FYUG Even Semester Exam, 2025 STATISTICS (2nd Semester) Course No.: STAIDC-151

Subject Name: Statistics

Paper Name: Index Number and Time Series Analysis

Paper Code: STAIDC-151

Semester: 2nd Semester (FYUG)

Full Marks: 70 | Pass Marks: 28

Time Duration: 3 hours

UNIT-I

Question 1 [1 x 4 = 4]

(a) Define index number: An index number is a statistical device for measuring the relative change in a large number of items (such as prices or quantities) over a period of time.

(b) Limitations: Limitations include errors in data collection, bias in choosing the base year, and the fact that they are often based on samples rather than complete data.

(c) Applications: Used to measure changes in the cost of living, inflation rates, and to monitor economic trends in production or trade.

(d) Paasche's Formula: P = [Sum(P1*Q1) / Sum(P0*Q1)] * 100.

(e) Base year: The year with which the changes in the current year are compared. It is usually a period of economic stability.

Question 2(a) [2]

"Index numbers are economic barometers." Explain.

Index numbers act as barometers because they measure the "pressure" of economic change. Just as a barometer measures atmospheric pressure to predict weather, index numbers measure changes in price levels, production, or employment to indicate the health and direction of an economy.

Question 3(b) [8]

Construct price index numbers for 1995 (base 1990).

Commodity	P0 (1990)	Q0 (1990)	P1 (1995)	Q1 (1995)	P1Q0	P0Q0	P1Q1	P0Q1
A	4	20	6	10	120	80	60	40
B	3	15	5	23	75	45	115	69
C	2	25	3	15	75	50	45	30
D	5	10	4	40	40	50	160	200
Total					310	225	380	339

Laspeyres' (L): (310 / 225) * 100 = 137.78

Paasche's (P): (380 / 339) * 100 = 112.09

Fisher's (F): Sqrt(L * P) = Sqrt(137.78 * 112.09) = 124.26

Marshal-Edgeworth: [Sum(P1Q0 + P1Q1) / Sum(P0Q0 + P0Q1)] * 100 = (690 / 564) * 100 = 122.34

UNIT-II

Question 4 [1 x 4 = 4]

(a) Time Reversal Test: A test requiring that if the time subscripts are interchanged, the resulting index should be the reciprocal of the original index.

(b) Factor Reversal Test: A test requiring that the product of the price index and the quantity index should equal the value ratio.

(c) Ideal Index: Fisher's index number.

(d) Fill in the blank: Fisher's index number is the geometric mean of Laspeyres' and Paasche's index numbers.

Question 6(a) [8]

Consumer Price Index (CPI): Steps and Uses.

CPI measures changes in the price level of a basket of consumer goods and services purchased by households.

Steps:

Selection of the consumer group (e.g., urban workers).
Conducting a family budget inquiry to determine the "basket."
Selection of the base year.
Collection of retail prices.
Selection of weighting system (e.g., Weighted Arithmetic Mean).

Uses: Wage negotiations, adjusting social security benefits, and determining the purchasing power of money.

UNIT-III

Question 7 [1 x 4 = 4]

(a) Time Series: A sequence of data points recorded at successive, equally spaced points in time.

(b) Time series vs Cross-section: Time series tracks one subject over time, while cross-section tracks many subjects at a single point in time.

(d) Trend: The long-term, underlying movement or direction in a time series.

(e) Applications: Weather forecasting, stock market analysis, and economic planning.

Question 8(b) [1]

Identify components for:

(i) Fire in a factory: Irregular variation.

(ii) Sale of umbrella in summer: Seasonal variation.

(iii) Fall in death rate due to science: Secular trend.

(iv) Flood due to heavy rain: Irregular variation.

UNIT-IV

Question 12(a) [8]

Method of Least Squares: Fit a straight line for the given production data.

Equation: Y = a + bX. For n=7, let middle year 1963 be X=0.

Year	Y (Prod)	x (X-1963)	x²	xY
1960	40	-3	9	-120
1961	60	-2	4	-120
1962	50	-1	1	-50
1963	70	0	0	0
1964	55	1	1	55
1965	75	2	4	150
1966	50	3	9	150
Total	400	0	28	65

Calculations:
a = Sum(Y) / n = 400 / 7 = 57.14
b = Sum(xY) / Sum(x²) = 65 / 28 = 2.32
Trend Equation: Y = 57.14 + 2.32x

UNIT-V

Question 13 [1 x 4 = 4]

(a) Seasonality: Periodic fluctuations in a time series that repeat at regular intervals (e.g., weekly, monthly).

(b) Method: Ratio-to-moving average method.

(c) Difference: Ratio-to-trend assumes trend is a specific function (like a line), while ratio-to-moving average uses a smoothed average to represent trend/cycle.

(d) Fill in blank: Moving average method eliminates short-term fluctuations.

Question 15(b) [8]

Ratio to Moving Average Method.

This method isolates seasonal variations by dividing the original data by a centered moving average.

Merits: Flexible, follows the trend effectively, and eliminates the cyclical component.

Demerits: Loses data at the beginning and end of the series, and involves complex calculations.

Exam Focus Enhancements

Exam Tips

For Index Number construction questions, always create a clear table for P0Q0, P1Q0, etc. to avoid calculation errors.
In Least Squares questions, always use a centered year (X=0) to simplify arithmetic.

Common Mistakes

Confusing the Time Reversal Test with the Factor Reversal Test.
Forgetting to multiply by 100 in Index Number formulas.

Important Formulas List

Fisher's Ideal Index: F = sqrt([Sum(P1Q0)/Sum(P0Q0)] * [Sum(P1Q1)/Sum(P0Q1)]) * 100
Least Squares Normal Equations: Sum(Y) = na + b*Sum(X) and Sum(XY) = a*Sum(X) + b*Sum(X²)

Answer Presentation Strategy

Use bullet points for theoretical questions and always define terms before writing formulas. For Time Series components, providing a one-sentence example (like the umbrella example) guarantees clarity for the examiner.