Question

The summer monsoon rains in India follow approximately a Normal distribution with mean 852 millimeters (mm) of rainfall and standard deviation 82 mm. Note: Use Table A to find the proportion or percentage below.

(a) In the drought year 1987, 697 mm of rain fell. In what percent of all years will India have 697 mm or less of monsoon rain?
(b) "Normal rainfall" means within 20% of the long-term average, or between 683 mm and 1022 mm. In what percent of all years is the rainfall normal?

Answer 1

a)

Here, μ = 852, σ = 82 and x = 697. We need to compute P(X <= 697). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (697 - 852)/82 = -1.89

Therefore,
P(X <= 697) = P(z <= (697 - 852)/82)
= P(z <= -1.89)
= 0.0294

b)

Here, μ = 852, σ = 82, x1 = 683 and x2 = 1022. We need to compute P(683<= X <= 1022). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (683 - 852)/82 = -2.06
z2 = (1022 - 852)/82 = 2.07

Therefore, we get
P(683 <= X <= 1022) = P((1022 - 852)/82) <= z <= (1022 - 852)/82)
= P(-2.06 <= z <= 2.07) = P(z <= 2.07) - P(z <= -2.06)
= 0.9808 - 0.0197
= 0.9611