Question

The annual per capita consumption of fresh apples​ (in pounds) in a nearby state can be approximated by a normal​ distribution, with a mean of 18.3 pounds and a standard deviation of 3.7 pounds. ​

(a) What is the smallest annual per capita consumption of apples that can be in the top​ 25% of​ consumptions?

​(b) What is the largest annual per capita consumption of apples that can be in the bottom​ 15% of​ consumptions?

Answer 1

Solution :

Given that,

mean = = 18.3

standard deviation = = 3.7

a)

P(Z > z ) = 25%

1 - P(Z < z ) = 0.25

P(Z < z ) = 1- 0.25

P(Z < z ) = 0.75

P(Z < 0.67 ) = 0.75

z = 0.75

Using z-score formula

X = z* +

X = 0.67*3.7 + 18.3

= 20.8

The smallest annual per capita consumption of apples that can be in the top​ 25% of​ consumptions is 20.8

b)

P(Z < z ) = 15%

P(Z < z ) = 0.15

P(Z < -1.04) = 0.15

z = -1.04

Using z-score formula

X = z* +

X = -1.04*3.7 + 18.3

= 14.5

The largest annual per capita consumption of apples that can be in the bottom​ 15% of​ consumptions is 14.5