Quantitative Analysis revision question and answers
This revision question and answers can be used for students pursuing the following Kasneb courses:
Define the following terms:
A stochastic process arises whenever we have a series of events in which each event is determined by chance. It is also known as probabilistic process. The development of a stochastic process, is governed by the laws of probability i.e. the development of such process is not certain.
This is a matrix containing probabilities of a process moving from a certain condition (or state) in the current stage (time period) to one of the possible states in the next stage. The general structure of a transition matrix is:
The elements in the matrix are transitional probabilities i.e. the probability of being in state j in the next period, given it is now in state I, denoted as P ij. The row sums in such a matrix always equals 1 because in moving from time t to t+1, the process must be in anyone of the states.
A recurrent state is a state in which the probability is one that it will be re-entered at some time in future but not necessarily in the next immediate time.
Steady state refers to a major property of Markov chains that in the long run,the process usually stabilizes. A stabilized system is said to approach steady state or equilibrium where the systems state probabilities have become independent of time. Thus a steady state is a time reached by the process where the probabilities no longer change with time, i.e. the process is in equilibrium.
What is a Lorenz curve? Explain any two areas Lorenz curve can be used. (6 marks)
A Lorenz carve is a descriptive technique that is used to show how equitably or inequitably incomes are distributed. It is constructed by plotting cumulative percentage of one variable against cumulative percentage total of the same variable. It can be used to show inequalities;
- In distribution of incomes among a population,
- In distribution of turnover (sales) among companies.
What is an index number? Explain two areas where index numbers are applied. (6 marks)
An index number is a measure of changes in a variable or a group of variables with respect to time, incomes, production or other characteristics. It can be used to compare the cost of living in the county over time or to compare the output of an agricultural product or mineral during a given period with its production in a previous period.
Explain what break-even analysis as used in Quantitative Techniques is (3 marks)
BE Analysis enables us to analyze the relationship between cost, volume and profits. It provides us with a model for determining the level of output (volume) at which profit will be Zero (i.e. when TR = TC). The B.E. model can also be used to help us to determine what would happen to profit if there were changes in costs (e.g. V.C or FC), volumes or even selling prices.
Puda Development Company (PDC) is a small real estate developer operating in the Eastland’s Valley. It has seven permanent employees whose monthly salaries are given below:
PDC leases a building for Sh. 20,000 per month. The cost of suppliers, utilities and leased equipment runs for another Sh. 30,000 per month. PDC builds only one style house in the valley. Land for each house costs. Sh. 550,000 and lumber, supplies and others run for another Sh. 280,000 per house. Total labour costs amount to Sh. 200,000 per house. The one sales representative of
PDC is paid a commission of Sh. 20,000 on the sale of each house. The selling price of the house is Sh. 1,150,000.
(i) Identify all the costs and denote the marginal revenue and marginal cost for each house. (4 marks)
Salaries (Sh „000):
100 + 60 + 45 + 55 + 40 + 30 + 20 = 350
Office lease and supply costs
= 20 + 30= 50
= 350,000 + 50,000= 400,000
- Land, Material, labour and sales commission per house is the variable
or marginal cost for the house. It is given as:
= 550,000 + 280,000 + 200,000 + 20,000
- The selling price of Sh. 1,150,000 is the marginal revenue per house.
(ii) Determine the monthly cost function; C(x), revenue function; R(x) and the profit function; P(x) (4 marks)
Total cost function;
TC = VC + FC = 1,070,000 + 400,000 = 1,470,000
TR = 1,150,000 (x) = 1,150,000x
Profit = TR – TC = 1,150,000x – 1,070,000x – 400,000 = 80,000x – 400,000
(iii) Determine the break-even point for monthly sales of the houses. (3 marks)
BER in number of houses;
At BEP TR = TC … substituting
-1,150,000x = 1,070,000x + 400,000
-80,000x = 400,000
x = 5 houses
(iv) Determine the monthly profit of 12 houses per month are build and sold. (2 marks)
The profit if 12 houses are built and sold is computed as equal to
= (80,000 x 12) – 400,000
= Sh. 560,000.
(c) What are some of the simplifying assumptions in part (b) above? (4marks)
Simplifying assumptions in Break even analysis above Analysis:
- There is a linear relationship between costs, revenues and volumes.
- The variable cost and marginal revenue per unit remains constant over the relevant range.
- The fixed costs remain the same over the relevant range of output.
- There‟s no uncertainty in the process of developing houses by Puda Development Company. Thus it is a determination model.
Explain the difference between mean squared error and mean absolute deviation as measures of forecast accuracy. (3 marks)
The mean squared error, like the absolute deviation attempts to measure the accuracy of forecasts. The mean squared error is computed by averaging the squared errors while the mean absolute deviation is obtained by taking the mean of the absolute error values.
Briefly but clearly explain the method of least squares.( 3 marks)
Method of least squares is used to determine the unique trend line forecast which minimizes the mean squared error between the trend line forecasts and the actual observed values of linear time series data.
Briefly but clearly explain the purpose of Input- Output analysis. (5 marks)
The main purpose of input-output analysis is to analyze the interdependence of various segments of an economy. According to this technique, an economy is dividend into two broad sectors, the sources, or output or producers‟ sector and the destinations or input or users‟ sectors.
The producers‟ sector includes:
(i) The section that manufacturers goods
(ii) The section that provides services.
These goods and services are intended for:
(i) Industrial usage and are called Industrial goods and services.
(ii) Final consumption and these are termed consumer goods and services.
The output for each industry from each industry is either for industrial use or for final consumption. Apart from industrial outputs, the other output constitutes payment for services rendered.
These services are obtained from:
– Depreciation of capital goods
– Household services e.g. wages and salaries.
The model aims at indicating how products produced from different sectors of the economy are consumed by the industrial sector and the final consumption sector. The model therefore indicates how the independent sectors of economy interact. The interaction of these sectors is a necessary information for planning purposes so as adequate stock of goods can be produced by the economy. A device to assist in such planning in the technical coefficients.
Explain briefly the application of calculus to economic models, especially in the context of maximizing contribution and minimizing costs.
The subject of calculus is an area of mathematics called the classical optimization model. It has many business and economic applications. It is concerned mainly with optimization i.e. the best combination of variables that will either maximize benefits or minimize losses. In the context of maximizing contribution, calculus is used in determining the level of production that will maximize profit or contribution. This is achieved through the concept of the rate of change in y, the dependent variable, given a change in x, the
independent variable. Once a functional relationship between these two variables is established, then through the process of differentiation an optimal output level is determined.
Explain null-hypothesis and the alternative hypothesis. (5 marks)
A hypothesis is some testable belief or opinion, and hypothesis testing is the process by which the belief is tested by statistical means. The process of hypothesis testing is also called significance testing. A null hypothesis is the statement of what we believe is the truth. On the other hand, an alternative hypothesis is the negation of what we believe. For instance, assume a machine fills packets with spice which are supposed to have a mean weight of 40 grams. A random sample of 36 packets is taken and the mean weight is found to be 42.4 grams with a standard deviation of 6 grams. We could be required to conduct a hypothesis test, also known as significance test, at say 5% level of significance. Once a level of significance is chosen then we are saying that our belief should be correct 95% of the times. We then set our hypothesis as:
It is the null hypothesis which is tested. If we find it to be true, then we accept it (H 0) and reject alternative hypothesis (HA). If H0
is accepted then our conclusion is that the test is as expected and nothing unusual has happened. If on the other hand, we reject the Null hypothesis H0, then we must accept the alternative, which we did not believe in. We say the test is significant.Assuming a normal distribution, 5% level of significance or 95% level of confidence means that out of our observations, 95% of them should be found with two standard deviation above or below the means.
Explain the circumstances in which the paired t-test should be used to test the difference between two means. (5 marks)
The paired “t” statistics is used to test the difference between two means in cases where the same persons are used to carry out the same job or same event but using different tools. In such a case, the difference between two means of the samples drawn from the same population cannot be used because the difference could be due to the faults in the tools.
Explain the difference between parametric tests and non-parametric tests. (5 marks)
A parametric equation is an equation indexed by a quantity called a parameter. To illustrate let
where t is a parameter. The term parameter is therefore the unknown to be found. Parametric test therefore deals with situations where the unknown function has been estimated using some parameter and we wish to test whether the parameter really represents the unknown. For example, the sample mean being used to estimate the population mean or the sample standard deviation being used to estimate the population standard deviation. The parametric tests are those tests which are carried out to draw inferences about the population using a parameter and the assumption is that the population from which the sample is drawn is normally
distributed or is near normal. On the other hand, where the experimenter does not know the exact form of the distribution of the population about which the inference is about to be made, then one needs to design statistical techniques which are applicable
regardless of the form of distribution. Such tests are called non-parametric tests, e.g. the likelihood ratio.
Explain the least squares method of linear regression.
The least squares method of regression is a technique of estimation in which the aim is to minimize the error margin or the error term. If the function is assumed linear between two variables then an equation in the form y= a + bx can be used a predictor of the observed relationship.
In the context of time series, explain the following terms:
(i) A basic trend.
Quantitative forecasting models assume that the time series follow some patterns which can be extrapolated into the future. Such patterns that can be used for future extrapolations are known as basic trends. For instance, a linear trend model forecasts a straight line trend for any period in the future. Exponential trend forecasts that the amount of growth will increase continuously.
(ii) Seasonal fluctuations.
If one were to closely study employment and output data in say the agricultural sector, one would easily note a definite relationship between output and season or between employment and the time of the year. Similarly, the sale of success cards will normally be associated with examination times and are therefore subject to large seasonal variations. Performances that vary with seasons are known as seasonal fluctuations.
(iii) Cyclical fluctuations.
Cyclical fluctuations consists of those fluctuations in a time series which do not repeat themselves periodically like seasonal variations. Cyclical fluctuations will show periods of rapid growth followed by those of slower growth and they will normally be of different duration and intensity.
(iv) Residual variations. (8 marks)
Residual variations are also known as random or irregular variations. This component of time series is due to purely random and irregular factors.
Explain the purpose of Venn diagrams.
The purpose of Venn diagrams is to visualize the relationships between event sets. Useful in the area of probability.
Distinguish between input-output analysis and Markov analysis.
Difference between input-output analysis and Markov analysis – input – output analysis shows the interdependence of sectors in an economy.
Foretasted levels of output required for each sector so as to satisfy both intermediate and final demand can be calculated if we are given the technical coefficient matrix and the foretasted levels of final demand. X = (1 – A) –1 D
Markov analysis is a probabilistic system whereby the state of a given phenomenon in future can be predicted from the current state and transition matrix (initial state vector)(Transition matrix) = (Future state vector).
Differentiate between paired t-test and two sample t-test.
- Paired t-test is used for the mean of differences where samples are not independent.
- Two-sample t-test is used to test for the difference in means where samples are independent.
Difference between multiplicative and additive models.
– Multiplicative model expresses the time series model as a product of the four principle components. That is Y = TSCR
– Additive model expresses the time series model as a sum of the four principle components. That is Y = T + C + R + S
b) State Conditions under which each model is used in your answer above.
-Multiplicative model is used if the four principle components are not independent.
– Additive model is used when the four principle components are independent.
State three purposes of seasonal index
-Used to evaluate seasonal effects on a time series.
– Used to adjust trend forecasts.
– Used to de-seasonalise data.