Question

Summer monsoons in Costa Rica are essential for the country's agriculture. Since the 1970's, records show that the amount of monsoon rainfall varies from year to year with a mean of 800mm and a standard deviation of 80mm.

A) At 70% of all years, between what two values is the amount of rainfall?

B) Least amount of monsoons in the driest 2.5% of all years is around?

C) What is the amount of monsoons exceeding 2.5% of all years?

Answer 1

Mean = 800 mm

standard deviation = 80mm

A) At 70% of all years, between what two values is the amount of rainfall?

Let, the values be x_l and x_u

P[ x_l < X < x_u ] = 70% = 0.7

P[ x_l < X < x_u ] = P[ ( x_l - mean )/sd < ( X - mean )/sd < ( x_u - mean )/sd ] = 0.7

P[ ( x_l - 800 )/80 < ( X - 800 )/80 < ( x_u - 800 )/80 ] = 0.7

P[ ( x_l - 800 )/80 < Z < ( x_u - 800 )/80 ] = 0.7

Also, P[ -1.4 < Z < 1.4 ] = 0.7

( x_l - 800 )/80 = -1.4

x_l - 800 = -1.4*80

x_l - 800 = -112

x_l = 800 - 112

x_l = 688 mm

( x_u - 800 )/80 = 1.4

x_u - 800 = 1.4*80

x_u - 800 = 112

x_u = 800 + 112

x_u = 912 mm

B) The least amount of monsoons in the driest 2.5% of all years is around?

Let, the value be x_l

P[ x_l < X ] = 2.5% = 0.025

P[ x_l < X ] = P[ ( x_l - mean )/sd < ( X - mean )/sd ] = 0.025

P[ x_l < X ] = P[ ( x_l - 800 )/80 < ( X - 800 )/80 ] = 0.025

P[ ( x_l - 800 )/80 < Z ] = 0.025

Also, P[ Z < -1.96 ] = 0.025

( x_l - 800 )/80 = -1.96

x_l - 800 = -1.96*80

x_l - 800 = -156.8

x_l = 800 - 156.8

x_l = 643.2 mm

C) What is the amount of monsoons exceeding 2.5% of all years?

Let, the value be x_u

P[ X > x_u ] = 2.5% = 0.025

P[ X > x_u ] = P[ ( X - mean )/sd < ( x_u - mean )/sd ] = 0.025

P[ X > x_u ] = P[ ( X - 800 )/80 < ( x_u - 800 )/80 ] = 0.025

P[ Z > ( x_u - 800 )/80 ] = 0.025

Also, P[ Z > 1.96 ] = 0.025

( x_u - 800 )/80 = 1.96

x_u - 800 = 1.96*80

x_u - 800 = 156.8

x_u = 800 + 156.8

x_u = 956.8 mm