Assignment Code: BCOC-134/TMA/2022-23

Course Code: BCOC-134

Assignment Name: Business Mathematics and Statistics

Year: 2022-2023

Verification Status: Verified by Professor

Section – A

Q. 1 Differentiate between Descriptive and Inferential statistics. (10)

Ans) Difference between Descriptive and Inferential statistics as follows:

Q. 2 Explain functions and importance of Statistics. (10)

Ans) The Important Functions of Statistics are:

1.  To Present Facts in a Proper Form: Statisticians make exact generalisations. In India, cotton yields 180 kg per hectare. This is more accurate and persuasive than saying India's cotton yield is poor.

2. To Simplify Unwieldy and Complex Data: Statistics simplify difficult facts. Data is often incomprehensible. Classifying facts by common traits is necessary to understand them. Take 1,000 industrial workers' weekly pay.

3. To Provide Techniques for Making Comparison: Statistics helps compare occurrences over time or location. Estimating national income is not a hobby. It's done to compare income over time to see if people's standard of life is rising. In 2005, India's per-capita income rose 10%. This data will illuminate the 2006 Indian standard of living.

4. To Study Relationship between Different Phenomena: Correlation and regression are used to study variables. Decision-making requires such interactions. For instance, product demand may affect price. Prices usually decrease product demand.

5. To Forecast Future Values: Some statistical methods predict variable values. A marketing manager can forecast next year's product demand based on 10 years of sales data.

6. To Measure Uncertainty: Probability theory estimates the likelihood of uncertain events. Probabilities aid decision-making. If you want to estimate your B. Com exam chances, look at the last 10 years' pass rates.

7. To Test a Hypothesis: Statistical approaches help generate new theories and test hypotheses. A business wants to know if its malaria medicine works. Chi-square Test could help.

8. To Draw Valid Inferences: Using sample data, statistical approaches can also infer universe (population) features.

9. To Formulate Policies in Different Fields: Statistics help create social, economic, and corporate policies. For family planning, the government uses essential statistics data. The government gives employees dearness allowance based on consumer price indices.

Importance of Statistics

Ancient statistics was employed only for statecraft. State administrative data included population, births, and deaths. Recently, statistics have expanded to include social and economic aspects. Over time, statistical methods expanded. It currently encompasses almost all sciences—social, physical, and natural. Statistics is used in agriculture, business, sociology, economics, biometry, and more. Thus, statistics is used in most human activities nowadays.

Statistics and State: The State used to keep order. For military and budgetary policymaking, it collected personnel, crime, income, and wealth data. The Welfare State has expanded the state's involvement. Thus, governments worldwide use price, output, consumption, income, and expenditure data to formulate economic and other policies. Planned economic development is used by developing nations like India to improve living conditions. The government must use accurate statistical data to make choices. To create its five-year plans, the government must know the availability of raw materials, capital goods, financial resources, and population distribution by age, sex, income, etc.

Statistics in Economics: Statistical analysis helps solve production, consumption, distribution, and other economic issues. Consumption data may show how different groups of society consume different goods. Economic policies require pricing, wage, consumption, savings, and investment data. Income inequality policies can benefit from national income and wealth data. Engel's Law of Consumption, Income Distribution Law, and others were developed using statistics in economics. Economic planning requires index numbers, time series analysis, regression analysis, etc. Workers receive DA or bonuses based on the consumer price index. Time series analysis can forecast demand. Statistics are increasingly utilised to test economic assumptions.

Statistics in Business and Management: Modern businesses are becoming more complex and demanding as they grow and compete. Professional management has emerged from huge companies' ownership-management separation. Managerial decision-making relies on timely access to statistical data. Thus, statistical data is increasingly employed in business and industry for sales, purchasing, production, marketing, finance, etc. Market and production research, investment policies, product quality control, economic forecasting, auditing, and many more sectors use statistical approaches. Managers must make uncertain decisions in all situations. Statistics can handle such scenarios. Thus, Wallis and Roberts state that “statistics may be seen as a corpus of strategies for making good decisions in the face of uncertainty.”

Q. 3 Discuss various preliminary adjustments that are necessary to do before analysing a time series data. (10)

Ans) Preliminary Adjustments before Analysing a Time Series Data:

1. Calendar Variations: The fact that the number of days in each month of the calendar varies slightly from month to month is a fact that is generally known and accepted. Because there are fewer days in February than there are in other months, for instance, the total quantity of goods that are produced during that month may be lower than the quantity that is produced during other months. When we take into account the holidays, we see an even greater degree of variation. As a consequence of this, it is necessary to make adjustments in order to take into account the myriad of calendar differences.

2. Price Changes: After taking into account the relevant price indices, it is necessary to perform the conversion from monetary values to real values. These real values can then be compared to the monetary values. This is necessary because fluctuations in the general price level are a natural occurrence that cannot be avoided and are therefore unavoidable.

3. Population changes: There is no question that the rate of population growth is getting faster and faster. As a direct result of this, the data need to be revised in order to take into account the shifts that have taken place in the population. In situations like these, it is possible to calculate per capita values if it is determined that doing so is necessary (dividing original figures by the total population).

Q. 4 Explain with example the properties of matrix multiplication. (10)

Ans) Associativity: Matrix multiplication is associative. For three matrices A, B and C of order m x n,n x p and p x q respectively,

(AB) C=A (BC)

Distributive over Addition: Matrix Multiplication is distributive over.

Matrix addition: For three matrices A, B and C of order m x n,n x p

and p x q respectively,

A (B+C) = AB+ AC

Identity: For any matrix A of order m x n, there is an identity matrix In

of order n x n and an identity matrix Im of order m x m such that Im A =

A =An

Q. 5 Why do we prefer standard deviation to quartile deviation as a measure of dispersion? (10)

Ans) Dispersion, also known as variability, is a measure of how spread out a set of data is. Standard deviation and quartile deviation are two measures of dispersion that are commonly used in statistical analysis. While both measures provide information about the spread of the data, standard deviation is often preferred over quartile deviation for several reasons.

Firstly, standard deviation is a more precise measure of dispersion than quartile deviation. Standard deviation considers all the data points and their deviations from the mean, whereas quartile deviation only looks at the range of the middle 50% of the data. This means that standard deviation gives a more accurate and detailed measure of the spread of the entire dataset, whereas quartile deviation only considers a portion of the data.

Furthermore, standard deviation is a widely used measure of dispersion, and as such, it is easier to compare and interpret results when standard deviation is used. It is a very common measure of dispersion and is widely used in statistics, probability theory, and other areas of research. This widespread use means that researchers and analysts are more familiar with the interpretation and properties of standard deviation, making it easier to communicate and compare results across studies.

Additionally, standard deviation is mathematically convenient. It has some mathematical properties that make it convenient to use in statistical analysis. For example, it has a simple formula that can be easily calculated, and it can be used in many statistical tests and models. This mathematical convenience makes standard deviation a more versatile and useful measure of dispersion in a wide range of statistical analyses.

Moreover, standard deviation allows for parametric analysis, which is particularly useful in hypothesis testing. Parametric analysis relies on the assumption that the data comes from a normal distribution. Standard deviation is useful in this context because it has the same mathematical properties as the normal distribution. In contrast, quartile deviation is not as useful in this context because it does not have the same mathematical properties as standard deviation.

However, there are situations where quartile deviation may be preferred over standard deviation. For example, if the data is skewed or has extreme outliers, quartile deviation may provide a more robust measure of dispersion. Additionally, quartile deviation may be more appropriate for data that is not normally distributed or has a non-parametric distribution. Quartile deviation may also be preferred if the researcher is interested in the central tendency of the data and not necessarily in the full spread of the data.

In conclusion, while both standard deviation and quartile deviation are measures of dispersion, standard deviation is often preferred over quartile deviation for several reasons. Standard deviation is a more precise measure of dispersion, it is widely used, mathematically convenient, and allows for parametric analysis. However, in some situations, quartile deviation may be more appropriate, such as when the data is skewed or has extreme outliers, or when the researcher is interested in the central tendency of the data. Ultimately, the choice of measure of dispersion will depend on the research question, the type of data, and the statistical analysis being performed.

Section – B

Q. 6 What are the rules of differentiation? Explain it with examples. (6)

Ans) Different rules of differentiation with examples.

Constant Function Rule: The derivative of a constant function is always zero. That is, if f(x) = c (where c is a constant), then f'(x) = 0.

Example

If f(x) = 5, then f'(x) = 0.

Power Function Rule: The derivative of a power function is equal to the product of the power and the function raised to the power minus one. That is, if f(x) = xn, then f'(x) = n * x(n-1)

Example

If f(x) = x3, then f'(x) = 3x2.

The Linear Function Rule: The derivative of a linear function is equal to its slope. That is, if f(x) = mx + b, then f'(x) = m.

Example

If f(x) = 2x + 3, then f'(x) = 2.

Rule of Sums and Differences: The derivative of a sum or difference of two or more functions is equal to the sum or difference of their derivatives. That is, if f(x) = g(x) ± h(x), then f'(x) = g'(x) ± h'(x).

Example

If f(x) = x2 + 3x - 1, then f'(x) = 2x + 3.

The Product Rule: The derivative of a product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function. That is, if f(x) = g(x) * h(x), then f'(x) = g'(x) * h(x) + g(x) * h'(x).

Example

If f(x) = x2 * sin(x), then f'(x) = 2x * sin(x) + x2 * cos(x).

The Quotient Rule: The derivative of a quotient of two functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. That is, if f(x) = g(x) / h(x), then f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / h(x)2.

Example

If f(x) = (x2 + 1) / x, then f'(x) = (2x * x - (x2 + 1) * 1) / x^2 = (x2 - 1) / x2.

The Chain Rule: The derivative of a composition of two functions is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. That is, if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).

Example

If f(x) = sin(x2), then f'(x) = cos(x2) * 2x.

Q. 7 At what rate percent compound interest per annum with Rs. 640 amount to 774.40 in 2 years? (6)

Q. 8 Solve the following equations by using Cramer’s Rule. (6)

x-2y = 11

Ans) Cramer's Rule is a method for solving a system of linear equations using determinants. To use Cramer's Rule, we need to write the system of equations in the form of:

ax + by = c

dx + ey = f

In this case, we have one equation:

x - 2y = 11

We can rewrite this equation as:

x + (-2y) = 11

So, a = 1, b = -2, and c = 11.

To use Cramer's Rule, we need to find the determinants of the coefficient matrix and the augmented matrix. The coefficient matrix is:

So, both Dx and Dy are equal to 0. This means that the system of equations has either no solution or an infinite number of solutions. In this case, the system of equations is inconsistent and has no solution.

Therefore, the solution using Cramer's Rule is that there is no solution to the given equation.

where Dx is the determinant of the matrix formed by replacing the first column of the coefficient matrix with the augmented column, Dy is the determinant of the matrix formed by replacing the second column of the coefficient matrix with the augmented column, and D is the determinant of the coefficient matrix.

Q. 9 State the characteristics of good measure of variation. (6)

Ans) A good measure of variation should have the following characteristics:

1. Simplicity: A good measure of variation should be easy to understand and calculate.

2. Sensitive: It should be able to detect even small differences in the data.

3. Applicability: The measure should be applicable to different types of data.

4. Stability: The measure should not change significantly with the addition or removal of outliers or extreme values.

5. Relative Independence: The measure should be independent of the unit of measurement used for the data.

6. Adequacy: The measure should be appropriate for the type of data being analyzed and the purpose of the analysis.

7. Compatibility: It should be compatible with other measures of central tendency and variability used in the analysis.

8. Reliability: The measure should produce consistent results when applied to the same data set multiple times.

9. Robustness: The measure should not be overly affected by extreme values or outliers.

10. Interpretability: The measure should be easily interpretable and understandable to the intended audience.

Q.10 What do you mean by method of factorization and method of substitution? (6)

Ans) In mathematics, the method of factorization and method of substitution are two common techniques used to solve equations. The method of factorization involves finding the factors of an equation and then using these factors to simplify the equation into more manageable parts. This method is particularly useful for solving quadratic equations, where the equation is of the form ax^2 + bx + c = 0. To solve such an equation using the method of factorization, we need to factorize the left-hand side of the equation and then set each factor equal to zero. This gives us the values of x that make each factor equal to zero, which are the solutions to the equation.

For example, consider the quadratic equation x2 + 5x + 6 = 0. We can factorize this equation into (x + 3)(x + 2) = 0. Setting each factor equal to zero gives us x = -3 and x = -2 as the solutions to the equation.

The method of substitution involves replacing one or more variables in an equation with other variables or expressions, in order to simplify the equation and make it easier to solve. This method is often used in algebraic equations that involve multiple variables, where we can use one of the variables to eliminate another variable.

For example, consider the system of equations:

2x + 3y = 7

4x - 5y = -1

To solve this system using the method of substitution, we can solve one of the equations for one of the variables in terms of the other variable, and then substitute this expression into the other equation. Let's solve the first equation for y in terms of x:

3y = 7 - 2x

y = (7 - 2x)/3

Now we can substitute this expression for y into the second equation:

4x - 5[(7 - 2x)/3] = -1

We can then simplify and solve for x:

4x - (35/3) + (10x/3) = -1

14x/3 = (32/3)

x = 32/14 = 16/7

Finally, we can substitute this value of x back into one of the equations to solve for y:

2(16/7) + 3y = 7

3y = 7 - (32/7)

y = 3/7

Therefore, the solutions to the system of equations are x = 16/7 and y = 3/7

Section – C

Q.11 Write short notes on the following: (5×2)

(a) Regression

Ans) Regression is a mathematical technique used to analyze the relationship between one or more independent variables and a dependent variable. It involves modeling the relationship between the variables using a mathematical function, which can then be used to predict the value of the dependent variable for any given value of the independent variable(s). The most common type of regression is linear regression, which involves fitting a straight line to the data points. The equation of the line represents the relationship between the independent variable and the dependent variable and can be used to make predictions about the dependent variable for any given value of the independent variable.

Nonlinear regression is another type of regression that is used when the relationship between the variables is not linear. In this case, a more complex mathematical function is used to model the relationship, which can take various forms such as quadratic, exponential, or logarithmic.

Regression analysis involves several steps, including data collection, model specification, parameter estimation, and model validation. Data collection involves gathering the necessary data for the analysis, while model specification involves selecting the appropriate type of regression model and specifying the functional form of the model. Parameter estimation involves estimating the parameters of the model using statistical methods, while model validation involves assessing the goodness of fit of the model and the statistical significance of the estimated parameters.

Regression analysis has numerous applications in many fields of study, including statistics, economics, finance, engineering, social sciences, and healthcare. It is a powerful tool for understanding the relationship between variables and making predictions about the behaviour of a system. It can also be used to identify important factors that influence the dependent variable and to test hypotheses about the relationship between the variables.

(b) Time series

Ans) Time series is a mathematical method used for analysing and forecasting data that is collected over time. It is a sequence of data points that are collected at regular intervals of time, such as hourly, daily, weekly, or monthly. Time series analysis is used in various fields, including economics, finance, engineering, and social sciences.  The purpose of time series analysis is to identify patterns and trends in the data and to make predictions about future values of the variable. It involves several key steps, including data collection, modeling, and forecasting.

Data collection involves gathering the necessary data for the analysis, while modeling involves selecting the appropriate mathematical model to represent the time series data. There are several types of models used for time series analysis, including autoregressive (AR), moving average (MA), and autoregressive integrated moving average (ARIMA).

Forecasting involves using the time series model to make predictions about future values of the variable. This can be done by extrapolating the pattern in the data, or by using advanced forecasting techniques such as exponential smoothing, trend analysis, or seasonal decomposition.  Time series analysis is used in various applications, such as in the analysis of financial data, weather patterns, and stock prices. It is also used in the analysis of economic trends and consumer behaviour.

Q.12 Differentiate between the following:

(a) Positive correlation and negative correlation

Ans) The differences between positive and negative correlation:

(b) Geometric mean and harmonic mean

Ans) The differences between geometric mean and harmonic mean: