Assignment Code: BCOS-134/TMA/2021-22

Course Code: BCOC-134

Assignment Name: Business Mathematics and Statistics

Year: 2021-2022

Verification Status: Verified by Professor

Maximum Marks: 100

Note: Attempt all the questions.

Section – A

Q1. In a partially destroyed laboratory record relating to correlation data, the following results are legible:

σ2x = 9,

Regression Equations 8X-10Y + 66 = 0, 40X-18Y = 214.

What were

(a) the mean values of X and Y,

(b) standard deviation of Y (σy )

(c) the co-efficient of correlation between X and Y ? (10)

Ans)

Q2. The median and the mode of the following distribution are known to be Rs. 335 and Rs. 340 respectively. Three frequency values from the table are however missing. (10)

Find the missing values when n = 230.

Ans)

Q3. From the following data, calculate Laspeyre’s, Paasche’s, and Fisher’s Ideal Index numbers. (10)

Ans)

Q4. Find the limit of the following function: (10) limh→ 0 (6+h)2−36 / h

Ans) limh→0 (6+h)2−36/ h

= limh→0 36+12h+h2−36 / h

= limh→0 h(12+h) / h

= limh→0 (12+h)

=12

Q5. Solve the following 2x2 system using Cramer’s Rule. (10)

12x+3y = 15

2x-3y = 13.

Ans) Step-by-step explanation:

D = 12 3. = (12*-3)- (2*3)= -42. D= -42

2 -3

Dx = 15. 3 = (15*-3)-(3*13)= -84. Dx= -84 ,

13. -3

X= Dx/D = 2

Dy = 12. 15

2 13 = (12*13)-+2*15)=126. Dy= 126

Y = Dy/D = -3

Section B

Q6. Define minor of a square matrix and cofactor of a square matrix? (6)

Ans) The minor of a square matrix is the number generated by removing a row and column corresponding to the element of a matrix from the determinant of a square matrix.

The signed minor is used to specify the cofactor of a square matrix. Cofactor of an element aij, denoted by Aij is defined by Aij = (–1)i+j Mij, where Mij is minor of aij.

Q7. What do you mean by method of factorization and method of substitution? (6)

Ans) The Factorization Method is a limit-finding technique that works by cancelling out common factors. It's usually used to convert an ambiguous form into one that can be evaluated directly. If f(x) and g(x) are two functions such that limx→a f(x) = limx→a g(x) = 0

and we have to find limx→a f(x) / g(x), then we obtain a 0/0 form, which is meaningless. Note that 0/0 is called as indeterminate form. Other indeterminate form is ±∞/±∞. Such limits are solved by method of factorization.

The Method of Substitution is a method of establishing limits that involves substituting the approaching value into the function and evaluating the result.

Q8. Describe cross elasticity of demand. Elaborate your answer. (6)

Ans) The responsiveness of demand for one product to a change in the price of a related product is known as cross elasticity of demand. Make a note of the word "related" product. It's worth noting that demand elasticity for unrelated products is zero. An increase in the price of black gramme, for example, will have no influence on ice cream demand.

By dividing the percentage change in demand for one product by the percentage change in the price of another product, we may calculate the cross elasticity of demand. Thus,

Cross elasticity of demand = % of change in the demand for Product A / % of change in the demand for Product B

or (dQ /dP′) ∗ (P′/Q), where Q= quantity demanded, P′=price of a related product of Q.

It's worth noting that demand cross elasticity is affected by whether the associated product is a substitute or a complement.

Q9. Explain relation between correlation coefficient and regression. (6)

Ans) The two types of studies that are based on the distribution of numerous variables are correlation and regression. They can be used to describe the type and strength of a relationship between two continuous quantitative variables. Despite the fact that these two mathematical concepts are studied at the same time, the distinction between correlation and regression is evident from the preceding discussion.

Correlation is employed when a researcher wishes to see if the variables under investigation are related and, if so, how strong their link is. Pearson's correlation coefficient is often recognised as the most accurate correlation measurement. A functional link between two variables is constructed in regression analysis in order to make future event estimates.

When there is an instant need for a direction to comprehend the link between two or more variables, correlation is used. When it's necessary to optimise and explain the numerical response from y to x, regression is used. To gain a rough understanding of how y influences x and to make an approximation of it.

Q10. Describe time series. Why do we analyse a time series? (6)

Ans) "A time series is a statistical series created by arranging quantitative data in the order of their occurrence."

For the following reasons, time series analysis is extremely useful not only to researchers but also to economics, businesspeople, and scientists.

1. It aids in comprehending the previous behaviour of the variables under investigation.

2. It aids in projecting future behaviour by using changes that have occurred in the past.

3. It aids in the planning of future actions.

4. It aids in determining current achievements.

5. Making comparisons between diverse time series and drawing important inferences from them is beneficial.

Section C

Q11. Write short notes on: (5X2)

a) Differentiation

Ans) Differentiation is a technique for determining a function's derivative. Differentiation is a mathematical procedure for determining the instantaneous rate of change of a function depending on one of its variables. The most common example is velocity, which is the rate of change of displacement with respect to time. Anti-differentiation is the inverse of finding a derivative.

If x is a variable and y is another variable, then the rate of change of x with respect to y is given by dy/dx. This is the general expression of derivative of a function and is represented as f'(x) = dy/dx, where y = f(x) is any function.

b) Quantity index

Ans) The amounts of a single commodity or a set of commodities are the main focus of consideration and comparison in these indices. For example, the focus could be on determining how the quantity of paddy produced in India has changed over time. This will necessitate the creation of a single commodity's quantity index. Alternatively, the focus could be on determining changes in food grain production in India; in this case, all commodities classified as food grains will be taken into account when generating the quantity index.

Q12. Differentiate between: (5X2)

a) Absolute and relative measure of dispersion

Ans) The differences between absolute and relative measure of dispersion are:

1. Absolute dispersion measurements concentrate on a single set of observations, whereas relative dispersion measures concentrate on two or more at once.

2. Absolute measures are calculated in the same units as the data, whereas relative measures can be calculated in any unit.

3. Absolute measures are used to calculate standard deviation and variance; thus you could hear about them in math class when talking about probability.

4. The Coefficients of standard deviation/variance, which are defined as a ratio between the standard deviation and the mean, are calculated using relative measurements. And it's not only standard deviation; I'm sure there's more, but I'm not sure what it is because I haven't studied probability in deep.

b) Average cost and marginal cost

Ans) The differences between average cost and marginal cost are:

1. The overall cost of products divided by the total number of goods is the average cost, whereas marginal cost is the cost of manufacturing one more unit or additional unit of product or service. The overall cost of production fluctuates as the output changes due to variations in the amount of production.

2. The average cost is used to determine the influence of a change in output level on total unit cost. The goal of marginal cost, on the other hand, is to determine whether producing an additional unit of products is helpful.

3. The average cost curve initially lowers due to falling fixed costs, but subsequently rises as average variable costs rise. The marginal cost curve, on the other hand, is concave with growing returns, then travels linearly and smoothly with a constant return, and eventually switches to convex when marginal cost shows a rise in return.

4. The average cost is made up of two parts: average fixed and average variable costs, while marginal cost is a single unit with no components.