(a) Define mutually exclusive events.
Two events are said to be mutually exclusive if they cannot occur at the same time. If event A happens, event B cannot happen, meaning their intersection is an empty set (P(A ∩ B) = 0).
(b) State the classical definition of probability.
If a random experiment has 'n' mutually exclusive, equally likely, and exhaustive outcomes, and 'm' of these are favorable to an event A, then the probability of A is P(A) = m / n.
(c) State the multiplication law of probability.
For two independent events A and B, the probability of both occurring is the product of their individual probabilities: P(A ∩ B) = P(A) × P(B).
(i) Probability of drawing one king and one queen from 52 cards.
Total ways to draw 2 cards: 52C2 = (52 × 51) / 2 = 1326.
(ii) Define conditional probability.
[span_13](start_span)Conditional probability is the probability of an event A occurring given that event B has already occurred, denoted as P(A|B) = P(A ∩ B) / P(B), where P(B) > 0.
[span_13](end_span)(iii) Probability that a leap year contains 53 Mondays.
[span_16](start_span)A leap year has 366 days = 52 weeks + 2 extra days.
[span_16](end_span)(a) What do you mean by average? Write one limitation of AM.
An average is a single value that represents the central position of a data set.
A key limitation of Arithmetic Mean (AM) is that it is highly affected by extreme values (outliers).(b) What is meant by dispersion?
Dispersion refers to the extent to which data points in a distribution differ from the average or from each other; it measures the "spread" of data.
(c) Find Standard Deviation: 8, 10, 12, 14, 16, 18, 20, 22.
Mean (x̄) = (8+10+12+14+16+18+20+22) / 8 = 120 / 8 = 15.
(i) Write five important measures of dispersion.
(ii) Find the Coefficient of Variation (CV).
Using the provided table for Marks 0-80
:(a) Regression equations.
X on Y: (X - x̄) = bxy(Y - ȳ)
(b) Compute Correlation Coefficient (r).
Formula: r = [nΣxy - (Σx)(Σy)] / √[nΣx² - (Σx)²][nΣy² - (Σy)²]
(iii) Given regression equations: 8x + 10y + 66 = 0 and 40x - 18y = 214.
1. Average value of x and y: Solve the two equations simultaneously.
Multiply 1st eq by 5: 40x + 50y = -330.
Subtract 2nd eq: (40x + 50y) - (40x - 18y) = -330 - 214.
68y = -544 → ȳ = -8.
Substitute ȳ: 8x + 10(-8) + 66 = 0 → 8x - 14 = 0 → x̄ = 1.75.
2. Correlation Coefficient (r):
From Eq 1: y = -0.8x - 6.6 (byx = -0.8).
From Eq 2: x = 0.45y + 5.35 (bxy = 0.45).
r² = byx × bxy = -0.8 × 0.45 = 0.36.
r = -0.6 (Negative because regression coefficients are negative).
(i) Laspeyres and Paasche's Price Index.
Using data for 1999 (Base) and 2000 (Current)
:(ii) Why Index Numbers are called 'Economic Barometers'?
[span_57](start_span)They are called economic barometers because they measure the "pressure" or changes in the economic variable (like price or production) over time, helping to sense the pulse of the economy and guide policy decisions.
[span_57](end_span)(i) Fit a linear trend by Least Squares method.
Years: 1975 to 1980 (n=6). Let x = Year - 1977.5.
Equation: Y = a + bx.
Solve Normal Equations: ΣY = na and ΣxY = bΣx².
(ii) Components of Time Series.
Would you like me to generate a Practice Quiz for the numerical sections of Index Numbers and Correlation to help you master these formulas?
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