Question

The annual precipitation data of a city is normally distributed with mean 1200 mm and standard deviation 200 mm. The probability that annual precipitation will be more than 1400 mm is

• A. 0.16
• B. 0.68
• C. 0.84
• D. 0.75

Hint:

For normal distribution, \(P(X > \mu + \sigma) \approx 0.16\).
Use the standard normal table for accurate values.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a normal distribution problem.
We need to find \(P(X > 1400)\).

Step 2: Key Formula or Approach:
Convert to standard normal: \(Z = \frac{X - \mu}{\sigma}\).

Step 3: Detailed Explanation:
Given \(\mu = 1200\), \(\sigma = 200\).
For \(X = 1400\): \[ Z = \frac{1400 - 1200}{200} = \frac{200}{200} = 1. \]
We need \(P(X > 1400) = P(Z > 1)\).
From standard normal table, \(P(Z < 1) = 0.8413\).
Thus, \(P(Z > 1) = 1 - 0.8413 = 0.1587 \approx 0.16\).
So the probability is approximately 0.16, which is option (A).
The empirical rule says 68% within 1 standard deviation, so 32% outside, and 16% above 1 standard deviation.