Match each standard normal interval in Column I with its probability in Column…
2024
Match each standard normal interval in Column I with its probability in Column II. Let Z follow the standard normal distribution N(0, 1).
Column I: Standard normal interval | Column II: Probability |
|---|---|
A. -1 ≤ Z ≤ ∞ | I. 0.98 |
B. -2 ≤ Z ≤ 1 | II. 0.48 |
C. -∞ ≤ Z ≤ 2 | III. 0.84 |
D. 0 ≤ Z ≤ 2 | IV. 0.82 |
Answer: A. A-III B-IV C-I D-II — ConceptFor a standard normal variable Z, the cumulative distribution function is Φ(z) = P(Z ≤ z). Symmetry gives Φ(-z) = 1 - Φ(z). For any interval, P(a ≤ Z ≤…
- A.
A-III B-IV C-I D-II
- B.
A-IV B-III C-II D-I
- C.
A-I B-IV C-II D-III
- D.
A-II B-III C-I D-IV
Attempted by 3 students.
Show answer & explanation
Correct answer: A
Concept
For a standard normal variable Z, the cumulative distribution function is Φ(z) = P(Z ≤ z). Symmetry gives Φ(-z) = 1 - Φ(z).
For any interval, P(a ≤ Z ≤ b) = Φ(b) - Φ(a), with Φ(∞) = 1 and Φ(-∞) = 0. Standard table values are Φ(0) = 0.5000, Φ(1) = 0.8413, and Φ(2) = 0.9772.
Application
For A, P(-1 ≤ Z ≤ ∞) = 1 - Φ(-1) = Φ(1) = 0.8413 ≈ 0.84, which pairs A with III.
For B, P(-2 ≤ Z ≤ 1) = Φ(1) - Φ(-2) = 0.8413 - 0.0228 = 0.8185 ≈ 0.82, which pairs B with IV.
For C, P(-∞ ≤ Z ≤ 2) = Φ(2) = 0.9772 ≈ 0.98, which pairs C with I.
For D, P(0 ≤ Z ≤ 2) = Φ(2) - Φ(0) = 0.9772 - 0.5000 = 0.4772 ≈ 0.48, which pairs D with II.
Cross-check
The whole left tail through Z = 2 must have the largest probability, so C pairs with 0.98.
By symmetry, the area from 0 to 2 is about half the central area from -2 to 2, so 0.48 for D is consistent.
Therefore, the matching is A-III, B-IV, C-I, D-II.