Question

Recreational fishers enjoy fishing for the Red Drum fish. In order to check on the Red Drum population, scientists often catch the fish, tag them, and then release them back into the water. The mean length of the Red Drum fish captured by the scientists is 15.4 inches, with a standard deviation of 4.5 inches. 1 A Current Texas Parks and Wildlife regulations (laws) only allow fishermen to keep Red Drum fish that are between 20 and 28 inches long. Fish that are any other length must be thrown back. Assume 2 that the distribution of lengths of Red Drum fish is approximately normally distributed. What is the probability that a Red Drum fish caught on the Texas coast can be brought home for dinner? Show probability notation in your answer.

Answer 1

0.15 is the probability that a Red Drum fish caught on the Texas coast can be brought home for dinner.

Explanation:

We are given the following information in the question:

Mean, μ = 15.4 inches

Standard Deviation, σ = 4.5 inches

We are given that the distribution of lengths of Red Drum fish is a bell shaped distribution that is a normal distribution.

Formula:

`z_(score) = \displaystyle(x-\mu)/(\sigma)`

P(Red Drum fish that are between 20 and 28 inches long)

`P(20 \leq x \leq 28) = P(\displaystyle(20 -15.4)/(4.5) \leq z \leq \displaystyle(28-15.4)/(4.5)) = P(1.0222 \leq z \leq 2.8)\\\\= P(z \leq 2.8) - P(z < 1.0222)\\= 0.997 - 0.847 = 0.15 = 15\%`

`P(20 \leq x \leq 28) = 15\%`

0.15 is the probability that a Red Drum fish caught on the Texas coast can be brought home for dinner.

Answer 2

The probability that a Red Drum fish caught on the Texas coast can be brought home for dinner is 0.1508.

Explanation:

Consider the provided information.

The mean length of the Red Drum fish captured by the scientists is 15.4 inches, with a standard deviation of 4.5 inches.

The Z score formula is:
`Z=(x-\mu)/(\sigma)`

From the above information.

σ=4.5, x=15.4

A Current Texas Parks and Wildlife regulations (laws) only allow fishermen to keep Red Drum fish that are between 20 and 28 inches long.

Therefore,

`P(20<X<28)=P(((20-15.4))/(4.5)<Z<((28-15.4))/(4.5))`

`P(20<X<28)=P((4.6)/(4.5)<Z<(12.6)/(4.5))`
`P(20<X<28)=0.8467-0.9974 =0.1508`

Hence, the probability that a Red Drum fish caught on the Texas coast can be brought home for dinner is 0.1508.