Answer to Question #245194 in Statistics and Probability for marry

Question

5. The number of oranges that grow on an orange tree in Florida is normally distributed with a mean of 136 with a standard deviation of 21 oranges.

a. What percent of the trees grow less than 150 oranges?

b. What percent of trees grow between 90 and 110 oranges?

c. How many oranges does a tree grow if it is in the top 20% of orange producers?

Answer 1

N(136, 441) = 136 + 21N(0,1)

So, the task is to calculate the next probabilities:

(a) P(136 + 21N(0,1) < 150) = P(N(0,1) < 0.67) = 0.75

75% grow less than 150 oranges

(b) P(90 < 136 + 21N(0,1) < 110) = P(-2.19 < N(0,1) < -1.24) = P(N(0,1) < -1.24) - P(N(0,1) < -2.19) = 0.09

9% grow between 90 and 110 oranges

(с) P(136 + 21N(0,1) > a) = 0.2 , then 0.84 ;

P(N(0,1) > ${\frac {a-136} {21}}$) =0.2 -> ${\frac {a-136} {21}}$ = 0.84 -> a = 153.64

A tree grows approximately 154 oranges.