MS-08 Management Programme ASSIGNMENT SECOND SEMESTER 2013 MS-08: Quantitative Analysis for Managerial Applications School of Management Studies INDIRA GANDHI NATIONAL OPEN UNIVERSITY MAIDAN GARHI, NEW DELHI 110 068
ASSIGNMENT QUESTIONS Course Code : MS - 8 Course Title : Quantitative Analysis for Managerial Applications Assignment Code : MS-8/SEM - II /2013 Coverage : All Blocks Note: Attempt all the questions and submit this assignment on or before 31 st October, 2013 to the coordinator of your study centre. 1. Statistical unit is necessary not only for the collection of data, but also for the interpretation and presentation. Explain the statement. 2. Find the standard deviation and coefficient of skewness for the following distribution Variable 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 Frequency 2 5 7 13 21 16 8 3 3. A salesman has a 60% chance of making a sale to any one customer. The behaviour of successive customers is independent. If two customers A and B enter, what is the probability that the salesman will make a sale to A or B. 4. To verify whether a course in Research Methodology improved performance, a similar test was given to 12 participants before and after the course. The original marks and after the course marks are given below: Original Marks 44 40 61 52 32 44 70 41 67 72 53 72 Marks after the course 53 38 69 57 46 39 73 48 73 74 60 78 Was the course useful? Consider these 12 participants as a sample from a population. 5. Write short notes on a) Bernoulli Trials b) Standard Normal distribution c) Central Limit theorem
ASSIGNMENT ANSWERS 1. Statistical unit is necessary not only for the collection of data, but also for the interpretation and presentation. Explain the statement. ANS. o Statistics may be defined as the science of collection, presentation analysis and interpretation of numerical data from the logical analysis. The word statistic is used to refer to - Numerical facts, such as the number of people living in particular area. - The study of ways of collecting, analyzing and interpreting the facts. o Statistics is the study of the collection, organization, analysis, interpretation and presentation of data. o It deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments. Statistical unit is necessary not only for the collection of data, but also for the interpretation and presentation. According to this statement there are four stages: 1) Collection of Data: o It is the first step and this is the foundation upon which the entire data set. o Careful planning is essential before collecting the data. 2) Presentation of data: o The mass data collected should be presented in a suitable, concise form for further analysis. o The collected data may be presented in the form of tabular or diagrammatic or graphic form. 3) Analysis of data: o The data presented should be carefully analyzed for making inference from the presented data such as measures of central tendencies, dispersion, correlation, regression etc. 4) Interpretation of data: o The final step is drawing conclusion from the data collected. o A valid conclusion must be drawn on the basis of analysis. A high degree of skill and experience is necessary for the interpretation. Statistical Methods broadly fall into three categories as shown in the following chart.
o Fact becomes knowledge, when it is used in the successful completion of a decision process. o Once you have a massive amount of facts integrated as knowledge, then your mind will be superhuman in the same sense that mankind with writing is superhuman compared to mankind before writing. o The following figure illustrates the statistical thinking process based on data in constructing statistical models for decision making under uncertainties. o The above figure depicts the fact that as the exactness of a statistical model increases, the level of improvements in decision-making increases. o That's why we need statistical data analysis. o Statistical data analysis arose from the need to place knowledge on a systematic evidence base. o This required a study of the laws of probability, the development of measures of data properties and relationships, and so on. o Statistical inference aims at determining whether any statistical significance can be attached that results after due allowance is made for any random variation as a source of error. o Intelligent and critical inferences cannot be made by those who do not understand the purpose, the conditions, and applicability of the various techniques for judging significance. o Statistics is a science assisting you to make decisions under uncertainties (based on some numerical and measurable scales). o Decision making process must be based on data neither on personal opinion nor on belief. o It is already an accepted fact that "Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write." So, let us be ahead of our time. o A body of techniques and procedures dealing with the collection, organization, analysis, interpretation, and presentation of information that can be stated numerically. o Descriptive statistics are used to organize or summarize a particular set of measurements. o In other words, a descriptive statistic will describe that set of measurements.
o Inferential statistics use data gathered from a sample to make inferences about the larger population from which the sample was drawn. o Initially derided by some mathematical purists, it is now considered essential methodology in certain areas. In number theory, scatter plots of data generated by a distribution function may be transformed with familiar tools used in statistics to reveal underlying patterns, which may then lead to hypotheses. Methods of statistics including predictive methods in forecasting are combined with chaos theory and fractal geometry to create video works that are considered to have great beauty. The process art of Jackson Pollock relied on artistic experiments whereby underlying distributions in nature were artistically revealed.[citation needed] With the advent of computers, statistical methods were applied to formalize such distribution-driven natural processes to make and analyze moving video art. Methods of statistics may be used predicatively in performance art, as in a card trick based on a Markov process that only works some of the time, the occasion of which can be predicted using statistical methodology. Statistics can be used to predicatively create art, as in the statistical or stochastic music invented by Iannis Xenakis, where the music is performance-specific. Thus we can say Statistical unit is necessary not only for the collection of data, but also for the interpretation and presentation. ==================================================================== ====================================================================
2. Find the standard deviation and coefficient of skewness for the following distribution Variable 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 Frequency 2 5 7 13 21 16 8 3 ANS. FOR STANDARD DEVVIATION Variable f M.P. X d = (X-22.5)/5 fd fd 2 C.F. 0-5 2 2.5-4 -8 32 2 5-10 10-15 15-20 5 7.5-3 -15 45 7 7 12.5-2 -14 28 14 13 17.5-1 -13 13 27 20-25 21 22.5 0 0 0 48 25-30 16 27.5 1 16 16 64 30-35 8 32.5 2 16 32 72 35-40 3 37.5 3 9 27 75 N=75 fd=-9 f d 2 =193 Now, For, standard Deviation, σ = fd2 N fd N 2 i
Now putting the values from above table in to above equation σ = 193 75 2 9 75 5 σ = 2. 57 0. 0144 5 σ = 1. 59862 5 Standard Deviation σ = 7. 999 COEFFICIENT OF SKEWNESS c.f. SK = Q3Q12Mean Q3 Q1 Q 1 = Size of N/4 th observation = 75/4= 18.75 th observation lies in the class 15-20 Now, Q 1 = L + N 4 pcf i f
Now putting the values in to above equation Q = 15 + 18.75 14 7 5 = 15 + 0.6785 x 5 Q 1 = 78.39 Now, Q 2 = Size of N/2 th observation = 75/2= 37.5 th observation lies in the class between 30-35 So Q 2 = Median = L + N 2 pcf i f Now putting the values in to above equation Q = 30 + 37.5 64 8 5 = 30 3.3125 X 5 Q 2 = 133.44 Now,
Q 3 = Size of 3N/4 th observation = 3(75)/4= 56.25 th observation lies in the class between 35-40 So, Q 3 = L + 3N 4 pcf i f Now putting the values in to above equation Q = 35 + 56.25 72 3 5 = 35 56.25 X 5 Q 3 = 148.75 Now, For Coefficient of skewness: c.f. SK = Q3Q12Mean Q3 Q1 Now putting the values in to above equation SK = 148.7578.392(133.44) 148.75 78.39 = 227.14 266.88 70.36 = -0.564
So, C.F of SK = - 0.564 ==================================================================== ====================================================================
3. A salesman has a 60% chance of making a sale to any one customer. The behaviour of successive customers is independent. If two customers A and B enter, what is the probability that the salesman will make a sale to A or B. ANS. The probability that the salesman make sale to any one customer = 60% = 0.60 The probability that the salesman fails to make sales to any one customer = 1-60 = 0.40 The behavior of successive customers is independent event therefore, 1) The probability that the salesman fails to make sales to any one customer in A & B = 0.40 X 0.40 = 0.16 = 16% 2) The probability that the salesman make sale to at least any one customer from A & B = 1 Probability that salesman sale to none of customer = 1 0.16 = 0. 84 = 84%
4. To verify whether a course in Research Methodology improved performance, a similar test was given to 12 participants before and after the course. The original marks and after the course marks are given below: Original Marks 44 40 61 52 32 44 70 41 67 72 53 72 Marks after the course 53 38 69 57 46 39 73 48 73 74 60 78 Was the course useful? Consider these 12 participants as a sample from a population. ANS. Let us take the hypothesis that there is no difference in the marks obtained before and after the course, i.e. the course has not been useful Applying t-test (difference formula):
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ANS. 5. Write short notes on a) Bernoulli Trials o In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure". o Repeated independent trials in which there can be only two outcomes are called Bernoulli trials in honor of James Bernoulli (1654-1705). Bernoulli trials lead to the binomial distribution. If a number of trials are large, then the probability of k successes in n trials can be approximated by the Poisson distribution. The binomial distribution and the Poisson distribution are closely approximated by the normal (Gaussian) distribution. These three distributions are the foundation of much of the analysis of physical systems for detection, communication and storage of information. o In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment, is conducted. o The mathematical formalization of the Bernoulli trial is known as the Bernoulli process. o Since a Bernoulli trial has only two possible outcomes, it can be framed as some "yes or no" question. For example: Did the coin land heads? Was the newborn child a girl? Therefore, success and failure are merely labels for the two outcomes, and should not be construed literally. The term "success" in this sense consists in the result meeting specified conditions, not in any moral judgment. Examples of Bernoulli trials include: Flipping a coin. In this context, obverse ("heads") conventionally denotes success and reverse ("tails") denotes failure. A fair coin has the probability of success 0.5 by definition. Rolling a die, where a six is "success" and everything else a "failure". In conducting a political opinion poll, choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum. The Assumptions of Bernoulli Trials i. Each trial results in one of two possible outcomes, denoted success (S) or failure (F). ii. The probability of S remains constant from trial-to-trial and is denoted by p. Write q = 1 p for the constant probability of F. iii. The trials are independent. o Independent repeated trials of an experiment with exactly two possible outcomes are called Bernoulli trials. Call one of the outcomes "success" and the other outcome "failure".
o Let p be the probability of success in a Bernoulli trial. Then the probability of failure q is given by q = 1 - p. o Random variables describing Bernoulli trials are often encoded using the convention that 1 = "success", 0 = "failure". o Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number n of statistically independent Bernoulli trials, each with a probability of success p, and counts the number of successes. o A random variable corresponding to a binomial is denoted by B (n, p), and is said to have a binomial distribution. o The probability of exactly k successes in the experiment B (n, p) is given by: o Bernoulli trials may also lead to negative binomial distributions (which count the number of successes in a series of repeated Bernoulli trials until a specified number of failures are seen), as well as various other distributions. o When multiple Bernoulli trials are performed, each with its own probability of success, these are sometimes referred to as Poisson trials.
ANS. b) Standard Normal distribution o The standard normal distribution is a special case of the normal distribution. o It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one. o The normal random variable of a standard normal distribution is called a standard score or a z score. o Every normal random variable X can be transformed into a z score via the following equation: z = (X - μ) / σ o Where X is a normal random variable, μ is the mean, and σ is the standard deviation. o In other words, A standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Areas under this curve can be found using a standard normal table (Table A in the Moore and Moore & McCabe textbooks). All introductory statistics texts include this table. o Some do format it differently. From the 68-95-99.7 rule we know that for a variable with the standard normal distribution, 68% of the observations fall between -1 and 1 (within 1 standard deviation of the mean of 0), 95% fall between -2 and 2 (within 2 standard deviations of the mean) and 99.7% fall between -3 and 3 (within 3 standard deviations of the mean). o No naturally measured variable has this distribution. o However, all other normal distributions are equivalent to this distribution when the unit of measurement is changed to measure standard deviations from the mean. (That's why this distribution is important--it's used to handle problems involving any normal distribution.) o Recall that a density curve models relative frequency as area under the curve. o Assume throughout this document then that we are working with a variable Z that has a standard normal distribution. o The letter Z is usually used for such a variable, the small letter z is used to indicate the generic value that the variable may take.
For Example- Question: What is the relative frequency of observations above -1.48? Identify the range of values described by "above -1.48" (shaded green). Identify the area you need to find (shaded blue). It appears to be about 95%. Use the value -1.48 to look up an area in your table. However, be careful. Doing so gives you 0.0694--this is nowhere near 0.95--our initial guess. That's because the table is oriented to find areas under the curve to the left of.. So, in fact, looking up -1.48 has found the answer to the question What is the relative frequency of measurements falling below -1.48. This range, z < -1.48 (in gray) and the associated area 0.0694 (in purple) are shown below. There are two ways to proceed. They are, of course, equivalent.
Since 0.0694 of the observations fall below -1.48, the remaining 0.9306 = 1-0.0694 must fall above -1.48. Since the total area under the curve is exactly 1, and the purple area is 0.0694, the blue area must be 1-0.0694 = 0.9306. In other words, subtraction from 1 is necessary. 93.06% of the observations fall above -1.48. (For any normal distribution, 0.9306 or 93.06% of the observations fall above 1.48 times the standard deviation below the mean.) Answer: 0.9306 or 93.06%. SAI ASSIGNMENTS: +919510895807 18
ANS. c) Central Limit theorem o In probability theory, the central limit theorem (CLT) states that, given certain conditions, the mean of a sufficiently large number of iterates of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed. o The central limit theorem has a number of variants. o In its common form, the random variables must be identically distributed. o In variants, convergence of the mean to the normal distribution also occurs for nonidentical distributions, given that they comply with certain conditions. o The Central Limit Theorem describes the characteristics of the "population of the means" which has been created from the means of an infinite number of random population samples of size (N), all of them drawn from a given "parent population". o The Central Limit Theorem predicts that regardless of the distribution of the parent population: 1) The mean of the population of means is always equal to the mean of the parent population from which the population samples were drawn. 2) The standard deviation of the population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size (N). 3) The distribution of means will increasingly approximate a normal distribution as the size N of samples increases. o Thus, the Central Limit Theorem explains the ubiquity of the famous bell-shaped "Normal distribution" (or "Gaussian distribution") in the measurements domain. o If x 1, x 2,..., x n are n random variables which are independent and having the same distribution with mean p. and standard deviation σ, then if, the limiting distribution of the standardised mean n SAI ASSIGNMENTS: +919510895807 19
o In practice, if the sample size is sufficiently large, we need not know the population distribution because the central limit theorem assures us that the distribution of x can be approximated by a normal distribution. o A sample size larger than 30 is generally considered to be large enough for this purposes. o Many practical samples are of size higher than 30. o In all these cases, we know that the sampling distribution of the mean can be approximated by a normal distribution with an expected value equal to the population mean and a variance which is equal to the population variance divided by the sample size n. o We need to use the central limit theorem when the population distribution is either unknown or known to be non-normal. ================================================================= ================================================================= SAI ASSIGNMENTS: +919510895807 20