Statstics - QAM Management

7/31/2019 Statstics - QAM Management

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Quantitative Analysis for Management - I

Theory without practice is sterile; practice without theory isblind

.Lenin Management has at its disposal several approaches in interpreting, analyzing, and solving business problems. Generally, the complexity of the problem indicates the appropriate method of analysis.

The three main approaches are: Conventional Approach Observational Approach Systematic Approach

Conventional Approach: It follows past techniques and solutions. Being so static, offers little or nothing to the advancement of management (Since, it is inopposition to the dynamics of business).

Observational Approach: Is the method of watching and learning from othermanagers in similar situations. It, too, is poor but improvements can be applied on occasionto improve particular techniques.

Systematic Approach: Utilizes the concept of theoretical systems, which may be

somewhat different from the actual problem under study. Can be useful in obtaining a final solution, since, it utilizes acombination of approaches, in particular the scientific method. (Even though scientific management was aimed initially at

manufacturing activities its basic methodology can beapplied to most current and future business problems).

Management education comprises . becoming aware of the availablemanagement techniques knowing the reasons and understanding the process of using them. A technique consists of concepts and principles as

also a sequence of steps. Available research on the efficacy of such

concepts and techniques provides the justification as


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well as the problem conditions for appropriate use.

A detailed discussion of the steps involved in atechnique, followed by a demonstration of the technique (in a laboratory setting,

simulated environment and/or real-life situation) help build the skill of using the technique. Management education, thus offers the what, why and how of management and also,

quite often, provides opportunities for test-application of what has been taught. Koontz, O Donnel & Weihrich, discussing the art and science of management, say

in their book: The most productive art is always based on anunderstanding of the science underlying it. Science and art are not mutually exclusive, but are complementary. Executives who attempt to manage without theory, or

knowledge structured by it, are left to rely on what they did, or saw others do or just take a shot in the dark trusting to luck.

A theory or organized knowledge base enables managers to design sound and workable

solutions to managerial problems. Good management practice is always based on a soundunderstanding of the relevant theory available at the time.

True, there is as yet not science in which everything is known or all relationshipsproved. Science c ouldnt, therefore, be a comprehensivetool for the artist. This is true whether one is diagnosing

an illness, designing bridges, or managing a company. Management education (be it the MBA, the EMBA or any

otherprogramme) must, therefore, offer a blend of theory and practice.

The role of the practitioner in the process will have greater impact, when thelearner is given both theory and practice as complementary.(Management has to be practical, of course. There is nothing as practical as a good

theory ).


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Quantitative Analysis in Management - II

The word Business Statistics is generally used in two senses:

- Statistical Data; and

- Statistical Methods.

1. Statistical Data: refers to facts expressed in QuantitativeForm , e.g., Statistics of Production, Consumption, Unemployment toNational Income, Population, etc.

Note - The area from which statistical data are collected is generally, referred to as thePopula tion or Universe .

2. Statistical Methods: refers to all such tools and techniques, which are used in : a. Collection;

b. Organization (Summarizing);c. Presentation;d. Analysis; ande. Interpretation..of Statistical Data

2.1 Collection - Constitutes the foundation of statistical analysis. The data is collected frompublished or unpublished sources or else may be collected by the investigator himself. 2.2 Organization: he first step in organizing a group of data is editing and the next is to

classify them according to some common characteristics and the last step inorganization is tabulation i.e., to arrange the data in columns and rows.2.3 Presentation: After the data have been collected and organized, they are presented in orderlymanner to facilitate statistical analysis. The two different ways in which the

collected data are presented are : Statistical Tables; and Diagrams/Graphs.2.4 Analysis :Business data are analyzed using techniques such

as : Measures of central tendency, Measures of variation, Correlation and Regression Analysis , etc.

2.5 Interpretation: It deals with drawing conclusions from the data collected and analyzed. SinceStatistical Methods help in taking decisions, statistics may rightly be regarded as A body of methods

for making wise decisions in the face of uncertainty

orStatistics is a method of decision-making in the faceof uncertainty, on the basis of numerical data, and at

calculated risks .

A. Collection of Business Data: Secondary Data


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Primary Data Internal Records

B. Types of Classification: Geographical ..area-wise, (cities, districts, etc.) Chronological ..on the basis of time.

Qualitative (Categorical Data) .according to some attributes. Quantitative . in terms of magnitudes.

C. Frequency Distribution: Is a tabular summary of a set of data that shows

the frequency or number of data items that fall in each of several distinct classes. A frequency distribution is also known as a frequency table. Some times, however, we require information on the number of observations whose numerical

value is less than a given value this information is contained in the

cumulative frequency distribution (cfd).

D. Variable: A frequency distribution refers to data classified on the basis of some variable that can be

measured such as prices, wages, age, number of units produced or consumed. The term Variable refers to the characteristic that varies in amount or magnitude in

a frequency distribution. A variable may be either continuous ordiscrete (also called discontinuous).

(a) Parts of a Table: Table NumberTitle of the TableCaptionStubBody of the TableHead NoteFoot Note

(a) Types of a Table:1. Simple and Complex Tables2. General Purpose and Special Purpose Tables

(c) Types of Diagrams:1. One-dimensional diagrams, e.g., bar diagrams:

i. Simple bar diagrams ii. Sub-divided bar diagrams iii. Multiple bar diagrams iv. Percentage bar diagrams


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2. Two-dimensional diagrams:i. Rectanglesii. Squaresiii. Circles

3. Pictograms and Cartograms

(d) Types of Graphs:

1. Graphs of time series or Line graphs:i. Range graphii. Band Graph

2. Graphs of frequency distribution:i. Histogram or Column diagramii. Frequency polygoniii. Frequency Curvesiv. Cumulative Freque ncy Curves or OGIVE


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X

W

W

Central Tendency/Dispersion/Skewness/Kurtosis

It may be appreciated that all individual observations comprising a set of data exhibit a tendencyto cluster or centre around a specific value. On the whole, they tend to be closer to one particular value than others. This peculiarcharacteristic of data is referred to as Central Tendency. By its very nature, the value around which individual observations come to cluster, is called theCentral Value. Thus, a measure of central tendency of .a set of data lies in obtaining this centralvalue. This idea behind determining such a typical value is to use it as representative of the entire setof data.

Three principal measures of central tendency that are widely used in statistical analysis are:

(i) arithmetic mean;

(ii) median; and

(iii) mode,

the last two being Location Measures. Two other, relatively much less used measures are:

(i) Geometric Mean (GM); and

(ii) Harmonic Mean (HM).

Mean ( )It is calculated by taking the sum of all observations comprising a given set of data anddividing the sum by the total number of observations).

It is reliable in that it reflects all the values in the data set. It may be affected by extreme value (very high or low values) (Out Liers). Tedious to compute, because one do use every data point. We are unable to compute mean for a data set that has open-ended classes.

Weighted Mean In situations, where the numbers are not equally important, we can assign to each a weight thatis proportional to its relative importance and calculate the weighted mean:

(i.e it enables us to calculate an average that takes into account the importance of each value tothe overall total.)

Note: Failure to weight the values when one is combing data is a common error.

Where Wi' s are the weights.


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Median It is the middle value in an ordered array of a set of observations. Extreme values do not affect the median. Easy to understand and can be calculated from any kind of data even for grouped data withopen-ended classes.

The bad news is that you do give up some accuracy by choosing a single value to represent adistribution, e.g., 2, 4, 5, 40, 100, 213 and 347. The median is 40, which has no apparentrelationship to any of the other values in the distribution.

Mode (Mo): (most frequently occurring value in a data set) If a distribution has two mod es is called Bimodal Distribu tion. Mode is not unduly affected by extreme values.

(Median is not as highly influenced by the frequency of occurrence of a single value as is theMode, nor is it pulled by extreme values as is the mean).

Warning Before you do any calculating, take a common sense look at the data themselves. If the distribution looks unusual, just about any thing you calculate from it . Will haveshortcomings.

Otherwise, there are no universal guidelines for applying the mean, median, or mode as themeasure of central tendency for different populations.

Geometric Mean (GM): We use the Geometric Mean to Show multiplicative effects over time in compound interest andinflation calculations.

It is specifically useful when it is desired to produce an average of percentage changes. ( e.g.rate of growth of population / rate of growth of Industrial production / rate of increase insales/costs/prices etc. in one period over the other).

Hint: Use GM whenever you are calculating the average percentage change in some variableovertime.

Harmonic Mean (HM): Is particularly useful in averaging rates and ratios. It is the most appropriate average when we are calculating the average speed of a vehicle, whenthe speed is expressed in kilo meters per hour, etc

Also useful in which values of a variable are compared with a constant quantity of anothervariable i.e rates,time,distance covered within certain time and quantities purchased or sold per unit

,etc.


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Dispersion: To increase our understanding of the pattern of data, we must also measure its

dispersion its spread or variability. This additional information enables us to judge the reliability of our measure of the

central tendency. Avid choosing the distribution with the greatest dispersion :

For Example: Widely dispersed earnings those varying from extremely high to low or even negative

levels indicate a high risk to stockholders and creditors. A drug that is average in purity but ranges from very pure to highly impure may

endanger lives.

Different methods of dispersion are:

- Range,- Quartile Deviation,- Mean Deviation, and- Standard Deviation.

Two most important measures of Dispersion are: The Variance and the Standard Deviation:Both of these tell us an average distance of any observation in the data set from the mean of the

distribution. i.e how for individual items in a distribution depart from the mean of the distribution.

(A ) data set with a large standard deviation has much dispersion with values widely scatteredaround its mean, and a data set with a small standard deviation has little dispersion with the valuestightly clustered around its mean).

Note: The square root of a +ve numbers may be either +ve or ve, because a2 = (-a2). Whentaking the square root of the Variance to calculate Standard Deviation, however, statisticiansconsider only the +ve square root.

Relative Measures of Dispersion : ( Give us a feel for the magnitude of the deviationrelative to the magnitude of the mean).

Different methods of Relative measures of dispersion are : Coefficient of Range, Coefficient of Quartile Deviation, Coefficient of Mean Deviation, Coefficient of Standard Deviation, Coefficient of Variation (CV)

(Do not compare the dispersion in data sets by using their standard deviations unless their

means are close to each other)


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************************************* Sample Mean:

Sample Standard Deviation : sSample Variance : s2

*************************************Population Mean :

Population Standard Deviation :Population Variance : 2

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

-1) as the denominator instead of n?

each sample, and average each of these together, then this average tends not to equal the populationvariance ( 2) unless we use (n-1) as the denominator.

-1) makes the sample variance (s2) abetter estimator of the population variance (2) in the sense that it is not biased downward. Since,the standard deviation is the (Positive) square root of the variance, we proceed using s as theestimator of

Sheppards Correction :

Variance and Standard Deviation computed from grouped data always containsome error because of grouping of individual

observations into different classes, called grouping. Sheppards correction is a factor used for correcting variance for grouping

errors. It is 1/12 of the square of the width of class interval C, which is deductedfrom the computed variance. That is,

Corrected Variance = Computed Variance C2/12(where C2/12 is known as Sheppards Correction for variance)

It is applicable only in the case of frequency distributions of continuousvariables.

Note: Statisticians are however, not in agreement over the utility of using the correctionfactor. This owes to the fear that it may lead to over-correction and may thus introducefresh error. What is generally agreed upon is that the correction factor is not to be used without

thorough examination of the problem.

Standard Score : A measure called the standard score gives us

the number of standard deviations of particular observation


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lies below or above the mean. If we let x symbolize the observation, the standard score computed from

population data is: Population Standard Score = (x ), [Where x =observation from the population; = population mean; and = population standarddeviation].

- For Example, Suppose we observe a vial of compound that is 0.108% impure.

Let Population Mean is 0.166 and Population Standard Deviation is 0.058. Anobservation of 0.108 would have standard score of:

(x )/ = (0.108 0.166)/0.058 = -1- An observation of 0.282% would have a standard score of = (0.282

0.166)/0.058 = +2- The standard score indicates that an impurity of 0.282% deviates from the mean

by 2(0.058)=0.116 unit, which is equal to +2 in terms of the number of standarddeviations away from the mean

Comparing the Mean/Median/Mode:

Symmetrical distributions that contain only one mode alwayshave the same value for the mean, the median and the mode.In these cases, we need not choose the measure of central tendencybecause the choice has been made.

Measures of Skewness: (Quantifies the extent of departurefrom symmetry, and also indicates the direction in which

the departure takes place). Positively (Right hand side) Skewed: There are just a few high extreme values on the

right side These extreme values raise the Mean of the data but do not affect the

median so the mean will be greater than the median. Negatively (Left hand side) Skewed: A few extremely low values are on the left

side which pulls the mean down so mean will be less than the median. The relative values of mode, median and mean can therefore tell us whether the

distribution is skewed to the left or right.


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Symmetric distribution (no skewness) Negatively skewed Positively skewed

Note:

When population is skewed to the right or left, the Median is often the best measure of locationbecause it is always between the Mean and the Mode.

The Median is not as highly influenced by the frequency of occurrence of a single value as is theMode, nor is it pulled by extreme values as is the mean.

Otherwise, there are no universal guidelines for applying the mean, median, or mode as themeasure of central tendency for different populations.

Each case must be judged independently, according to the guidelines we have discussed.


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Kurtosis: Can be used to show the degree of concentration, either thevalues concentrated in the area around the mode (a peaked curve) or decentralized from the

mode to both tails of the frequency curve (a flat- topped curve). Kurtosis in Greek meansBulginess.

In statistics Kurtosis refers to the degree of flatness or peaked-ness in the region about the mode of a frequency curve.

`The degree of Kurtosis of a distribution is measured relative to the peaked- ness of a normal curve. If a curve is more peaked than the normal curve it is called LeptoKurtic. If more flat topped than the normal curve then it is called or Flat Topped. The normal curve itself is known as Mesokurtic.


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Example( Of : A company in table uses three grades of labour to produce two end products.. and wants to knowthe average cost of labour per hour for each of the product.

Grades Labour Hourly Wage Labour Hours Per Unit OutputProduct-1 Product-2

Unskilled $5 1 4

Semi-skilled 7 2 3

Skilled 9 5 3Total (Hours of Labour) 8 10

Using this average rate, we would compute the labour cost of one unit of

Product-1 to be: 7(1+2+5) = $56 and

Product-2 to be: 7(4+3+3) = $70

(But these answers are incorrect)

To be correct, the answers must take into account that different amounts of each grade of labour are used:i.e., find weighted average:Xw = [( W x X)/ W].We weight the hourly wage for each grade by its proportion of the total labour requiredto produce the product:

Product-1 W

1 Unskilled out of 8 hours 1/8

2 Semi-skilled out of 8 hours 2/8

5 Skilled out of 8 hours 5/8

Similarly,

Product-2 W

1 Unskilled out of 8 hours 4/10

2 Semi-skilled out of 8 hours 3/10

5 Skilled out of 8 hours 3/10

For Product-1:

Xw = [(1/8 x 5) + (2/8 x 7) + (5/8 x 9)]/(1/8 + 2/8 + 5/8)

= $ 8 per hour

For Product-2:

Xw = [(4/10 x 5) + (3/10 x 7) + (3/10 x 9)]/(4/10 + 3/10 + 3/10)

= $6.8 per hour


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5. The classes will overlap the span of the actual data values by 42 37 = 5 units.


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6. Determine the class boundaries :

Class boundaries must be selected so that the classes cover all the actual data values and so that each datavalue falls in a distinct class.The lower boundary for the first class is an arbitrary but convenient value below the lowest data value.Then find the upper boundary for the first class by adding the class width to the lower boundary.Find the boundaries for the remaining classes by successively incrementing by the class width until all the

class boundaries have been determined.Find the frequency for each class and present the results in a table.

7. This 5 is adjusted in the lower (first class) not fully but: half only: (5) = 2.5 units b elow 121 .

Class Limits Class Boundaries Frequency

119-125 118.5-125.5 1126-132 125.5-132.5 4133-139 132.5-139.5 26140-146 139.5-146.5 59147-153 146.5-153.5 15154-160 153.5-160.5 1

N = 106(Sturge s Rule: K = 1 + 3.322 [log (n)], where n = Number of Observations )

HISTOGRAM:

1** When raw data are grouped into classes, a certain amount of information is lost, since no distinction is madebetween observations falling in the same class.

2** The larger the class interval or width is, the greater is the amount of information lost.3** For the employee absence data a class interval of 15 is so large that the corresponding histogram gives very

little idea of the shape of the distribution ( figure: 1a ).4** A class interval of 2, in co ntrast, gives a ragged histogram ( figure :1c )little information has been

lost, but the presentation is somewhat misleading.


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\

5** Thus, the rule for determining the approximate number of classes, will likely produce histograms between

the extremes of giving too much detail or giving too little ( figure : 1b ).. however, the number of classes may need to be adjusted for some sets of data in order to obtain a clearerpresentation of the information.


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n : Ogive . 1***Just as a frequency distribution can be represented graphically by a histogram, a cumulative frequencydistribution is represented graphically by an OGIVE.

Figure-2

2*** From Ogive , we may, for example, get an approximation for the number of observations whose numericalvalue is less than 150 by finding the height of the curve over that point.

3*** With the help of the dashed line in Figure-2(a), we approximate that on 98 days, fewer than 150 employeeswere absent.

4** If we convert the cumulative frequencies to cumulative relative frequencies, by dividing each cumulativefrequency by the number of observations in the data set (there are 106 observations), then we have a CumulativeRelative Frequency Distribution (CRFD), Figure-2(b).

5** The primary use of a CRFD is to approximate percentiles.

6** A Percentile is essentially a value below which a given proportion of the values in a data set fall.

7** With the help of the dashed lines of Figure-2(b), our approximations of the 25th, 50th and75th percentiles are 139, 141 and 145.

Figure-2


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8** In other words, we approximate that the value 139 exceeds 25% of all theabsence values.

9** The 25th, 50th and 75th percentiles are termed the first, second, and third

quartiles, respectively.10** Thus the first, second, and third quartiles (Q1, Q2 and Q3) are approximately 139, 141 and 145.11** The terms quantile and fractile are general terms that refer to any of the percentile or

quartile values that measures positions in the data set.12** Fractile is a term used more by Statisticians than by the rest of us, who are more familiar with 100 fractiles,

or percentiles, especially when our percentile score on the CAT is involved.13** When we get that letter indicating that our percentile score was 95, we know that 95 percent of those taking

the test did worse than we did.14** For example, the 0.50 quantile or fractile is the same as the 50th percentile, the median, and the second

quartile.

15** Hence, the approximation for the 0.50 quantile is 141.

(Di): There are 9 deciles that divide the frequency distribution into 10 equalparts].

(Pk): There are 99 percentiles which divide the frequency distribution into 100equal parts].


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PROBABILITYProbability Theory

Probability Theory(Probability Theory is of great value in Managerial Decision Theory)

and debatable subject in recent years.

For example, We often find people making such statements as:

It is likely that it may rain. We probably will get the contract. It is possible that the price of our shares may go down further, etc.

specifying the meaning of these statements.

h we are unable to forecast the future with complete certainty.Theory.

.ability constitutes the foundation of statistical theory and application.

-making under uncertainty.-making in business and in government with the means for quantifying

the uncertainties which affect his choice of appropriate actions.

uncertainty in business situations in such a way that he can assess systematically the risk involved in eachalternative, and consequently act to minimize risks.

imals (0.25, 0.50, 0.125) between zero and 1.

something will always happen.comes of doing something.

Theory as an experiment.

Using this Formal Language: We could ask the question in a coin-tossExperiment what is the probability of the event Head (i.e., 1/2). The set of all possible outcomes of an experiment is called the sample space.In coin-toss experiment, the sample space is:S = {Head, Tail}.In the Card-Drawing Experiment, the sample space has 52 members.

Types of Events:1. Favourable Events2. Mutually Exclusive Events3. Equally Likely Events

4. Collectively Exhaustive Events5. Compound Events6. Dependent Events7. Independent Events


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favourable event.re mutually exclusive if there is no outcome that belongs to both sets. Can two or more of

these events occur at on e time, if the answer is Yes, the events are not mutually exclusive .

said to be collectively exhaustive .equally likely if they have equal chance of happening in an experiment .

compound events. Compound Events can be either

Dependent or Independent .

event has no effect on the Probability of the occurrence of any other event.

upon or affected by theoccurrence of some other event.

Types of ProbabilityThere are three basic ways of classifying probability.1. Classical Approach2. Relative Frequency Approach or Empirical Approach3. Subjective Approach

All of them define Probability as a ratio or proportion.

1. Classical Approach

Above to be va lid .. each of the possible outcomes must be equally likely. we can compute

the probability of an event without even performing experimental trials

e.g., P (Head ) = 1/2

If there are a possible outcomes favourable to the occurrence of an event E, and b possible outcomesunfavourable to the occurrence of E and all these possible outcomes are equally likely and mutually exclusive, thenthe probability that a will occu r, denoted by P(E) is:

( E ) a= a/(a+b) = Number of outcomes favourable to concurrence of ETotal Number of Outcomes

unbiased dice, and standard decks of cards, we can state the answer in advances (a priori) without toss a coin, rollinga die, or drawing a card i.e., we need not to perform experiments to make our probability statements.


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Drawbacks of Classical ApproachNot applicable to the less orderly decision problems we encounter in management.

trouble.-life situations, disorderly and unlikely as they often are, make it useful to define Probabilities in other ways.

Principles of Insufficient Reasons

the name of Bayes (Englishman 1702 -1761).

His famous hypothesis states that:

An interesting application of this principle is the toss of a fair coin:

i) How do we know that a fair coin has a probability of 0.5 showing head and probability of 0.5 showing tail?

ii) Since, there is no logical reason for a coin coming up one way versus another, the probabilities must beequal or 0.5

because there are only two possibilities.

2. Relative Frequency of Occurrence Defined Through Experimentation or Empirical Appro ach: Suppose we begin asking ourselves complex questions such as:What is the probability that I will live up to 85? What is the Probability that the location of a new paper plant on the river near our town will cause asubstantial fish kill?

What is the Probability that a production process used by a particular firm will produce a defective? We see that we may not be able to state in advance,without experimentation, what these probabilities are?

ested in a theoretical foundation for calculating risk of losses in lifeinsurance and commercial insurance, began defining probabilities from statistical data collected on births anddeaths. Today this approach is called relative Frequency of Occurrence . It defines probability as either:(1) The observed relative frequency of an event in a very large number of trials, or(2) The proportion of times that an event occurs in the long run when conditions are stable.

s of past occurrences as probabilities.

will happen again in the future.


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i.e., the Probability is determined objectively by repetitive empiricalobservations.- If says that the only valid procedure for determining event probabilities is through repetitive experiments.

3. Subjective Probabilities (Based on Intuition, Judgement, Educated Guess, Experience): Are based on the beliefs of the person making the probability assessment.In fact, can be defined as the probability assigned to anevent by an individual based on whatever evidence is available, this evidence may be just aneducated guess.Subjective Probability assignments are frequently found when events occur only once or at most a very fewtimes.

For Example, You have to select one person out of three:

Each has an attractive appearance,

A high level of energy;Abounding self-confidence,A record of past accomplishments, andA state of mind that seems to welcome challenges.

Assuming this question and choosing among the three will require you to assign a subjective Probability toeach persons.

Since most higher level social and managerial decisions are concerned with specific, unique solutions,rather than with a long series of identical situations, decision-maker at this level

make considerable use of subjective probabilities.The subjective approach to assigning probabilities was introduced in 1926 by Frank Ramsey in his book, theFoundation of Mathematics and other Logical Essays. The concept was further developed by Bernard Koopman, Richard Good, and Leonard Savage, namesthat appear regularly in advanced work in this field.

The two people who made opposing bets on the outc omes of the Football game would understand quite well whathe meant.

Probability Rules

Most Managers who use probabilities are concerned with two conditions:

1. The case where one event or another will occur.2. The situation where two or more events will both occur.


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We are interested in the first case when we ask : What is the Probability that todays demand will exceedour inventory?

To illustrate the second situation, we could ask : What is the probability that todays demand will exceedou r inventory and that more than 10 percent of our sales force will not report for work.

Commonly used Symbols in Probability Theory, we use symbols to simplify the presentation of ideas.

P(A) = is the Probability of Event A happening P(B) = is the Probability of Event B happening etc.

Types of Probabilities :

Simple/Marginal ProbabilityConditional ProbabilityJoint ProbabilityObjective ProbabilitySubjective ProbabilityPrior ProbabilityPosterior Probability

Simple or Unconditional Probability, are also known as Marginal Probability:

A Single Probability means that only one event can take place. It is called a simple or unconditionalprobability.Symbolically , P(A) is a Simple Probability of event A

Properties of Probabilities : Property 1 : If A is an event, then : 0 P (A) 1

Property 2 : The sample space contains all possible outcomes of the experiment. Thus .. P(S) =1 Property 3 : If event A is mutually Exclusive of event B, then :

P ( A or B ) = P(A) + P(B)

Venn Diagram = Is a Pictorial representation Developed by English Mathematician, John Venn.In this, the entire sample space is represented by a rectangle and events are represented by parts of the

rectangle.

Theorems in Probability Theory :Addition TheoremMultiplication TheoremBayes Theorem

Addition Theorem :

P(A or B)=P(A)+P(B)if events are mutually exclusive

Venn Diagram .


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P(A or B)=P(A)+P(B) P(AB)if events are not strictly mutually exclusive

Multiplication Theorem :

P(A and B) = P(A) P(B)..if events are Independent P(A and B) = P(A) P(B/A).if events are Dependent

Venn Diagram

A and B are independent events if : P(B/A) = P(B)

Or, in words, A and B are independent events whenever A gives no information about the probability of B.

Complementary Events : If ______ is the complement of event A , then : P(____) = 1-AP(A)


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Probabilities Under Statistical Independence and Dependence

Joint Probability Table

Bayes Theorem :

We often have to obtain the additional information in the form of samples, surveys, tests, or experiments that canbe used to revise probabilities. (e.g., a consultant may be retained to provide information).

Or, tests such as blood tests, credit tests, or tests of raw materials may be used to obtain sample information thatcan be used to revise probabilities.

The sample information provides conditional probabilities.

Bayes theorem combines the prior probabilities and the conditional probabilities to give revised or posteriorprobabilities that reflect the sample information.

The revised probabilities can enrich the analysis of probability problems and eventually to improved decisions .


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..

Bayes Theorm:

B1, B2, Bn are mutually exclusive and collectively exhaustive events with prior probabilities : P(B1),P(B2), .P( Bn) , and

B2,.A given Bn are: P(A /B1)

, P(A/B2)P( A/ Bn).

n , are known.

P(B1 / A) = P(B1 ) P( A / B1 ) // P(B1 ) P( A / B1 ) P(B2 ) P( A / B2 ) ......P(Bn ) P( A / Bn )

PROBABILITY TREE:

1. Probability Tree Diagrams depict events or sequences of events as branches of a tree.2. Each branch of the tree is labeled to indicate which event it represents.3. Along each branch is given the probability of the event occurrence, given

that the sequences of the events have reached that point in the tree.4. Thus all the branches that emanate from one point must be:

(i) Mutually exclusive; and(ii) Collectively exhaustive.

i.e.,they must represent distinct events, and they must account for all possible events that can occur at that point.

Example: The chance that someone has a particular disease is 0.02. A test that identifies this disease gives a positivereaction in 97% of the people who have the disease, but it also gives a positive reaction in 5% of the people who donot have the disease. Given that you have just received a positive reaction from this test, what is the probabilitythat you have the disease?

Solution:the end of the top branch or the third one down.

at the ends of these two branches.

sum.


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P(Disease / Positiv e) = 0.0194/(0.0194 + 0.0490) = 0.0194/0.0684 = 0.2836

the disease and A denote the positive test.

Example: A company has 245 customers that are classified in the table below by the frequency withwhich they place a regular or an irregular order and by their payment terms, cash or credit.

I) Probability that a randomly selected customer is either a regular or a credit customer;ii) Probability that a randomly selected customer is both a credit customer and orders irregularly.


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1. Rank the following personal computers with respect to their usage in your office, assigning the number 1 to themost used system, 2 to the next most used, and so on. If a particular system is not used at all in your office, put azero against it.---IBM/AT --- Compaq--- IBM/ XT --- Dell--- Apple --- Other (Specify)--- MacIntosh

--- Zenith

Interval scale :

is used when responses to various items that measure a variable can be tapped on a five-point (or seven-point or any other number of points) scale, which can thereafter be summated across the items. See example of aLikert scale below.

Using the scale below, please indicate your response to each of the items that follow by circling the numberthat best desribes your feeling


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Ratio scales :

usually get used in organizational research when exact figures on objective (as opposed to subjective) factorsare called for, as in the following questions:

1. How many other organizations did you work for before joining this system? ----

2. Please indicate the number of children you have in each of the following categories:

--- below 3 years of age

--- between 3 and 6

--- over 6 years but under 12

--- 12 years and over

3. How many retail outlets do you operate? ---- The responses could range from 0 to any figure.Nominal Ordinal Interval Ratio Scales


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Research Methods

Table of Contents in Research (Project ) Report

1) Problem Statement2) Review of Literature3) Hypothesis Formulation4) Research Design & Sample Design5) Data Collection6) Data Processing7)Data Analysis & Hypothesis Testing8) Interpretation9) Report Writing

Some Important Questions

Why should we do research?What research should be done?Is it worth doing the research?How should the research be designed to achieve the research objectives?What will we do with the research?

Research Process

Five Key Steps1. Define Problem2. Formulate Research Design3. Collect Data4. Prepare and Analyze Data5. Prepare and Present Report

Problem Definition Process

Defining the problem is the most important stepTwo types of problems:Management Problem action orientedResearch Problem information oriented


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Sampling Techniques

Sampling Fundamentals

When Is Census Appropriate? Population size itself is quite small Information is needed from every individual in the population

Cost of making an incorrect decision is high Sampling errors are high

Sampling Fundamentals (Contd.)

When Is Sample Appropriate?

Population size is large Both cost and time associated with obtaining information from the population is high Quick decision is needed More time can be spent on each interview, thereby increasing response quality

Easier to manage surveys of smaller samples Population being dealt with is homogeneous Used if census is impossible

Error in Sampling

Total Error - Error in Sampling

Difference between the true value and the observed value of a variable

Sampling Error Error is due to sampling

Non-sampling Error Error is observed in both census and sample

Measurement Error Data Recording Error Data Analysis Error Non-response Error

Terms Used in Sampling

Universe Sampling unit-home/office/person Sampling frame


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Sampling Process

Identifying the Target PopulationDetermining the Sampling Frame Reconciling the Population, Sampling Frame DifferencesSelecting a Sampling FrameProbability Non-ProbabilitySampling Sampling

Determining the Relevant Sample SizeExecute SamplingData Collection From Respondents Handling the Non-Response ProblemInformation for Decision-Making

Sampling Techniques

Probability Sampling All population members have a known probability of being in the sample

Simple Random Sampling Each population member, and each possible sample, has equal probability of being selected

Stratified Sampling The chosen sample is forced to contain units from each of the segments or strata of the population

Types of Stratified SamplingProportionate Stratified Sampling

Number of objects/sampling units chosen from each group is proportional to number in population Can be classified as directly proportional or indirectly proportional stratified sampling

Disproportionate Stratified Sampling Sample size in each group is not proportional to the respective group sizes Used when multiple groups are compared and respective group sizes are small

Directly Proportionate Stratified Sampling

Inversely Proportional Stratified Sampling

If the opinion of the exporters are valued more than that of non -exporters, more firms should be sampled from theexporters group. In that case, inversely proportional stratified sampling may be used.

If a sample size of 60 is desired, a 10 percent inversely proportional stratified sampling is employed.


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In inversely proportional stratified sampling, the selection probabilities are computed as follows:


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Denominator 600/200 + 600/400 = 3 + 1.5 = 4.5

Exporters proportion and sample size 3/ 4.5 = 0.667; 0.667 * 60 = 40

Non -exporters proportion and sample size 1.5 / 4.5 = 0.333; 0.333 * 60 = 20

Cluster Sampling

Involves dividing population into subgroups

Random sample of subgroups/clusters is selected and all members of subgroups are interviewed

Very cost effective

Useful when subgroups can be identified that are representative of entire population

Comparison of Stratified and Cluster Sampling

Stratified sampling Cluster sampling -Homogeneity within group -Homogeneity between groups-Heterogeneity between groups -Heterogeneity within groups-All groups are included -Random selection of groups-Sampling efficiency improved by -Sampling efficiency improved by decreasingincreasing accuracy at a faster rate cost at a faster rate than accuracythan cost

Systematic Sampling Involves systematically spreading the sample through the list of population members commonly used in telephone surveys

Non-Probability Sampling Costs and trouble of developing sampling frame are eliminated Results can contain hidden biases and uncertainties

Types of Non Probability Sampling

Judgmental

"Expert" uses judgment to identify representative samplesSnowball

Form of judgmental sampling Appropriate when reaching small, specialized populations Each respondent, after being interviewed, is asked to identify one or more others in the appropriate group

Judgmental "Expert" uses judgement to identify representative samples

Snowball

Form of judgmental sampling Appropriate when reaching small, specialized populations Each respondent, after being interviewed, is asked to identify one or more others in the appropriate group.


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Convenience Used to obtain information quickly and inexpensively

Quota Minimum number from each specified subgroup in the population Often based on demographic data

Non Response Problems

Respondents may: Refuse to respond Lack the ability to respond Be inaccessible

Sample size has to be large enough to allow for non response Those who respond may differ from non respondents in a meaningful way, creating biases Seriousness of non-response bias depends on extent of non response

Solutions to Non-response Problem Improve research design to reduce the number of non-responses Repeat the contact one or more times (call back) to try to reduce non-responses Attempt to estimate the non-response bias

Basic Statistics Population Sample

Mean XVariance 2 s2

Standard Deviation s

Sample Size N n


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DECISION MAKING

Decision Analysis

( A)*** Meaning :-- 1. Making decisions is an integral and continuous aspect of human life.2. Decision Theory in business today play an important role in helping managers

makes decisions.3. The act of decision making enters into almost all of a managers activities. Indeed, a

noted social scientist has used the term decision making as though it weresynonymous with managing:

Managers must reach decisions about objectives and plans for their organizational units

They must decide how to direct, how to organize, how to control.

They must not only make many decisions but guide subordinates in reaching decisions of their own.

What does this process of decision making involve?

What is a decision?

How we can analyze and systematize the solving of certain types of decision problems?

**** The answer to these and similar questions form the subject matter of decision theory.

4. Decision theory is rich, complex, growing and interdisciplinary subject.

5. Since we live in a world, where the course of future events cannot bepredicted with absolute certainty, the best we can do is to reach approximate solutions based upon thelikelihood of possible future events.

6. We assign a certain value to Probability, which can range from 0 to 1. Zerobeing completely pessimistic and one being completelyoptimistic about the occurrence of a particular event under certain conditions.

7. Hence,**** Decision making is the act of selecting a preferred courseof action among alternatives

8. From this definition one can conclude that decisions are involved inalmost all facets of life.

9. Once it is known that a choice must be made, there is no escape from decision making. One maypostpone the final choice, but this too is a decision----not to make a choice at present.

C)*** Types of Decisions:- Decisions are of three kinds from the view point of decision theory, viz:

a) Strategic decisionsb) Managerial decisionsc) Operating decisions

D*** Steps in Decision-Making:-i. Recognition of the Need for a Decision.

ii. Identification of objectives i.e. analyzing the problem and gathering information.iii. Search for Reasonable Alternatives i.e. developing alternative solutions.iv. Evaluation of Alternatives.


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v. Selection of Alternatives i.e. Deciding upon the best solution; andvi. Implementation i.e. converting the decision into effective action.

E.***DM Under Certainty, Risk, Uncertainty and ompetitive Conditions:-

General Features:a. Alternative courses of actions or acts or strategies..the objective is to

select the best of these alternatives from among the total set. (A strategy is a set of decisionsthat completely determines a course of action ).

b. Conditions outside the control of the decision maker thatwill determine the consequences of a particularact. These conditions, sometimes termed events or statesof nature, must be mutually exclusive and all possibleconditions must be listed (collectively exhaustive ).

c. Payoff or loss, as a measure of the benefit to the decisionmaker of particular state of nature resulting from a particular course of action.

d. Some criterion or measure of what constitutes the objective being soughtby the decision maker.

e. An environment representing the extent of knowledge about the state of nature thatwill occur.

a. The first three elements listed above are often displayed in a decision matrix to organize some of thefeatures of a situation into an orderly format.b. The alternatives, Si, are listed as column headings, theevents, Ni, are listed as rows headings, or vice versa.

c. The consequences, Pij (Payoffs or Losses), are then displayed within thebody of the matrix at the intersection of theappropriate alternative and the appropriate event.

Illustrative Decision Matrix:

Alternatives Actsor Strategies

States of Nature (Events)

N1 N2 Nn

S1 P11 P12 P1n

S2 P21 P22 P2n

. . . .

. . . .


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Sm Pm1 Pm2 Pmn

G. Decision Environmenta. Certainty: The decision maker knows which state of nature will occur.

b. Risk: The decision maker does not know which state of nature will occur but can estimatethe probability that any one state will occur.

c. Uncertainty: The decision maker lacks sufficient information even toestimate the probabilities of the possible states of nature.

Selection of Best Criterion

Depending on how much we know about the states of nature or the businessenvironment, we can refer to decisions making under:

1. Certainty

2. Risk

3. Uncertainty

4. Competitive conditions


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Statstics - QAM Management