Question

4. The height of maple trees are distributed normally with a mean of 31 meters and a standard deviation of 4 meters.

a) What is the probability of a tree being taller than 33 meters? Represent this graphically as well as numerically.

b) A group of 20 trees is selected at random. What is the probability that the average height of these 20 trees is more than 33 meters? Represent this graphically as well as numerically.

c) A group of 20 trees is selected at random. What is the probability that the average height of these trees is between 30 and 32 meters? Represent this graphically as well as numerically.

Answer 1

Solution :

Given that,

mean = = 31

standard deviation = = 4

a ) P (x > 33 )

= 1 - P (x < 33 )

= 1 - P ( x -  / ) < ( 33 - 31/ 4)

= 1 - P ( z < 2 / 4 )

= 1 - P ( z < 0.5)

Using z table

= 1 - 0.6915

= 0.3085

Probability = 0.3085

b ) n = 20

= 31

= / n = 4 20 = 0.8944

b ) P ( > 33 )

= 1 - P ( < 33 )

= 1 - P ( - /) < ( 33 - 31/ 0.8944)

= 1 - P ( z < 2 / 0.8944 )

= 1 - P ( z < 2.24)

Using z table

= 1 - 0.9875

= 0.0125

Probability = 0.0125

c ) P (30 < < 32 )

P ( 30 - 31 / 0.8944) < ( - / ) < ( 32 - 31 / 0.8944)

P ( -1/ 0.8944 < z < 1 / 0.8944 )

P (-1.12 < z < 1.12 )

P ( z < 1.12 ) - P ( z < -1.12)

Using z table

= 0.8686 - 0.1314

= 0.7372

Probability = 0.7372