KDF2C QUANTITATIVE TECHNIQUES FOR BUSINESSDECISION. Unit : I - V

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1 KDF2C QUANTITATIVE TECHNIQUES FOR BUSINESSDECISION Unit : I - V

2 Unit I: Syllabus Probability and its types Theorems on Probability Law Decision Theory Decision Environment Decision Process Decision tree 2

3 Probability TM The numerical chance that a specific outcome will occur. That is, it is a mathematical measure of Measuring the certainty or uncertainty of an event Sometimes you can measure a probability with a number: "10% chance of rain", or you can use words such as impossible, unlikely, possible, even chance, likely and certain. Example: "It is unlikely to rain tomorrow". 3

4 Approaches to Probability TM 1.Relative frequency event probability = x/n, where x=no.of occurrences of event of interest, n=total no. of observations 2.Subjective probability individual assigns prob. based on personal experience, anecdotal evidence, etc 3.Classical approach : every possible outcome has equal probability. The probability for the occurrence of an event A is defined as the radio between the number of favorable outcomes for the occurrence of the event and the total number of possible outcomes, Probability of an event = Number of favorable outcomes/total number of outcomes P(E) = m/n 4

5 TM The Additive Rule for Unions: For any two events, A and B, the probability of their union, P(A B), is P( A B) P( A) P( B) P( A B) AUB A B When two events A and B are mutually exclusive, P(AB) = 0 and P(AB) = P(A) + P(B). 5

6 TM Example: Additive Rule Example: Suppose that there were 120 students in the classroom, and that they could be classified as follows: A: Statistics Student P(A) = 50/120 B: female P(B) = 60/120 Allied Statistics Non Statistics Male Female P(selecting a statistics female student) = P(AB) = P(A) + P(B) P(AB) = 50/ /120-30/120 = 80/120 = 2/3 6

7 TM Independent Event &Conditional Probabilities Two events, A and B, are said to be independent if the occurrence or nonoccurrence of one of the events does not change the probability of the occurrence of the other event. In terms of conditional, Two events A and B are independent if and only if P(A B) = P(A) or P(B A) = P(B)Otherwise, they are dependent. The probability that A occurs, given that event B has occurred is called the conditional probability of A given B and is defined as P( A B) P( A B) if P( B) P( B) given 0 7

8 TM Example 1 1. Toss a fair coin twice. Define A: head on second toss B: head on first toss P(A B) = ½ P(A not B) = ½ P(A) does not change, whether B happens or not A and B are independent! 2.Toss a pair of fair dice. Define A: red die show 1 B: green die show 1 P(A B) = P(A and B)/P(B) =1/36 / 1/6=1/6=P(A) 8

9 The Multiplicative Rule for TM Intersections For any two events, A and B, the probability that both A and B occur is = P(A) P(B given that A occurred) If the events A and B are independent, then the probability that both A and B occur P( AB) P( A). P( B / A) if A & B are independent, 1. Example: In a certain population, 10% of the people can be classified as being high risk for a heart attack. Three people are randomly selected from this population. What is the probability that exactly one of the three are high risk? Define H: high risk N: not high risk P(exactly one high risk) = P(HNN) + P(NHN) + P(NNH) = P(H)P(N)P(N) + P(N)P(H)P(N) + P(N)P(N)P(H) = (.1)(.9)(.9) + (.9)(.1)(.9) + (.9)(.9)(.1)= 3(.1)(.9)2 =.243 P( AB) P( A). P( B) 9

10 The Law of Total TM Probability Let S 1, S 2, S 3,..., S k be mutually exclusive and exhaustive events (that is, one and only one must happen). Then the probability of any event A can be written as P(A) = P(A S 1 ) + P(A S 2 ) + + P(A S k ) = P(S 1 )P(A S 1 ) + P(S 2 )P(A S 2 ) + + P(S k )P(A S k ) 10

11 TM Bayes Rule,...k, i S A P S P S A P S P A S P i i i i i 2 1 for ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( Proof i i i i i i i i i i i i S A P S P S A P S P A P AS P A S P S A P S P AS P S P AS P S A P 11

12 Decision Theory Decision Theory represents a general approach to decision making which is suitable for a wide range of operations management decisions, including: Capacity Planning Product and Service Design Location Planning Equipment Selectiom 12

13 Decision Theory Process Step 1 Step 2 Step 3 Identify possible future conditions called states of nature Develop a list of possible alternatives, one of which may be to do nothing Determine the payoff associated with each alternative for every future condition Step 4 Step 5 If possible, determine the likelihood of each possible future condition Evaluate alternatives according to some decision criterion and select the best alternative 13

14 Decision Environments Certainty - Environment in which relevant parameters have known values Risk - Environment in which certain future events have probable outcomes Uncertainty - Environment in which it is impossible to assess the likelihood of various future events Decision Making under Uncertainty Maximin - Choose the alternative with the best of the worst possible payoffs Maximax - Choose the alternative with the best possible payoff Laplace - Choose the alternative with the best average payoff of any of the alternatives Minimax Regret - Choose the alternative that has the least of the worst regrets 14

15 Format of a Decision Tree Decision Point Chance Event Payoff 1 Payoff B Payoff 3 Payoff 4 2 Payoff 5 Payoff 6 KDF2C- QUANTITATIVE TECHNIQUES FORBISINESS BUSINESS DECISION 15

16 Expected Value of Perfect Information Expected value of perfect information: the difference between the expected payoff under certainty and the expected payoff under risk Expected value of perfect information Expected payoff under certainty = - Expected payoff under risk 16

17 Unit II: Syllabus Sampling theory Point And Interval Estimation Interval Estimate For Mean Interval Estimate For Variance Interval Estimate For Proportion 17

18 TM Sampling Definition: The process of drawing a sample from a population Is called sampling. 18

19 TM Random Sampling Methods simple random sample (each sample of the same size has an equal chance of being) stratified sample (divide the population into groups called strata and then take a sample from each stratum) cluster sample (divide the population into strata and then randomly select some of the strata. All the members from these strata are in the cluster sample.) systematic sample (randomly select a starting point and take every n-th piece of data from a listing of the population) 19

20 Non Random sampling methods TM Convenience sampling: Drawn at the convenience of the researcher. Common in exploratory research. Does not lead to any conclusion. Judgment sampling: Sampling based on some judgment, gut-feelings or experience of the researcher. Common in commercial marketing research projects. If inference drawing is not necessary, these samples are quite useful. Quota sampling:an extension of judgmental sampling. It is something like a two-stage judgmental sampling. Quite difficult to draw. Snowball sampling :Used in studies involving respondents who are rare to find. To start with, the researcher compiles a short list of sample units from various sources. Each of these respondents are contacted to provide names of other probable respondents 20

21 Point and Interval Estimation Point Estimate A sample statistic used to estimate the exact value of a population parameter Confidence interval (interval estimate) A range of values defined by the confidence level within which the population parameter is estimated to fall. Confidence Level The likelihood, expressed as a percentage or a probability, that a specified interval will contain the population parameter. 21

22 Estimations Lead to Inferences Take a subset of the population Try and reach conclusions about the population 22

23 The Central Limit Theorem Revisited If repeated sample of size is drawn from any population of whatever form and this the population has a mean standard error of the m mean and Standard deviation,then as N becomes large, the sampling distribution of the sample mean approaches normality with mean The standard deviation of a sampling distribution SE. y y N 23

24 Confidence Levels& Its Construction Confidence Level The likelihood, expressed as a percentage or a probability, that a specified interval will contain the population parameter. 95% confidence level there is a.95 probability that a specified interval DOES contain the population mean. In other words, there are 5 chances out of 100 (or 1 chance out of 20) that the interval DOES NOT contain the population mean. 99% confidence level there is 1 chance out of 100 that the interval DOES NOT contain the population mean. The standard error of the mean makes it possible to state the probability that an interval around the point estimate contains the actual population mean. 24

25 Estimating standard errors y y N Since the standard error is generally not known, we usually work with the estimated standard error: s Y s Y N 25

26 Determining a Confidence Interval (CI) CI Y Z ( s Y ) where: Y = sample mean (estimate of ) Z = Z score for one-half the acceptable error = estimated standard error s Y 25

27 Confidence Interval Width More precise, less confident More confident, less precise 26

28 Interval Estimate of a Population Mean When Known Interval Estimate of mean x x z /2 n where: X is the sample mean 1 -α is the confidence coefficient z α /2 is the z value providing an area of α /2 in the upper tail of the standard normal probability distribution s is the population standard deviation n is the sample size 28

29 Sample Size for an Interval Estimate of a Population Mean Margin of Error E z /2 n Necessary Sample Size n z ( / 2 ) E

30 Interval Estimate of a Population Mean: variance is Unknown Interval Estimate x t /2 s n where: 1 - α = the confidence coefficient t α /2 = the t value providing an area of α/2 in the upper tail of a t distribution with n - 1 degrees of freedom s = the sample standard deviation 30

31 Interval Estimate of a Population Proportion Interval Estimate p z / 2 p( 1 p) n where: 1 - α is the confidence coefficient z α /2 is the z value providing an area of α /2 in the upper tail of the standard p normal probability distribution is the sample proportion 31

32 Unit III: Syllabus Testing of Hypothesis Testing Process Testing of Population Mean/Means Testing of Population Variance Testing of Proportion/Proportions Chi square test for independence of attributes ANOVA one way and Two way Classifications 32

33 Hypothesis Testing Introduction Hypothesis: A conjecture about the distribution of some random variables. A hypothesis can be simple or composite. A simple hypothesis completely specifies the distribution. A composite does not. There are two types of hypotheses: The null hypothesis, H 0, is neutral statement The alternative hypothesis, H a, is the alternative statement of null hypothesis Standard Error The standard deviation of the sample statistics is known as Standard Error Large Sample- the sample size n is greater than 30 Small Sample the sample size n is less than or equal to 30 33

34 Testing Process Hypothesis testing is a proof by contradiction. 34

35 Inference on the Mean of a Population,Variance Known Assumptions 35

36 Testing about sample mean with population mean 36

37 Hypothesis Testing about the Difference in Two Sample Means Population 1 X 1 X 1 X x 2 n x n 2 X X X X 1 2 X 2 Population 2 37

38 Hypothesis Testing about the Difference in Two Sample Means For Large sample, the test statistic is Z X X n n 1 2 X 1 X X X X 1 X 2 n1 n2 X X 1 2 For Small sample,the test statistic is t X X s n s n

39 Example: Hypothesis Testing for Differences Between Means Computer Analysts n1 X S S n 2 X S S Registered Nurses

40 Example: Hypothesis Testing for Differences between Means Rejection Region.01 2 Z c Nonrejection Region Critical Values Rejection Region Z c If Z < or Z > 2.33, reject H o. If Z 2.33, do not reject H o. Z X X S n S n Since Z = 3.36 > 2.33, reject H o. 40

41 Testing on single Proportion: 41

42 Testing of Differences in Sample Proportions For large samples 1. 5, , , and where q ˆ = 1 - pˆ 2 2 the difference in sample proportions is normally distributed with pˆ pˆ pˆ pˆ n n n n 2 pˆ qˆ pˆ qˆ P P 1 2 P Q n and P Q n Z p 1 p 2 n1 n2 P1 P2 Q Q p p P P P Q P 1 Q n1 n2 proportion from sample proportion from sample 1 size of sample 1 size of sample 2 proportion from population 1 proportion from population P P

43 Example(Problem) H H o a : : 2 Z. 005 n1 X p 1 P P P P n2 X p P X X n 1 n Rejection Region Z c Z Nonrejection Region 0 c Critical Values 2 Z Rejection Region.005 ˆ ˆ 1 2 p p P P PQ n n 1 2 Since Z = , do not reject H o. 43

44 Hypothesis Testing about the Difference in Two Population Variances F Test for testing the Two Population Variance F 2 S1 2 S 2 F ( n 11),( n 21) Where S 12 is the variance of sample 1 S 2 2 is the variance of sample 2 44

45 2 Test Chi Square Test for independence of attributes Chi Square Test for single variance Chi square contingency table Chi Square Test for Goodness of Fit So far, we have assumed the population or probability particular problem is known. distribution for a There are many instances where the underlying distribution is not known, and we wish to test a particular distribution. Use a goodness-of-fit test procedure based on the chi-square distribution. 45

46 Analysis of Variance Analysis of Variance is a widely used statistical technique that partitions the total variability in our data into components of variability that are used to test hypotheses. In ANOVA, we compare the between-group variation with the within-group variation to assess whether there is a difference in the population means Types 1.One way ANOVA 2.Two way ANOVA 46

47 One Way ANOVA The one-way analysis of variance (ANOVA) is used to determine whether the mean of a dependent variable is the same in two or more unrelated, independent groups. 47

48 Two Way ANOVA The two-way ANOVA compares the mean differences between groups that have been split on two independent variables (called factors). The primary purpose of a two-way ANOVA is to understand if there is an interaction between the two independent variables on the dependent variable 48

49 Unit IV: Syllabus Correlation Regression Multiple and Partial Correlation Multivariate Analysis 49

50 TM CORRELATION Correlation coefficient is a statistical measure of the degree to which changes to the value of one variable predict change to the value of another. When the fluctuation of one variable reliably predicts a similar fluctuation in another variable, there s often a tendency to think that means that the change in one causes the change in the other. Sub KDF2C- Code QUANTITATIVE - Sub Name TECHNIQUES FOR BUSINESS DECISION Slide number / Total 50 slides

51 TM Properties of Correlation Coefficient Coefficient of Correlation lies between -1 and +1. Coefficients of Correlation are independent of Change of Origin. Coefficients of Correlation possess the property of symmetry. Coefficient of Correlation is independent of Change of Scale. Co-efficient of correlation measures only linear correlation between X and Y. Types of Correlation 51

52 TM Methods of studying Correlation Scatter diagram. Karl pearson's coefficient of correlation. Spearman's Rank correlation coefficient. 52

53 TM Karl pearson's coefficient of correlation The formula to calculate Karl pearson s coefficient correlation is as follows: r n( x 2 n ) ( xy ( x) 2 x)( n( y y) 2 ) ( y) 2 The following video explains the correlation problem 53

54 TM Spearman's Rank correlation coefficient. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them. The following video explains the rank correlation problem 54

55 TM Regression In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships among variables. It includes many techniques for modeling and analysing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables (or 'predictors'). 55

56 TM Regression Lines If Y depends on X then the regression line is Y on X. Y is dependent variable and X is independent variable The regression equation Y on X is Y = a + bx, is used to estimate value of Y when X is known. If X depends on Y, then regression line is X on Y and X is dependent variable and Y is independent variable. The regression equation X on Y is X = c + dy is used to estimate value of X when Y is given Here a, b, c and d are constant. The following link explains the regression problem in detail 56

57 TM Partial Correlation Coefficient Definition: Partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. The following represents Geometrical interpretation of partial correlation for the case of N=3 observations and thus a 2-dimensional hyperplane 57

58 TM Multiple Correlation Coefficient The coefficient of multiple correlation is a measure of how well a given variable can be predicted using a linear function of a set of other variables. It is the correlation between the variable's values and the best predictions that can be computed linearly from the predictive variables. 58

59 TM Multivariate Analysis Multivariate Data Analysis refers to any statistical technique used to analyze data that arises from more than one variable. This essentially models reality where each situation, product, or decision involves more than a single variable. 59

60 Unit V: Syllabus Operations Research Linear Programming Problem Transportation Problem Assignment Problem 60

61 Operations Research Definition: Operations research (OR) is an analytical method of problem-solving and decision-making that is useful in the management of organizations. 61

62 LINEAR PROGRAMMING PROBLEM LPP deals with the optimization (maximization or minimization) of a function of decision variables (The variables whose values determine the solution of a problem are called decision variables of the problem) known as objective function. 62

63 LINEAR PROGRAMMING PROBLEM Objective function: The linear function Z=C1X1+C2X2+..+CnXn (in the general LPP) is called the objective function of the LPP. Solution of LPP: Any (X1,X2, Xn)which satisfies the constraints of the LPP is called a solution to the LPP. 63

64 PROCEDURE FOR FORMING A LPP MODEL: 1.identify the unknown decision variables. 2.identify the restrictions and constraints. 3.identify the objective function. 4.express the complete formulation. 64

65 CANONICAL AND STANDARD FORM OF LPP: CANONICAL AND STANDARD FORM OF LPP: Maxi Z=C1X1+C2X2+ +CnXn s.to a 11 X 1 +a 12 X 2 +.+a 1n X n b 1 a 21 X 1 +a 22 X 2 +.+a 2n X n b 2.. a m1 X 1 +a m2 X 2 +.+a mn X n b m X1,X2..Xn 0 This form of LPP is known is known as canonical form of LPP. 65

66 STANDARD FORM OF LPP: STANDARD FORM OF LPP: Maxi Z=C1X1+C2X2+ +CnXn s.to a 11 X 1 +a 12 X 2 +.+a 1n X n = b 1 a 21 X 1 +a 22 X 2 +.+a 2n X n = b 2.. a m1 X 1 +a m2 X 2 +.+a mn X n = b m X1,X2..Xn 0 This form of LPP is known is known as Standard form of LPP. 66

67 Artificial variables techniques Artificial variables techniques. There are many LPP s were slack variables can not provide initial basic Feasible solution to solve Such LPP. There are two methods (closely related) namely. 1.The Big M-methods or the method of penalties due to A.Charnes. 2.The two phase method due to Dantzig, order and Wolfe. 67

68 Transportation Problems The transportation problem is a special type of LPP where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations. Because of its special structure the usual simplex method is not suitable for solving transportation problems 68

69 Assignment Problem Assignment Problem Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a time, though with varying degree of efficiency. Let Cij be the cost if the ith person is assigned to the jth job. The problem is to find an assignment so that the total cost of performing all jobs is minimum. Problems of this kind are known as assignment problem. 69