ms-08 jan june 2016 solved assignment
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SAMPLE OF MBA SOLVED ASSIGNMENT
JAN-JUNE 2016
Course Code MS - 08 Course Title Quantitative Analysis for Managerial
Applications Assignment Code MS-08/TMA/SEM - I/2016 Assignment Coverage All Blocks
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Q1.Explain the concept of Maxima & Minima and discuss its managerial applications.
Concept of Maxima & Minima and its managerial applications-
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Maxima Problems
In maxima problems you are trying to maximize something. A typical problem involves a farmer with 100 feet of fence who wants to enclose as much area as possible. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ 50 - P. If 50 - P = 0, then P = 50. If 50 feet of fence runs parallel to the barn, the area will be maximized. If P = 50, the area is 50 x 25 = 1250. If P is a little longer, the area is 52 x 24 = 1248. If P is a little shorter, the area is 48 x 26 = 1248. Clearly, P = 50 gives a maximum area.
Minima Problems
Minima problems involve equations where we want to minimize something. Examples include finding the shortest ladder that can go over a 10-foot fence to rest on a wall two feet behind the fence. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------of the radius, then use this to express the formula for the surface area of a cylinder, differentiate and set to zero and solve.
Maxima and Minima Problems
The general solution is to express the quantity you -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------). If the double derivative is negative, you have found a maximum value. If the double derivative is positive, you have found a minimum.
The terms maxima and minima refer to extreme values of a function, that is, the maximum and minimum values that the
function attains. Maximum means upper bound or largest possible quantity. The absolute maximum of a function is the
largest number contained in the range of the function. That is, if f(a) is greater than or equal to f(x), for all x in the domain of
the function, then f(a) is the absolute maximum.
For example, ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- . If f(a) is less than or equal to f(x), for all x in the domain of the function, then f(a) is an absolute minimum. As an example, f(x) = 32x2 - 32x - 6 has an absolute minimum of -22, because every value of x produces a value greater than or equal to -22.
In some cases, a function will have no absolute maximum or minimum. For instance ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ; if f(b) is less than or equal to f(b ± h), then f(b) is a relative minimum. For example, -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------. It also has a relative maximum (point B), but no absolute maximum.
Finding the maxima and minima, both absolute and relative, of various functions represents an important class of problems solvable by use of differential calculus------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- negative slope, and the function is said to be decreasing. Precisely at the point where the function changes from increasing to decreasing or from decreasing to increasing, the tangent line is horizontal (has slope 0), and the derivative is zero. (With reference to figure 1, the function is decreasing to the left of point A, as well as between points B and C, and increasing between points A and B and to the right of point C).
In order to find maximum and minimum points, first find the values of the independent variable for which the derivative of
the function is zero, then substitute them in the original function to obtain the corresponding maximum or minimum values
of the function. Second, inspect the behavior of the derivative to the left and right of each point. If the derivative is negative
on the left and positive on the right, the point is a minimum. If the derivative is positive on the left and negative on the right,
the point is a maximum. Equivalently, find the second derivative at each value of the independent variable that corresponds
to a maximum or minimum; if the second derivative is positive, the point is a minimum, if the second derivative is negative
the point is a maximum.
A wide variety of problems can be solved by finding maximum or minimum values of functions. For example, --------------------------------------------------------------------------------------------------------------------------------------------------------
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- . Substitution of this value into the equation of the semicircle gives y = 1⁄2(r √ +2), that is, y = x. Thus, the maximum area of a rectangle inscribed in a semicircle is A = 2xy = r2.
There are numerous practical applications in which it is desired to find the maximum or minimum value of a particular quantity. Such applications exist in economics, business, and engineering. Many can be solved using the methods of differential calculus described above. For example, in any manufacturing business it is usually possible to express profit as a
function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. In other cases, the shape of a container may be determined by minimizing the amount of material required to manufacture it. The design of piping systems is often based on minimizing pressure drop which in turn minimizes required pump sizes and reduces cost. The shapes of steel beams are based on maximizing strength.
Finding maxima or minima also has important applications in linear algebra and game theory. For example, linear programming consists of maximizing (or minimizing) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- automobile given certain known constraints on the cost of each part, and the time spent by each laborer, all of which may be interdependent. Regardless of the application, though, ---------------------------------------------------------------------------------- is expressing the problem in mathematical terms.
Numerical Example-
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Q2.The average sales of a product for a particular week, excluding Sunday, were reported by the
city departmental store as 150 units. Sunday being a national festival, there was heavy rush of sales
which inflated average sales for the entire week to 210 units. Find the sales for Sunday.
Solution-
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Q3.A box contains 2 red, 3 black, and 5 white balls. If 3 balls are drawn at random without
replacement, find the probabilities that a) all 3 are black, b) two are red and one black, c) at least
one is white.
Solution-
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4. A firm is manufacturing two brands, a and B, of battery cells. It claims that the average life of
brand A cells is more than that of brand B cells by 100 hrs, the variances of the two brands being
the same. To maintain this standard, two independent samples of 12 cells of each brand are
selected on the 20th of every month and a t value of the difference of sample means computed. The
firm is satisfied with its claim if the computed t value falls between ±t0.025. A sample of 12 cells of
brand A gives a mean life of 1200 hrs and variance of 49 hrs, and that of 12 cells of brand B gives
a mean life of 1095 hrs and variance of 64 hrs. Comment on the outcome of the sample results.
Solution-
--------- ------- ------ -------- Variance
(hrs.) ------
A -------- 12 --------- ------- --------
----- 20th ----- ---------- 64 --------
A/Q
Here, A : B
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Hence,
------------------------- = 1095
= 12 --------------------------------
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Test Procedure-
If we assumed that and --------------------------------------------------------------------
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--------------------- any of-
Depending upon desire of the researches or -------------------------------------------------------
-------------------------------------------------------------------------- data is collected and a t
statistics generated using the formula-
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The histograms, also the variance are relatively similar ----------------------
------------------------------------------------------------------------------------ variance data
analysis tool to test the following null hypothesis.
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And Brand B cell is 4.25 of select the market.
Hence,
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Q5.Discuss the stochastic models developed by Box and Jenkins for time series analysis.
Box - Jenkins Analysis refers to a systematic method of identifying, fitting, checking, and using integrated autoregressive,
moving average (ARIMA) time series models. The method is appropriate for time series of medium to long length (at least
observations).
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A statistical process is stationary if the probability distribution is the same for all starting values of t. This implies that the
mean and variance are constant for all values of t. A series that exhibits a simple trend is not stationary because the values of
the series depend on t. A stationary stochastic process is completely defined by its mean, variance, and autocorrelation
function. One of the steps in the Box - Jenkins method is to transform a non-stationary series into a stationary one.
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A crucial step in an appropriate model selection is the determination of optimal model parameters. One criterion is that the
sample ACF and PACF, calculated from the training data should match with the corresponding theoretical or actual values.
Other widely used measures for model identification are Akaike Information Criterion (AIC) and Bayesian Information
Criterion (BIC) which are defined below.
Here n is the number of effective observations, used to fit the model, p is the number of parameters in the model and is
the sum of sample squared residuals. The optimal model order is chosen by the number of model parameters, which
minimizes either AIC or BIC. Other similar criteria have also been proposed in literature for optimal model identification.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Auto-regressive Models
In such models, the current value of the process is expressed as a finite, linear aggregate of previous values of the process and a random shock or error at. Let us denote the value of a process at equally spaced times t, t-1, t - 2... by Zt, Zt-1, Zt-2 ……
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is called an auto-regressive (AR) process of order p. The reason for this name is that equation (1) represents a regression of the variable Zt on successive values of itself. The model contains p + 2 unknown parameters m,
a which in practice have to' be estimated from the data.
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Moving Average models
Another kind of model of great importance is the moving average model where Zt is made linearly dependent on a
finite number q of previous a's (error terms)
Thus,
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Mixed Auto-regressive-moving average models :
It is sometimes advantageous to include both auto-regressive and moving average terms in the model. This leads to the mixed
auto-regressive-moving average (ARMA) model.
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In using such models in practice p and q are not greater than 2. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The Box-Jenkins stochastic models represent a flexible class of models that can be used to represent the short-term behavior
of a wide class of time series. Stochastic models are useful as a means for developing optimal short-term forecasters solely in
terms of the variables of primary interest. In some instances, these stochastic forecasters are about as accurate as those based
on elaborate econometric models. This situation would hold to an even greater extent with multivariate stochastic models.
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-----------------------------------------------------------------------------------------------------------------------------------------Finally, they are especially well-suited to the problem of simulating near future realizations, or outcomes, of a time series.
The Box-Jenkins stochastic-dynamic models include a useful class of models intermediate between the "purely stochastic and
the "purely" econometric models. With this class it may be possible to approach the increased precision of an econometric
model, without the need for including a large number of exogenous variables in the model.
The applications of these models to control problems have been noted. The important characteristic of the Box-Jenkins
method is not, however, that it might produce a forecaster that is as accurate as one based on an econometric model (it
probably won't).
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-----------------------------------------------------------------------------------------------------------------------------------------Because of the recent introduction of the Box-Jenkins method, there is not substantial literature available comparing this
method to other methods currently in wide use. It is hoped that this situation will change quickly, as the business community
increases its use of the Box-Jenkins method.
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