Quantitative Technics

Question # 01 (a)

Explain the use of Quantitative Techniques in Business and Management? Answer: Executives at all levels in business and industry come across the problem of making decisions at every stage in their day to day activities. Quantitative techniques provide the executive with scientific basis for decision making and enhance their abilities to make long range plans and to solve every day problems of running a business and industry with greater efficiency & confidence. Some of the areas where statistics can be used are as follows:

1.Management: I)

Marketing • •

Analysis of marketing research Statistical records for building and maintaining an extensive market

•

Sales forecasting

Production

II) •

Production Planning, control and analysis

•

Evaluation of machine performance

•

Quality Control requirements

•

Inventory Control measures

III) Finance, Accounting & Investment •

Financial forecasting, budget preparation

•

Financial investment decisions

•

Selection of securities

•

Audit Planning & functions

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•

Credit Policies, credit risk and delinquent accounts

IV) Personnel •

Labor turn over rate

•

Employment trends

•

Performance appraisal

•

Wage rate and incentive plans

2.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 economics and economic geography

•

Operational studies of public Utilities

•

Formulation of appropriate economic policies and evaluation of their effects

3.Research & Development •

Development of new products

•

Optimal use of resources

•

Evaluations of existing products

4.Natural Science

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•

Diagnosing the disease based on data like temperature, pulse rate, blood pressure etc.

•

Judging the efficacy of a particular drug for curing a certain disease

•

Study of Plant Life

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Question # 02 (a)

Different types of functions are introduced and used in CALCULUS, briefly explain them. Answer: There are some different types of functions are introduced which are particularly useful in CALCULUS. 1.)

Linear Functions: A linear function is one in which the power of independent variable is 1, the general expression of linear function having only one independent variable is:

Where a and b are given real numbers and x is an independent variable taking all numerical values in an interval. A function with only one independent variable is also called single variable function. Further , a single-variable function can be linear and non-linear, for example

(non-linear single variable function)

A linear function with one variable can always be graphed in a two dimensional plane (or space). 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.

2.) Polynomial Functions: A function of the from

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+

+……+

Where a1 ‘s(i=1,2,…….,n+ 1) are real numbers, integer is called a

+

0 and n is a positive

Polynomial of degree n

a) If n=1, then the polynomial function is of degree 1 and is called a linear function. That is, for n=1, function (2-4) can be written as:

y = a1 x2 + a2 x1 + a3 (a1 This is usually written as

y=a+bx(

3. Absolute Value Functions The functional relationship expressed by

y=

is known as an absolute value

function where |x| is known as magnitude (or absolute value) of x. By absolute value we mean that whether x is positive or negative, its absolute value remains positive. For example |7|=7 and |-6|=6 M. Aslam Javed, Roll # AB-523810, 1st Semester COL MBA

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Plotting of the graph of the function

y=

assigning various values of x and then

calculating the corresponding values of y, is shown in the following table x y

... 2 ... 2

3 3

-3 ... 3 ...

-2

-1

0

1

2

1

0

1

4. 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 function of y, so that for given values of y, the value of x can be uniquely determined as per the functional relationship. This is called the inverse function and is also denoted by x=f (y), for example consider the linear function y= ax + b expressing this in terms of x, we get

x=y–b

=

a where c=1/a and d=-b/a

This is also a linear function and is denoted by

(y)

5. Step Function For different values of an independent variables x in an interval, the dependent variable y= f(x) takes a constant value, but takes different values in different intervals. In such cases the given function y= f(x) is called a step function. For example

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y = f(x) =

6. Algebraic and Transcendental Functions Functions can also be classified with respect to the mathematical operations (addition, subtraction, multiplication, powers and roots) involved in the functional relationship between dependent variable and independent variable(s). When only finite numbers of terms are involved in a functional relationship and variables are effected only by the mathematical operations, than the function is called an Algebraic Function, otherwise transcendental function. The following functions are algebraic functions of x. i. ii. iii.

–

1

+2

The sub classes of transcendental functions are follows: a) Exponential Functions b) Logarithmic Functions

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Question # 3(a): Given the following input-output, calculate the gross output so as to meet the final demand of 200 units of Agriculture and 800 units of Industry. Consumer Producer Sector Agriculture Industry

Sector Agriculture 300 400

Industry 600 1200

Final

Total

Demand

Output

100 400

1000 2000

Solution:

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M. Aslam Javed, Roll # AB-523810, 1st Semester COL MBA

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Question # 03(b)

Discriminate between the census and sampling methods of data collection and there merits and demerits. Why is the sampling method unavoidable in certain situations? Answer: When secondary data is not available for the problem under study, a decision may be taken to collect primary data through original investigation. This original investigation may be obtained either by census method or sampling method. When the investigator collects data about each and every item in the population, it is known as the Census Method. But when the investigator studies only a representative part of the total population and makes M. Aslam Javed, Roll # AB-523810, 1st Semester COL MBA

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inferences about the population on the basis of that study, it is known as the Sampling Method. In both the situations, the investigator is interested in studying some characteristics of the population. The advantage of the census method is that information about every item in the population can be obtained. Also the information collected is more accurate. The main limitations of the census method are that it requires a great deal of money and time. Moreover, in certain practical situations of quality control, such as finding the tensile strength of a steel specimen by stretching it till it breaks is not even physically possible to check each and every item because quality testing result in the destruction of the item itself. In most cases, it is not necessary to study every unit of the population to draw some inference about it. If a sample is representative of the population then the study of the sample will yield correct information about the total population. It should be noted that out of the census and sampling methods, the sampling method is much more widely used in practice.

Question # 04 (a) The following data relates to the sales of 100 companies is given below. 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 & Rs: 36 Lakhs.

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Sales (Lakhs)

No. of Companies 5 12 13 20 18 15 10 7

5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45

============================================== ============================== Solution: Sales (Lakhs) 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45

No. of Companies (f) 5 12 13 20 18 15 10 7

Sales Less Less Less Less Less Less Less Less

than than than than than than than than

Cumulative Frequency 10 15 20 25 30 35 40 45

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5 17 30 50 68 83 93 100

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Question # 4(b) Briefly explain the following important concepts: • • • • •

Continuous Data Discrete Data Frequency Distribution Qualitative Data Quantitative Data

Answer:

Continuous Data When the data is described by a continuous variable it is called continuous data. Continuous data can take all values of the variable. For example, the data relating to weight, distance, and volume are examples of continuous data.

Discrete Date The term discrete data refers to quantitative data that is limited to certain numerical values of a variable. For example, the number of employees in an organization or the number of machines in a factory, are examples of discrete data.

Frequency Distribution When the data is arranged into groups or categories according to conveniently established divisions of the range of the observations, such an arrangement in tabular form is called frequency distribution. In a frequency distribution, raw data is represented by distinct groups which are known as

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classes. The number of observations that fall into each of the class is known as frequency. Thus a frequency distribution has two parts, on its left there are classes and on its right there are frequencies.

Qualitative Data The term qualitative is used to describe certain types of information. Qualitative data is described in terms of quality i.e. informal or relative characteristics such as warmth, and flavor. When the data is classified according to some attributes (distinct categories) which are not capable of measurement is known as qualitative classification.

Quantitative Data When the data is classified according to some characteristics that can be measured, it is called quantitative data. For example, the employees of a company may be classified according to their monthly salaries. Since quantitative data is characterized by different numerical values, the data represents the value of a variable. Quantitative data may be further classified into one or two types; discrete or continuous.

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Question # 05 (a) What are quantiles? Explain and illustrate the concepts of quartiles, deciles and percentiles. Answer: Quantiles: Quantiles are the related positional measures of central tendency. These are useful and frequency employed measures of non-central location. When the number of observations is quite large, the principle according to which a distribution or an ordered data set is divided into two equal parts, may be extended to any number of divisions. The three values which divide the distribution into four equal parts are called Quartiles. These values are denoted by Q1, Q2 and Q3 respectively. Q1 is called the first or lower quartile and Q3 is known as the third or upper quartile. Quartiles, deciles, percentiles and other values obtained by equal subdivision of the given set of data, are collectively called Qauntiles or sometimes Fractiles. The Quantiles should be calculated when the number of observations is quite large.

Deciles: Deciles are those values which divide the total data into ten equal parts. Since nine points divide the distribution into ten equal parts, we shall have nine deciels denoted D1, D2 ……….. D9.

Percentile: Percentiles are those values which divide the total data into hundred equal parts, Since ninety nine points divide the distribution into hundred equal parts, we shall have ninety nine percentiles denoted by P1, P2, P3………………. P99.

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