Question

The summer monsoon rains in India follow approximately a Normal distribution with mean 852 millimeters (mm) of rainfall and standard deviation 82 mm. (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? 2.94% 47.06% 2.5% 97.06% (b) "Normal rainfall" means within 20 S of the long-term average, or between 682 mm and 1022 mm. In what pereent of all years is the rainfall normal? 3.84% 20% 99.18% 96.16%

Answer 1

a) The summer monsoon rains in India follow a normal distribution with mean `mu = 852` millimeters (mm) of rainfall and standard deviation `sigma = 82` mm. In the drought year 1987, `X = 697` mm of rain fell.The probability that the rainfall is less than or equal to 697 mm is calculated as follows:Since the distribution is normal, we have: `z = (X - mu)/sigma = (697 - 852)/82 = -1.89`Using a standard normal table, we find that the probability of a value less than or equal to -1.89 is `0.0294` or `2.94%`.Therefore, in `2.94%` of all years, India will have 697 mm or less of monsoon rain. (b) "Normal rainfall" means within `20` mm of the long-term average, or between `682` mm and `1022` mm. The probability that the rainfall is normal is calculated as follows:We have to find the probability that the rainfall is between `682` mm and `1022` mm. Since the distribution is normal, we have: `z_1 = (682 - 852)/82 = -2.07` and `z_2 = (1022 - 852)/82 = 2.07`Using a standard normal table, we find that the probability of a value less than or equal to -2.07 is `0.0196` and the probability of a value less than or equal to 2.07 is `0.9804`.Therefore, the probability that the rainfall is between `682` mm and `1022` mm is `0.9804 - 0.0196 = 0.9608` or `96.08%`.Therefore, in `96.08%` of all years, the rainfall is normal.