be studied taking the background of the countries culture, philosophy or religion into consideration. Thus the statistical study should be supplemented by other evidences. References:

www.textbooksonline.tn.nic.in/

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Question 2 a). Different types of functions are introduced and used in CACULUS, briefly explain them? Marks: 10 b). Graph and find domain of 2x2

if x < 0

3x + 1

if x > 0

f(x) =

Marks: 10

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a).

TYPES OF FUNCTION Different types of functions that are introduced and used in calculus are:

i).

i).

Linear Function

ii).

Polynomial Function

iii).

Absolute Value Function

iv).

Inverse Function

v).

Step Function

vi).

Algebraic Function

LINEAR FUNCTION A linear function is one in which the power of independent variable is 1, it is also called single variable function. The general expression of linear function having only one independent variable is: y =

f(x) = a + bx

where a and b are given real numbers and x is an independent variable taking all numerical values in an interval. Single variable function can be linear and non-linear, for example y =

3 + 2x

(Linear single variable function)

y =

2 + 3x – 5x2 + x2

(non-linear single variable function)

and

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A linear function with one variable can always be graphed in a two dimensional plane. This graph can always be plotted by giving different values to x and calculating corresponding values of y. The graph of such functions is always a straight line. ii).

POLYNOMIAL FUNCTION Polynomial functions are functions with x as an input variable, made up of several terms, each term is made up of two factors, the first being a real number coefficient, and the second being x raised to some non-negative integer power. Polynomial functions are functions that have this form: y = f(x) = anxn + an-1xn-1 + ... + a1x1 + a0 The value of n must be a nonnegative integer. That is, it must be whole number; it is equal to zero or a positive integer. an, an-1, ..., a1, a0 are called coefficients. These are real numbers. The degree of the polynomial function is the highest value for n where an is not equal to 0. If n = 1, then the polynomial function is of degree 1 and is called a linear function and if n = 2 then the polynomial function is of degree 2 and is called quadratic function and usually written as: y =

iii).

ax2 + bx + c

ABSOLUTE VALUE FUNCTION The absolute value function is the real-valued function defined as follows. y = │x │ Where │x │ is known as magnitude or absolute value of x. By absolute value means that whether x is positive or negative, its

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absolute value will remain positive. For example │7 │= 7 and │-6 │ = 6. The graph of given function is like this:

iv).

INVERSE FUNCTION Take the function y = f(x). Then the value of y can be uniquely determined for given values of x as per the functional relationship. . Sometimes it is required to consider x as a of y, so that for given values of y, the values of x can be determined as per the functional relationship. This is called the inverse function and is also donated by x = f -1 (y). For example consider the linear function: y=

ax + b

Expressing this in terms of x, we get x =

y - b /a

=

y/a - b/a

=

cy – d

where c = 1/a, and d = - b/a This is also a linear function and is denoted by x = f -1 (y). v).

STEP FUNCTION For different values of an independent variable x in an interval, the dependant variable y = f(x) takes a constant value, but takes Page 10 of 29

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different values in different intervals. In such cases the given function y = f(x) is called a step function.

vi).

ALGEBRAIC & TRANSCENDENTAL FUNCTIONS Functions can also be classified with respect to the mathematical operations (addition, subtraction, multiplication, division, powers and roots) involved in the functional relationship between dependant variable and independent variables. When only finite number of terms are involved in a functional relationship and variables are affected only by the mathematical operations, then the function is called an algebraic function otherwise transcendental function. The following functions are algebraic functions of x. i).

y = 2x3 + 5x2 - 3x + 9

ii).

y = √x + 1/x2

iii).

Y = x3 - 1/ √x + 2

The subclasses of transcendental functions are: i).

Exponential Function

ii).

Logarithmic Function

b).

DOMAIN: Domain consists of all real numbers except zero. Now consider f (x) = 2x2

where x < 0

Supposed the values of x are -1, -3, -5, then

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f (-1) = 2(-1)2 = 2 f(-3) = 2(-3)2 = 18 f(-5) = 2(-5)2 = 50 Its graph is as follows;

60

f (x) = 2x2

50 40

Y-Axis

30 20 10 0 -6

Now consider

-5

-4 -3 X-Axis

-2

f (x) = 3x + 1 if x > 0

Suppose value of x are 1, 3, 5, so f (1) =

3(1) + 1

=

4

f (3) =

3(3) + 1

=

10

f (5) =

3(5) + 1

=

16

Its graph will be as follows;

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-1

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18 16 14 12 10 8 6 4 2 0

Y-Axis

f (x) = 3x + 1

6

5

4 X-Axis

3

Reference: http://www.themathpage.com http://oregonstate.edu Quantitative Techniques (AIOU)

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2

1

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Question 3 a). Given the following input-output table, calculate the gross output so as to meet the final demand of 200 units of Agriculture and 800 units of industry.The following data relates to the sales of 100 companies is given below: Consumer Sector Producer Sector Agriculture Industry

Agriculture

Industry

Final Demand

Total Output

300 400

600 1200

100 400

1000 2000

Marks: 10 b). Discriminate between the census and sampling methods of data collection and compare their merits and demerits. Why is the sampling method unavoidable in certain situations? Marks: 10

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Solution:

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b).

CENSUS Census is a complete enumeration of an entire population of statistical units in a field of interest. It is also called complete enumeration survey. For example, population census canvases every household in a country to count for the number of permanent residents and other characteristics; census of manufacturing canvases all establishments engaging in manufacturing activities. Data from the census serve as base-year or benchmark data. Requirement: A complete and up-to-date register of all statistical units in the field of inquiry is required. Advantages: Census provides the most reliable statistics if done professionally and with integrity. Disadvantages: Very costly to enumerate and to process data. Timeliness is low: data is available for use only many months, even years after. Census is normally carried every five or ten years.

SAMPLING METHOD When the investigator studies only a representative part of the total population and makes inferences about the population on the basis of that study. It is known as sampling method or Survey. In both methods, the investigator is interested in studying some characteristics of the population. Advantages: Provide more up-to-date statistics, which are reliable if scientifically designed and professionally implemented, less costly than census. Sampling errors can also Page 16 of 29

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be obtained. Surveys are normally carried out weekly, monthly, quarterly or annually. Disadvantages: Timeliness requires prompt data processing, thus less information may be asked. SITUATIONS IN WHICH SAMPLING METHOD IS UNAVOIDABLE Sampling method is unavoidable in following situations • • • •

Unlimited population Distractive population nature Unapproachable population e.g. Mobilink users In quality control, such as finding the tensile strength of a steel specimen by stretching it till it breaks. Another example is in process checking in the manufacturing of pharmaceuticals where it is not possible to check the each and every tablet or injection. Secondly quality testing results in destruction of items itself.

*******************************

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Question 4 a). The following data relates to the sales of 100 companies is given below: Sales (Lakhs) 5-10 10-15 15-20 20-25

No. of Complaints 5 12 13 20

Sales (Lakhs) 25-30 30-35 35-40 40-45

No. of Complaints 18 15 10 7

Draw less than and more than ogives. Determine the number of companies whose sales are (i) less than Rs.13 Lakhs (ii) more than Rs.36 Lakhs and (iii) between Rs.13 lakhs and Rs.36 lakhs. Marks: 10 b). Briefly explain the following important concepts: i). Continuous Data ii). Discrete Data iii). Frequency Distribution iv). Qualitative Data v). Quantitative Data Marks: 10

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a).

OGIVE “An ogive is a specialized line graph which shows how many items there are which are below a certain value.” The horizontal axis shows the upper class boundaries marked on the scale, just like a histogram. The vertical axis shows the cumulative frequency, which is just a fancy name for a running total. Some line graphs are actually ogives. First we calculate the cumulative frequency. We start at 5. Next line: 5 + 12 = 17 Next line: 17 + 13 = 30 Next line: 30 + 20 = 50 and so on. Sales No. of X Companies (Lakhs) (f) 5-10 5 Less than 10 10-15 12 Less than 15 15-20 13 Less than 20 20-25 20 Less than 25 25-30 18 Less than 30 30-35 15 Less than 35 35-40 10 Less than 40 40-45 7 Less than 45

Cumulative Frequency 5 17 30 50 68 83 93 100

Decumulative Frequency More than 5 More than 10 More than 15 More than 20 More than 25 More than 30 More than 35 More than 40

100 95 83 70 50 32 17 7

i).

Less than Rs. 13 Lakhs

=

12 companies

(Graph - I)

ii).

More than Rs. 36 Lakhs =

15 companies

(Graph - II)

iii).

B/w 13 and 36 Lakhs

73 companies

(Graph - I)

=

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b). i).

CONTINUOUS DATA Continuous data is information that can be measured on a continuum or scale. Continuous data can have almost any numeric value and can be meaningfully subdivided into finer and finer increments, depending upon the precision of the measurement system. In contrast to discrete data like good or bad, off or on, etc., continuous data can be recorded at many different points (length, size, width, time, temperature, cost, etc.). Let's say you are measuring the size of a marble. To be within specification, the marble must be at least 25mm but no bigger than 27mm. If you measure and simply count the number of marbles that are out of spec (good v/s bad) you are collecting attribute data. However, if you are actually measuring each marble and recording the size (i.e. 25.2mm, 26.1mm, 27.5mm, etc) that's continuous data, and you actually get more information about what you're measuring from continuous data than from attribute data. Data can be continuous in the geometry or continuous in the range of values. The range of values for a particular data item has a minimum and a maximum value. Continuous data can be any value in between.

ii).

DISCRETE DATA Discrete data is information that can be categorized into a classification. Discrete data is based on counts. Only a finite number of values is possible, and the values cannot be subdivided meaningfully. For example, the number of parts damaged in shipment. Example . A 5 question quiz is given in a Math class. The number of correct answers on a student's quiz is an example of discrete data. The number of correct answers would have to be one of the Page 22 of 29

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following: 0, 1, 2, 3, 4, or 5. There are not an infinite number of values, therefore this data is discrete. Also, if we were to draw a number line and place each possible value on it, we would see a space between each pair of values. iii).

FREQUENCY DISTRIBUTION Frequency distribution is a way of summarizing a set of data. It is a record of how often each value (or set of values) of the variable in question occurs. It may be enhanced by the addition of percentages that fall into each category. A frequency table is used to summarize categorical, nominal, and ordinal data. It may also be used to summarize continuous data once the data set has been divided up into sensible groups.

Example Suppose that in thirty shots at a target, a marksman makes the following scores: 522344320303215 131552400454455 The frequencies of the different scores can be summarized as:

Score 0 1 2 3 4 5

Frequency 4 3 5 5 6 7

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Frequency (%) 13% 10% 17% 17% 20% 23%

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iv).

QUALITATIVE DATA Qualitative data is extremely varied in nature. It includes virtually any information that can be captured that is not numerical in nature. Qualitative data are generally (but not always) of less value to scientific research than quantitative data, due to their subjective and intangible nature. It is possible to approximate quantitative data from qualitative data Qualitative methods are ways of collecting data which are concerned with describing meaning, rather than with drawing statistical inferences. What qualitative methods (e.g. case studies and interviews) lose on reliability they gain in terms of validity. They provide a more in depth and rich description.

v).

QUATITATIVE DATA Information that can be counted or expressed numerically. This type of data is often collected in experiments, manipulated and statistically analyzed. Quantitative data can be represented visually in graphs and charts. Quantitative methods are those which focus on numbers and frequencies rather than on meaning and experience. Quantitative methods (e.g. experiments, questionnaires and psychometric tests) provide information which is easy to analyze statistically and fairly reliable. Quantitative methods are associated with the scientific and experimental approach and are criticized for not providing an in depth description.

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Reference: http://www.stats.gla.ac.uk/ en.wikipedia.org www.isixsigma.com

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Question 5 a). What are the Quantiles? Explain and illustrate the concepts of Quartiles, deciles and Percentiles? Marks: 10 b). The geometric mean of 10 observations on a certain variable was calculated to be 16.2. It was later discovered that one of the observations was wrongly recorded as 10.9, when in fact it was 21.9. Apply appropriate correction and calculate the correct geometric mean. Marks: 10

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a).

QUANTILE “Quantiles are points taken at regular intervals from the cumulative distribution function (CDF) of a random variable” For example quantiles divide distribution into four parts.

QUARTLE “A quartile is any of the three values which divide the sorted data set into four equal parts, so that each part represents one fourth of the sampled population.” Each quartile contains 25% of the total observations. Generally, the data is ordered from smallest to largest with those observations falling below 25% of all the data analyzed allocated within the 1st quartile, observations falling between 25.1% and 50% and allocated in the 2nd quartile, then the observations falling between 51% and 75% allocated in the 3rd quartile, and finally the remaining observations allocated in the 4th quartile. Its formula is: Quartiles

=

l + h(2n/4 - pcf) f

where

l h f n pcf

= lower class boundary of specific class = class interval = frequency of specific class = total frequecy i.e. ∑ f = n = cumulative frequency of preceding class

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DECILES Any one of the numbers or values in a series dividing the distribution of the individuals in the series into ten groups of equal frequency. Since nine points divide the distribution into ten equal parts, we shall have nine deciles denoted by D1, D2, ………. D9. The formula of decile is: Decile (DL) =

l + h(2n/10 - pcf) f

where

l h f n pcf

= lower class boundary of specific class = class interval = frequency of specific class = total frequecy i.e. ∑ f = n = cumulative frequency of preceding class

PERCENTILES Those values which divide the total data into hundred equal parts are called percentiles. Since 99 points divide the distribution into hundred equal parts, we shall have 99 percentiles denoted by P1,P2, ………… P99. Its formula is: Percentile (Pi)

=

l + h(2n/100 - pcf) f

where

l h f n pcf

= lower class boundary of specific class = class interval = frequency of specific class = total frequecy i.e. ∑ f = n = cumulative frequency of preceding class Page 28 of 29

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b).

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