Question

Suppose a team of biologists has been studying the Pinedale children’s fishing pond. Let represent the length of a single trout taken at random from the pond. This group of biologists has determined that has a normal distribution with mean µ = 10.2 inches and standard deviation σ = 1.4 inches.

a. What is the probability that a single trout taken at random from the pond is between 8 and 12 inches long?

b. What is the probability that the mean length of 16 trout taken at random is between 8 and 12 inches?

Answer 1

Solution :

Given that,

mean = = 10.2

standard deviation = = 1.4

a ) P (8 < x < 12 )

P ( 8 - 10.2 / 1.4) < ( x -  / ) < ( 12- 10.2 / 1.4)

P ( - 2.2 / 1.4 < z < 1.8 / 1.4 )

P (-1.57 < z < 1.28 )

P ( z < 1.28 ) - P ( z < -1.57)

Using z table

= 0.8997 - 0.0582

= 0.8415

Probability = 0.8415

b ) mean = = 160.2

P (8 < x < 12 )

P ( 8 - 16 / 1.4) < ( x -  / ) < ( 12- 16 / 1.4)

P ( - 8 / 1.4 < z < - 4 / 1.4 )

P (-5.71 < z < -2.86)

P ( z < -2.86 ) - P ( z < -5.71)

Using z table

= 0.0021- 0

= 0.0021

Probability = 0.0021