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

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                            MBA and MBA (Banking & Finance)

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.