https://file.aiquickdraw.com/mathbot/file/1730381144817ycmo6k0i.jpg

Let’s go through the questions one by one based on the provided information. The fish lengths are normally distributed with a mean \( \mu = 47.4 \) cm and a standard deviation \( \sigma = 3.45 \) cm.

### 1. State the random variable.
   - The random variable is **the length of a fish caught** because we're analyzing the probability of different lengths.

### 2. Find the probability that a fish caught has a length less than 51 cm.
   To calculate this probability \( P(X < 51) \), we need to standardize the value:
   
   \[
   Z = \frac{X - \mu}{\sigma} = \frac{51 - 47.4}{3.45} \approx 1.0435
   \]
   Using a standard normal table or a calculator, we can find \( P(Z < 1.0435) \approx 0.8508 \) (rounded to four decimal places).

   **Answer:** \( P(X < 51) \approx 0.8508 \).

### 3. Find the probability that a fish caught has a length of more than 44 cm.
   To calculate \( P(X > 44) \), we again standardize:

   \[
   Z = \frac{X - \mu}{\sigma} = \frac{44 - 47.4}{3.45} \approx -0.9855
   \]
   Using a standard normal table, we find \( P(Z > -0.9855) \approx 0.8378 \).

   **Answer:** \( P(X > 44) \approx 0.8378 \).

### 4. Find the probability that a fish caught has a length between 43.5 cm and 51.5 cm.
   We calculate \( P(43.5 \leq X \leq 51.5) \) by finding the \( Z \)-scores for both bounds and then finding the probability in between:
   
   For \( 43.5 \):
   \[
   Z = \frac{43.5 - 47.4}{3.45} \approx -1.1304
   \]
   For \( 51.5 \):
   \[
   Z = \frac{51.5 - 47.4}{3.45} \approx 1.1884
   \]
   Using the standard normal table:
   \[
   P(-1.1304 < Z < 1.1884) \approx P(Z < 1.1884) - P(Z < -1.1304) \approx 0.8830 - 0.1292 = 0.7538
   \]

   **Answer:** \( P(43.5 \leq X \leq 51.5) \approx 0.7538 \).

### 5. Is it unusual for a fish caught to have a length of more than 54 cm?
   To check if it's unusual, we calculate \( P(X > 54) \).

   \[
   Z = \frac{54 - 47.4}{3.45} \approx 1.9130
   \]
   Using the standard normal table, \( P(Z > 1.9130) \approx 0.0278 \), which is less than 5%.

   **Answer:** Yes, it is unusual because the probability is less than 5%.

### 6. What length are 15% of all Atlantic cod longer than?
   To find this length, we need the 85th percentile (since 100% - 15% = 85%).

   From the standard normal table, \( Z \approx 1.036 \) for the 85th percentile.

   \[
   X = Z \cdot \sigma + \mu = 1.036 \cdot 3.45 + 47.4 \approx 51.0
   \]

   **Answer:** Approximately 51.0 cm.

Would you like further details on any step? Here are some additional questions for practice:

1. How would you calculate the probability for an interval not symmetric around the mean?
2. What is the probability that a fish is shorter than 43 cm?
3. How do you determine if an event is unusual based on probability?
4. What is the Z-score for a length of 49 cm?
5. How would you calculate the 95th percentile length for these fish?

**Tip:** Remember that probabilities in a normal distribution are calculated using the area under the curve, which corresponds to Z-scores in the standard normal distribution.

The size of fish is very important to commercial fishing. Suppose the length of a certain fish caught in nets has a mean of 47.4 cm and a standard deviation of 3.45 cm. Assume the length of fish is normally distributed.

1. State the random variable.
2. Find the probability that a fish caught has a length less than 51 cm. Round to four decimal places.
3. Find the probability that a fish caught has a length of more than 44 cm. Round to four decimal places.
4. Find the probability that a fish caught has a length between 43.5 and 51.5 cm. Round to four decimal places.
5. Is it unusual for a fish caught to have a length of more than 54 cm? Why or why not?
6. What length are 15% of all Atlantic cod longer than? Round to one decimal place.

Solve probability questions on fish length distribution with a mean of 47.4 cm and standard deviation of 3.45 cm. This problem covers normal distribution, Z-score calculations, and interpretation of probability values. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation