If 'Z' represents the standard normal variate, then arrange the following…
2024
If 'Z' represents the standard normal variate, then arrange the following probabilities in increasing order:
A. P(0 ≤ Z ≤ 1)
B. P(−∞ ≤ Z ≤ 0)
C. P(1 ≤ Z ≤ 3)
D. P(2 ≤ Z ≤ +∞)
Choose the most appropriate answer from the options given below:
Answer: D. D, C, A, B — ConceptFor a standard normal variable Z, an interval probability is the area under the normal curve between its bounds. Using the cumulative distribution…
- A.
A, B, C, D
- B.
B, C, D, A
- C.
C, D, A, B
- D.
D, C, A, B
Show answer & explanation
Correct answer: D
Concept
For a standard normal variable Z, an interval probability is the area under the normal curve between its bounds. Using the cumulative distribution function Φ, P(a ≤ Z ≤ b) = Φ(b) − Φ(a); symmetry gives Φ(0) = 0.5.
Application
For P(0 ≤ Z ≤ 1), use Φ(1) − Φ(0) = 0.84134 − 0.50000 = 0.34134.
For P(−∞ ≤ Z ≤ 0), use Φ(0) = 0.50000.
For P(1 ≤ Z ≤ 3), use Φ(3) − Φ(1) = 0.99865 − 0.84134 = 0.15731.
For P(2 ≤ Z ≤ +∞), use 1 − Φ(2) = 1 − 0.97725 = 0.02275.
Cross-check
The standard normal curve is symmetric about zero, so the probability to the left of zero is one-half. Tail area decreases as the cutoff moves farther right, which is consistent with the interval beyond 2 having the smallest probability.
Therefore, 0.02275 < 0.15731 < 0.34134 < 0.50000, so the increasing sequence is D, C, A, B.