Question No. 5 - MMPC-005: Quantitative Analysis for Managerial Applications - MBA and MBA (Banking & Finance)

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MMPC-005: Quantitative Analysis for Managerial Applications

MMPC-005/TMA/JULY/2022

Question No. 5. Write short notes on any two of the following:-   
(a) Mathematical Properties of Arithmetic Mean 

(b) Stratified Sampling 

Stratified sampling is more complex than simple random sampling, but where applied properly, stratification can significantly increase the statistical efficiency of sampling.

The concept: 

Suppose we are interested in estimating the demand of non aerated beverages in a residential colony. We know that the consumption of these beverages has some relationship with the family income and that the families residing in this colony can be classified into three categories-viz., high income, middle income and low income families. If we are doing a sampling study we would like to make sure that our sample does have some members from each of the three categories-perhaps in the same proportion as the total number of families belonging to that category-in which case we would have used proportional stratified sampling. On the other hand, if we know that the variation in the consumption of these beverages from one family to another is relatively large for the low income category whereas there is not much variation in the high income category, we would perhaps pick up a smaller than proportional sample from the high income category and a larger than proportional sample from-the low income category. This is what is done in disproportional stratified sampling. The basis for using stratified sampling is the existence of strata such that each stratum is more homogeneous within and markedly different from another stratum. The higher the homogeneity within each stratum, the higher the gain in statistical efficiency due to stratification. 

What are strata?: 

The strata are so defined that they constitute a partition of the population-i.e., they are mutually exclusive and collectively exhaustive. Every element of the population belongs to one stratum and not more than one stratum, by definition. This is shown in Figure II in the form of a Venn diagram, where three strata have been shown. A stratum can therefore he conceived of as a sub-population which is more homogeneous than the complete population-the members of a stratum, are similar to each other and are different from the members of another stratum in the characteristics that we are measuring. 

Proportional stratified sampling: 

Sampling Methods After defining the strata, a simple random sample is picked up from each of the strata. If we want to have a total sample of size 100, this number is allocated to the different strata-either in proportion to the size of the stratum in the population or otherwise. If the different strata have similar variances of the characteristic being measured, then the statistical efficiency will be the highest if the sample sizes for different strata are in the same proportion as the size of the respective stratum in the population. Such a design is called proportional stratified sampling and is shown in Table 4 below. 

Disproportional stratified sampling: If the different strata in the population have unequal variances of the characteristic being measured, then the sample size allocation decision should consider the variance as well. It would be logical to have a smaller sample from a stratum where the variance is smaller than from another stratum where the variance is higher. In fact, if

Stratified sampling in practice: 

Stratification of the population is quite common in managerial applications because it also allows to draw separate conclusions for each stratum. For example, if we are estimating the demand for a non-aerated beverage in a residential colony and have stratified the population based on the family income, then we would have data pertaining to each stratum which might be useful in making many marketing decisions. Stratification requires us to identify the strata such that the intra-stratum differences are as small as possible and inter-strata differences as large as possible. However, whether a stratum is homogeneous or not-in the characteristic that we are measuring e.g. consumption of non-aerated beverage in the family in the previous example-can be known only at the end of the study whereas stratification is to be done at the beginning of the study and that is why some other variable like family income is to be used for stratification. This is based on the implicit assumption that family income and consumption of non-aerated beverages are very closely associated with each other. If this assumption is true, stratification would increase the statistical efficiency of sampling. In many studies, it is not easy to find such associated variables which can be used as the basis for stratification and then stratification may not help in increasing the statistical efficiency, although the cost of the study goes up due to the additional costs of stratification. 

(c) Exponential Distribution 

Time between breakdown of machines, duration of telephone calls, life of an electric bulb are examples of situations where the Exponential distribution has been found useful. In the previous unit, while discussing the discrete probability distributions, we have examined the Poisson process and the resulting Poisson distribution. In the Poisson process, we were interested in the random variable of number of occurrences of an event within a specific time or space. Thus, using the knowledge of Poisson process, we have calculated the probability that 0, 1, 2 …. accidents will occur in any month. Quite often, another type of random variable assumes importance in the context of a Poisson process. We may be interested in the random variable of the lapse of time before the first occurrence of the event. Thus, for a machine, we note that the first failure or breakdown of the machine may occur after 1 month or 1.5 months etc. The random variable of the number of failures within a specific time, as we have already seen, is discrete and follows the Poisson distribution. The variable, time of first failure, is continuous and the Exponential p.d.f. characterises the uncertainty. If any situation is found to satisfy the conditions of a Poisson process, and if the average occurrence of the event of interest is m per unit time, then the number of occurrences in a given length of time t has a Poisson distribution with parameter mt, and the time between any two consecutive occurrences will be Exponential with parameter m. This can be used to derive the p.d.f. of the Exponential distribution.

 
If we assume that the occurrence of an event corresponds to customers arriving for servicing, then the time between the occurrence would correspond to the inter-arrival time (IAT), and m would correspond to the arrival rate. Exponential has been used widely to characterise the IAT distribution. The Exponential p.d.f. is also used for characterising service time distributions. The parameter 'm' in that case, corresponds to the service rate. We take up an example to show the probability calculations using the Exponential p.d.f. In the final section of this unit, we will be illustrating through an example, the use of the Exponential distribution in decision making.

Example:

The distribution of the total time a light bulb will burn from the moment it is first put into service is known to be exponential with mean time between failure of the bulbs equal to 1000 hrs. What is the probability that a bulb will burn more than 1000 hrs.

Solution:

 

(d) Time Series Analysis

Time series analysis is one of the most powerful methods in use, especially for short term forecasting purposes. From the historical data one attempts to obtain the underlying pattern so that a suitable model of the process can be developed, which is then used for purposes of forecasting or studying the internal structure of the process as a whole. We have already seen in earlier unit that a variety of methods such as subjective methods, moving averages and exponential smoothing, regression methods, causal models and time series analysis are available for forecasting. Time series analysis looks for the dependence between values in a time series (a set of values recorded at equal time intervals) with a view to accurately identify the underlying pattern of the data.

In the case of quantitative methods of forecasting, each technique makes explicit assumptions about the underlying pattern. For instance, in using regression models we had first to make a guess on whether a linear or parabolic model should be chosen and only then could we proceed with the estimation of parameters and model-development. We could rely on mere visual inspection of the data or its graphical plot to make the best choice of the underlying model. However, such guess work, through not uncommon, is unlikely to yield very accurate or reliable results. In time series analysis, a systematic attempt is made to identify and isolate different kinds of patterns in the data. The four kinds of patterns that are most frequently encountered are horizontal, non-stationary (trend or growth), seasonal and cyclical. Generally, a random or noise component is also superimposed.

We shall first examine the method of decomposition wherein a model of the time-series in terms of these patterns can be developed. This can then be used for forecasting purposes as illustrated through an example. A more accurate and statistically sound procedure to identify the patterns in a time-series is through the use of auto-correlations. Auto-correlation refers to the correlation between the same variable at different time lags and was discussed in Unit 18. Auto-correlations can be used to identify the patterns in a time series and suggest appropriate stochastic models for the underlying process. A brief outline of common processes and the Box-Jenkins methodology is then given. 

Question No. 3 - MMPC-005: Quantitative Analysis for Managerial Applications - MBA and MBA (Banking & Finance)

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MMPC-005: Quantitative Analysis for Managerial Applications

MMPC-005/TMA/JULY/2022

Question No. 3. What do you understand by Primary Data? What are the various methods of collecting primary data? Also, mention what points to be kept in mind while designing the questionnaire?

Data used in statistical study is termed either “Primary” or “secondary” depending upon whether it was collected specifically for the study in question or for some other purpose. When the data used in a statistical study was collected under the control and supervision of the investigation, such type of data is referred to as “Primary data”. When the data was not collected by the investigator, but is derived from other sources then such data is referred to as “secondary data”.

The difference between primary and secondary data is only in terms of degree. For example, data which is primary in the hands of one become secondary in the hands of another. Suppose in investigator wants to study the working conditions of labour in a big industrial concerned. If he collects the data himself or through his agent, then this data is referred to as primary data. But if this data is used by someone else, then this data becomes secondary data.

METHOD OF COLLECTING PRIMARY DATA

Primary data may either be collected through the observation method or through the questionnaire method. In the observation, the investigator asks no questions, but he simply observes the phenomenon under consideration, and records the necessary data. Sometimes individuals make the observation; on other occasion, mechanical and electronic devices do the job. 

In the observation method, it may be difficult to produce accurate data. Physical difficulties on the part of the observer may result in errors. Because of these limitations in the observation method, the questionnaire method is most widely used for collecting data. In the questionnaire method, the investigator draws up a questionnaire containing all the relevant questions which he wants to ask from his respondents, and accordingly records the responses. Questionnaire method may be conducted through personal interview, or by mail or telephone.

Personal Interviews In this method the interviewer sits face-to-face with the respondent and records his responses. In this method, the information is likely to be more accurate and reliable because the interviewer can clear up doubts and cross-checks the respondents. This method is time consuming and can be very costly if the number of respondents is large and widely distributed.

Mail Questionnaire In this method a list of questions (questionnaire) is prepare and mailed to the respondents. The respondents are expected to fill in the questionnaire and send it back to the investigator. Sometimes, mail questionnaire are placed in respondents’ hands through other means such as attaching them to consumers’ products or putting them in newspapers or magazines. This method can be easily adopted where the field of investigation is very vast and the respondents are spread over a wise geographical area. But this method can be adopted only where the respondents are literate and can understand written question and answer them.

Telephone  In this method the investigator asks the relevant questions from the respondents over the telephone. This method is less expensive but it has limited application since only those respondents can be interviewed who have telephones; moreover, very few questions can be asked on telephone 

 

The questionnaire method is a very efficient and fast method of collecting data. But it has a very serious limitation as it may be extremely difficult to collect data on certain sensitive aspects such as income, age or personal life details, which the respondent may not be willing to share with the investigator. This is so with other methods also different people may interpret the questions differently and consequently there may be errors and inaccuracies in data collection.

DESIGNING OF QUESTIONNAIRE

The success of collecting data through a questionnaire depends mainly on how skillfully and imaginatively the questionnaire has been designed. A badly designed questionnaire will never be able to gather the relevant data. In designing the questionnaire, some of the important points to be kept in mind are: 

1. Covering letter : Every questionnaire should be contain a covering letter. The covering letter should highlight the purpose of study and assure the respondent the all responses will be kept confidential. It is desirable that some inducement or motivation is provided to the respondent for better response. The objectives of the study and questionnaire design should be such that the respondent derives a sense of satisfaction through his involvement. 

2. Number of questions should be kept to the minimum: The fewer the question, the greater the chances of getting a better responses and having all the questions answered. Otherwise the respondent may feel disinterested and provide inaccurate answers particularly towards the end of the questionnaire. Informing the question, the investigator has to take into consideration several factors such as the purpose of study, the time and resources available. As a rough indication, the number of questions should be between 15 to 40. In case the number of questions is more than 25, it is desirable that the questionnaire be divided into various part to ensure clarity. 

3. Questions should be simple, short and unambiguous: The questions should be simple, short, easy to understand and such that their answers are unambiguous. For example, if the question is ‘Are you literate? The respondent may have doubts about the meaning of literacy. To some literacy may mean a university degree whereas to others even the capacity to read and write may mean literacy. Hence it is desirable to specify whether you have passed (a) high school (b) graduation (c) post graduation etc. Questions can be of Yes/No type, or of multiple choice depending on the requirement o the investigator. Open-ended questions should generally be avoided. 

4. Questions of sensitive or personal nature should be avoided: The questions should not be such as would require the respondent to disclose any private, personal or confidential information. For example, questions relating to sales, profits, material happiness etc. should be avoided as far as possible. If such questions are necessary in the survey, an assurance should be given to the respondent that the information provided shall be kept strictly confidential and shall not be used at any cost to their disadvantage. 

5. Answers to questions should not require calculations: The questions should be framed in such a way that their answers do not require any calculations.

6. Logical arrangement Collection of Data The questions should be logically arranged so that there is a continuity of responses and the respondent does not feel the need to refer back to the previous questions. It is desirable that the questionnaire should begin with some introductory questions followed by vital questions crucial to he survey and ending with some light questions so that the overall impression of the respondent is a happy one. 

7. Cross-check and Footnotes: The questionnaire should contain some such questions which act as a cross-check to the reliability of the information provided. For example, when a question relating to income is asked, it is desirable to include a question : “are you an income tax assessee?”

For the purpose of clarity, certain questions which might create a doubt in the mind of respondents, it is desirable to give footnotes. The purpose of footnotes is to clarify all possible doubts which may emerge from the questions and cannot be removed while answer them. For example, if a question relates to income limit like 1000-2000, 2000—3000; etc., a person getting exactly Rs. 2,000 should know in which income class he has to place himself.

Question No. 2 - MMPC-005: Quantitative Analysis for Managerial Applications - MBA and MBA (Banking & Finance)

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MMPC-005: Quantitative Analysis for Managerial Applications

MMPC-005/TMA/JULY/2022

Question No. 2. Why is forecasting so important in business? Explain the application of forecasting for long term decisions. 

The future is inherently uncertain and since time immemorial man has made attempts to unravel the mystery of the future. In the past it was the crystal gazer or a person allegedly in possession of some supernatural powers who would make predications about the things-to be-major events or the rise and fall of kings. In today's world, predictions are being made daily in the realm of business, industry and politics. Since the operation of any capital enterprise has a large lead time (1-5 years is typical), it is clear that a factory conceived today is for some future demand and the whole operation is dependent on the actual demand coming up to the level projected much earlier. During this period many circumstances, which might not even have been imagined, could come up. For instance, there could be development of other industries, or a major technological breakthrough that may render the originally conceived product obsolete; or a social upheaval and change-of government may redefine priorities of growth and development; or an unusual weather condition like drought or floods may alter completely the buying potential of the originally conceived market. This is only a partial list to suggest how uncertainties from a variety of sources can enter to make the task of prediction of the future extremely difficult.

It is proper at this stage to emphasise the distinction between prediction and forecasting. Forecasting generally refers to the scientific methodology that often uses past data along with some well-defined assumptions or 'model' to come up with a 'forecast' of future demand. In that sense, forecasting is objective. A prediction is a subjective estimate made by an individual by using his intuitive 'hunch' which may in fact come out true. But the fact that it is subjective (A's prediction may be different from B's and C's) and nonrealisable as a Well-documented computer programme (which could be used by anyone) deprives it of much value. This is not to discount. the role of intuition or subjectivity in practical decision-making. In fact, for complex long term decisions, intuitive methods such as the Delphi technique are most popular. The opinion of a well informed, educated person is likely to be reliable, reflecting the well-considered contribution of a host of complex factors in a relationship that may be difficult to explicitly quantify. Often forecasts are modified based on subjective judgment and experience to obtain predictions used in planning and decision making.

The primary purpose of forecasting. is to provide valuable information for planning the design and operation of the enterprise. Planning decisions may be classified as long term, medium term and short term. Long term decisions include decisions like plant expansion or. new product introduction which may require new technologies or a. complete transformation in social or moral fabric of society. Such decisions are generally, characterised by lack of quantitative information and absence of historical data on which to base, the forecast of future events. Intuition and the collected opinion of. experts in the field generally play a significant role in developing forecasts for such decisions.

Technological Forecasting 

Technological growth is often haphazard, especially in developing countries like India. This is because Technology seldom evolves and there are frequent technology transfers -due to imports of knowhow resulting in a leap-frogging phenomenon. In spite of this, it is generally seen that logarithms of many technological variables show linear trends with time, showing exponential growth. Some extrapolations reported by Rohatgi et al. are 

• Passenger kms carried by Indian Airlines (Figure I) 

• Fertilizer applied per hectare of cropped area (Figure II) 

• Demand and supply of petroleum crude (Figure III) 

• Installed capacity of electricity generation in millions of KW (figure IV). 

Figure I: Passenger Km Carried by Indian Air Lines 

Delphi 

This is a subjective method relying on the opinion of experts designed to minimise bias and error of judgment. A Delphi panel consists of a number of experts with an impartial leader or coordinator who organises the questions. Specific questions (rather than general opinions) with yes-no or multiple type answers or specific dates/events are sought from the experts. For instance, questions could be of the following kind : 

• When do you think the petroleum reserves of the country would be exhausted? (2020,2040, 2060) 

• When would the level of pollution in Delhi exceed danger limit? (as defined by a particular agency)? 

• What would the population of India be in 2020, 2040 and 2060? 

• When would fibre optics become a commercial viability for communication? 

A summary of the responses of the participants is sent to each expert participating in the Delphi panel after a statistical analysis. For a forecast of when an event is likely to happen, the most optimistic and pessimistic estimates along with a distribution of other responses is given to the participant. On the basis of this information the experts may like to revise their earlier estimates and give revised estimates to the coordinator. It may be mentioned that the identities of the experts are not revealed to each other so that bias or influence by reputation is kept to a minimum. Also the feedback response is statistical in nature without revealing who made which forecast. The Delphi method is an iterative procedure in which revisions are carried out by the experts till the coordinator gets a stable response.

The method is very efficient, if properly conducted, as it provides a systematic framework for collecting expert opinion. By virtue of anonymity, statistical analysis and feedback of results and provision for forecast revision, results obtained are free of bias and generally reliable. Obviously, the background of the experts and their knowledge of the field is crucial. This is where the role of the coordinator in identifying the proper experts is important.

Opinion Polls 

Opinion polls are a very common method of gaining knowledge about consumer tastes, responses to a new product, popularity of a person or leader, reactions to an election result or the likely future prime minister after the impending polls. In any opinion poll two things are of primary importance. First, the information that is sought and secondly the target population from whom the information is sought. Both these factors must be kept in mind while designing the appropriate mechanism for conducting the opinion poll. Opinion polls may be conducted through 

• Personal interviews. 

• Circulation of questionnaires. 

• Meetings in groups. 

• Conferences, seminars and symposia. 

The method adopted depends largely on the population, the kind of information desired and the budget available. For instance, if information from a very large number of people is to be collected a suitably designed questionnaire could be mailed to die people concerned. Designing a proper questionnaire is itself a major task. Care should be taken to avoid ambiguous questions. Preferably, the responses should be short one word answers or ticking an appropriate reply from a set of multiple choices. This makes the questionnaire easy for the respondent to fill and also easy for the analyst to analyse. For example, the final analysis could be summarised by saying 

• 80% of the population expressed opinion A, 

• 10% expressed opinion B, 

• 5% expressed opinion C, 

• 5% expressed no opinion. 

Similarly in the context of forecasting of product demand, it is common to arrive at the sales forecast by aggregating the opinion of area salesmen. The forecast could be modified based on some kind of rating for each salesman or an adjustment for environmental uncertainties. Decisions in the area of future R&D or new technologies too are based on the opinions of experts. The Delphi method treated in this Section is just an example of a systematic gathering of opinion of experts in the concerned field. The major advantage of opinion polls lies in the fact that a well formed opinion considers the multifarious subjective and objective factors which may not even be possible to enumerate explicitly, and yet they may have a bearing on the concerned forecast or question. Moreover the aggregation of opinion polls tends to eliminate the bias that is bound to be present in any subjective, human evaluation. In fact for long term decisions, opinion polls of opinions of the experts constitute a very reliable method for forecasting and planning. 

Question No. 4 - MMPC-005: Quantitative Analysis for Managerial Applications - MBA and MBA (Banking & Finance)

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MMPC-005: Quantitative Analysis for Managerial Applications

MMPC-005/TMA/JULY/2022

Question No. 4. The means of two large samples of sizes 1000 and 2000 are 67.5 and 68.0 respectively. Test the quality of the means of the two populations each with standard deviation of 2.5. (z table value at α0.05= -1.96).  

Question No. 1 - MMPC-005: Quantitative Analysis for Managerial Applications - MBA and MBA (Banking & Finance)

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MMPC-005: Quantitative Analysis for Managerial Applications

MMPC-005/TMA/JULY/2022

Question No. 1. The income of a group of 10,000 persons was found to be normally distributed with mean Rs.750 per month and a standard deviation of Rs. 50, show that of this group about 95% has income exceeding Rs. 668 and only 5% had income exceeding Rs. 832. (area between 750 and 668 = 0.4495, area between 750 and 832 = 0.4495).