For the spot speed study distribution, x: N[55, 49], what is the speed of the next vehicle with a probability of 97·5% ? (Take equivalent static on the standard normal distribution (z) = 1·96)
- A64 km/h
- B68 km/hCorrect
- C72 km/h
- D76 km/h
Solution
Concept
Normal distribution of spot speeds
Formula
x = μ + z σ
Step-by-step solution
1. N[55, 49]: mean = 55 km/h, variance = 49, σ = 7 km/h. 2. For 97.5% (one-sided), z = 1.96. 3. x = 55 + 1.96 × 7 = 68.7 km/h ≈ 68 km/h (the nearest option).
Explanation
With a probability of 97.5 percent the next vehicle's speed will not exceed μ + 1.96σ. This is the upper limit of the two-sided 95% confidence band.
Why the other options are wrong
64 km/h is only about μ + 1.3σ, while 72 and 76 km/h are beyond μ + 2.4σ.
Shortcut
x = μ + 1.96 σ.
Common mistake
Taking the variance 49 as the standard deviation.
Correct answer: B) 68 km/h