(a) (i) What is the meaning of positive and negative correlation? Explain with suitable example. (ii) Define regression. What…

(a) Correlation and regression (5 + 10 = 15 marks)

(i) Correlation measures the direction and strength of the linear relationship between two variables. Positive correlation: both variables move in the same direction, as advertising expenditure and sales rise together. Negative correlation: they move in opposite directions, as the price of a product rises and quantity demanded falls. Pearson's coefficient r = Σ(x − x̄)(y − ȳ)/√[Σ(x − x̄)² Σ(y − ȳ)²] lies between −1 (perfect negative) and +1 (perfect positive), with 0 meaning no linear relationship. A scatter diagram sloping upwards indicates positive and downwards indicates negative correlation. Correlation shows association, not causation.

(ii) Regression is a statistical technique for estimating or predicting the average value of one variable (dependent) from the known value of another (independent), by fitting a line to the data. There are two regression lines: Y on X, Y − Ȳ = b_yx(X − X̄), and X on Y, X − X̄ = b_xy(Y − Ȳ). The regression coefficients are

b_yx = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² = r σ_y/σ_x, and b_xy = Σ(x − x̄)(y − ȳ)/Σ(y − ȳ)² = r σ_x/σ_y.

Properties: (1) r² = b_yx × b_xy, so r is the geometric mean of the two coefficients, with their common sign; (2) both coefficients have the same sign, since each has the sign of the covariance; (3) both cannot exceed 1, so if one is greater than 1 the other is less than 1; (4) the arithmetic mean of the coefficients is at least r; (5) they are independent of a change of origin but not of scale; (6) the two lines intersect at (X̄, Ȳ); and (7) the coefficient measures the change in the dependent variable per unit change in the independent variable.

Example: advertising X (₹ lakh) = 1, 2, 3, 4, 5 and sales Y (₹ crore) = 2, 4, 5, 4, 5. X̄ = 3, Ȳ = 4, Σ(x − x̄)(y − ȳ) = 6, Σ(x − x̄)² = 10, Σ(y − ȳ)² = 6. So b_yx = 0.6, b_xy = 1.0 and r = √(0.6 × 1.0) = 0.775. The line Y on X is Y = 2.2 + 0.6X: each additional lakh of advertising is associated with ₹0.6 crore more sales; for X = 6, predicted Y = 5.8.

(b) Order quantity with price discounts (15 marks)

Annual demand D = 800, ordering cost S = ₹300, carrying cost H = ₹15 per unit per year (fixed, not a percentage of price). The EOQ is √(2DS/H) = √(2 × 800 × 300/15) = √32,000 = 178.9 units, the same for every price because H is fixed.

Check feasibility. At ₹66 (range 1-199), 178.9 is feasible. At ₹60 (range 200-599), 178.9 is below the range, so the lowest feasible quantity is 200. At ₹55 (range 600+), the feasible quantity is 600. Compare total annual costs, TC = DP + (D/Q)S + (Q/2)H:

• Q = 178.9 at ₹66: purchase 800 × 66 = 52,800; ordering (800/178.9) × 300 = 1,341.6; carrying (178.9/2) × 15 = 1,341.6; total = ₹55,483.
• Q = 200 at ₹60: purchase 48,000; ordering (800/200) × 300 = 1,200; carrying (200/2) × 15 = 1,500; total = ₹50,700.
• Q = 600 at ₹55: purchase 44,000; ordering (800/600) × 300 = 400; carrying (600/2) × 15 = 4,500; total = ₹48,900.

The total cost is least at Q = 600, so the firm should order 600 units each time (1.33 orders a year), saving ₹1,800 a year against 200 units and about ₹6,583 against the ₹66 EOQ.

(c) Managing data resource and database approach (5 + 15 = 20 marks)

Managing the data resource means treating data as a valuable organisational asset and managing it, through planning, organisation, control and technology, so that accurate, timely, secure and consistent data are available to all who need them. It includes data planning and standards, database administration, defining data structures and metadata in a data dictionary, ensuring quality and integrity, security and privacy, and backup and recovery, using a database management system (DBMS) and data warehouses.

In a file processing approach each application (payroll, sales, inventory) keeps its own files, with its own record formats, so data are duplicated and tied to programs. A database management approach keeps data in a shared, integrated database controlled by a DBMS. Advantages:

• Reduced redundancy: data are stored once, saving storage and effort.
• Consistency and integrity: a single update reaches every user, avoiding conflicting versions, and integrity rules can be enforced centrally.
• Data independence: programs are separated from the physical storage and structure, so changes to data structure or hardware do not require rewriting applications; file systems have program-data dependence.
• Data sharing and integration across departments, giving management a single view of the business.
• Ad hoc query and reporting: users can ask questions with query languages such as SQL, without programmers, while file systems need custom programs.
• Better security and access control by user and by field, with backup, recovery and concurrency control for multi-user access.
• Standards and administration: a database administrator applies uniform standards, improves data quality and governance.
• Faster application development and lower maintenance, since new applications reuse existing data.
• Better decision support through data warehouses and analytics.

The costs are higher expense for the DBMS and hardware, need for skilled staff, complexity, and greater damage from failure if backups are inadequate; for the business these are outweighed by the benefits.