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
A. Current Texas Parks and Wildlife regulations (laws) only allow fishermen to keep Red Drum fish that that the distribution of lengths of Red Drum fish is approximately normally distributed. What is the the Red Drum fish captured by the scientists is 15.4 inches, with a standard deviation of 4.5 inches are between 20 and 28 inches long. Fish that are any other length must be thrown back. Assume probability that a Red Drum fish caught on the Texas coast can be brought home for dinner? Show probability notation in your answer.
B. what is the probability that a randomly selected Red Drum fish is longer than 25 inches? would a 25-inch Red Drum fish be unusually long? Show probability notation in your answer.
C. What is the likelihood that a random Red Drum fish is shorter than 9 inches? Would a Red Drum fish that was 9 inches long be unusually short?
D. If you wanted to find the length that separates the top 2% (longest) of Red Drum fish from the lower 98% of the Red Drum fish, what Z-score would you use? D E How long would the top 2% of Red Drum fish be?

Answer 1

Answer:

A. P(20 ≤ X ≤ 28) =  0.1513

B. P(X > 25) = 0.01659

C. P(X < 9) = 0.07780

D.  Z-score: 2.054

E. X= 24.625 inches

Step-by-step explanation:

Hello!

The study variable is X: length of a red Drumfish. (inches)

X≈N(μ;δ²)

μ= 15.4 inches

σ= 4.5 inches

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.

A. 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.

According to the regulations listed on the text, if the fish is between 20 and 28 inches, It can be kept, so the probability asked in this question is:

P(20 ≤ X ≤ 28)

You can rewrite is as:

⇒ P(X ≤ 28) - P(X ≤ 20)

To know the probability you have to use the population information and the standard normal distribution, the first step is to standardize both probabilities:

The formula of the standard normal is Z=(X-μ)/σ

⇒ P(Z ≤ (28-15.4)/4.5) - P(Z ≤ (20-15.4)/4.5) = P(Z ≤ 2.8) - P(Z ≤ 1.02)

Now you look at the table of the Z-distribution for the cumulative probabilities for both Z-values

⇒ 0.99744 - 0.84614= 0.1513

B. What is the probability that a randomly selected Red Drum fish is longer than 25 inches? would a 25-inch Red Drum fish be unusually long? Show probability notation in your answer.

You have to calculate the probability of a red Drum to be longer than 25 inches, symbolically:

P(X > 25)

Since the tables present cumulative probabilities P(Z<Z$_{1-\alpha }$), and keeping in mind that the maximum value that a probability can take is 1, you can rewrite the asked probability as:

⇒ 1 - P(X ≤ 25)

Now you can standardize the value:

⇒ 1 - P(Z ≤ (25 - 15.4)/4.5) = 1 - P(Z ≤ 2.13)

Then you look for the accumulated probability in the table:

⇒ 1 - 0.98341= 0.01659

The probability of finding a red Drum fish larger than 25 inches is 0.01659, this is very low. You could say that it is quite unusual to find a fish this long.

C. What is the likelihood that a random Red Drum fish is shorter than 9 inches? Would a Red Drum fish that was 9 inches long be unusually short?

P(X < 9)

Since we are calculating a cumulative probability "less than" the value of interest, you can directly standardize it and obtain the value from the table:

⇒ P(Z < (9-15.4)/4.5) = P(Z < -1.42) = 0.07780

With this kind of probability, you could also say that is unusual to find a fish this short.

D. If you wanted to find the length that separates the top 2% (longest) of Red Drum fish from the lower 98% of the Red Drum fish, what Z-score would you use?

In this situation, instead of having a fish length, you are given the probability and need to do a reverse search of the Z value. This means, instead of looking at the Z value in the margins of the table and obtaining the probability value from the body, you have to look for the probability in the body of the table and look for the corresponding number in the margins.

So, let's call this particular Z value "a"

This value has below 98% of the population and above 2% of the population, symbolically: 98% ≤ a ≤ 2%, in the attached image you can see this graphically.

Symbolically:

P(Z > a) = 0.02

-or-

P(Z < a) = 0.98 (This is the one to use to look in the Z-table, remember that the table shows cumulative probabilities from least to greatest)

The Z-score is 2.05 (See 2nd attachment)

E. How long would the top 2% of Red Drum fish be?

In this case, you have to use the Z score obtained in D. and reverse the standardization of the variable value. This means, that you need to clear the value of X from the formula:

Z= ( X - μ ) / σ

⇒ (Z * σ) = X - μ

⇒ X= (Z * σ) + μ

⇒ X= (2.05 * 4.5) + 15.4 = 24.625

The top 2% of the red Drumfish are 24.625 inches or larger.

I hope it helps!