Statistics and Probability

0.70

Step 1: Use the Addition Theorem: P(A∪B) = P(A) + P(B) - P(A∩B). Step 2: Substitute values: P(A∪B) = 0.37 + 0.42 - 0.09. Step 3: P(A∪B) = 0.79 - 0.09 = 0.70. Step 4: Common mistake is adding all three values (getting 0.88) instead of subtracting P(A∩B), or forgetting to subtract the intersection.

9.12

Step 1: Original σ ≈ 2.28 for data 4, 7, 8, 10, 11. Step 2: Key property: When all data values are multiplied by a constant k, the new standard deviation = k × original σ. Step 3: New σ = 4 × 2.28 = 9.12. Step 4: This is because multiplying each value by k scales the deviations from the mean by k, and hence σ scales by k. Common mistake: thinking σ stays the same (that's only for addition/subtraction of a constant).

1/5

Step 1: n(S)=50, n(A)=28 (NCC), n(B)=30 (NSS), n(A∩B)=18. Step 2: P(A) = 28/50, P(A∩B) = 18/50. Step 3: P(NCC but not NSS) = P(A∩B̄) = P(A) - P(A∩B) = 28/50 - 18/50 = 10/50 = 1/5. Step 4: Common mistake is using 28 - 30 = incorrect subtraction, or forgetting to divide by n(S).

√2

Step 1: For n=5, variance σ² = (n²-1)/12 = (25-1)/12 = 24/12 = 2. Step 2: Standard deviation σ = √(variance) = √2. Step 3: Verify: Data = 1,2,3,4,5; mean = 3; Σ(xi-x̄)² = 4+1+0+1+4 = 10; σ² = 10/5 = 2; σ = √2 ✓. Step 4: Common mistake is computing variance as n²/12 without subtracting 1.