According to a report by the U.S. Fish and Wildlife Service, the mean length of six-year-old rainbow trout in the Arolik River in Alaska is  millimeters with a standard deviation of 41  millimeters. Assume these lengths are normally distributed.

(a) What proportion of six-year-old rainbow trout are less than 451  millimeters long?

(b) What proportion of six-year-old rainbow trout are between  420 and  518 millimeters long?

(c) Is it unusual for a six-year-old rainbow trout to be less than 412  millimeters long?

Round the answers to at least four decimal places.

To solve this problem, we'll use the properties of the **normal distribution**. We'll apply the **z-score formula** to standardize the values and use the **standard normal distribution table** (or a calculator) to find the required proportions.

The z-score formula is:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \(x\) is the value for which we're calculating the z-score.
- \(\mu\) is the mean of the distribution.
- \(\sigma\) is the standard deviation.

Given dataTo solve this problem, we'll use the properties of the **normal distribution**. We'll apply the **z-score formula** to standardize the values and use the **standard normal distribution table** (or a calculator) to find the required proportions.

The z-score formula is:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \(x\) is the value for which we're calculating the z-score.
- \(\mu\) is the mean of the distribution.
- \(\sigma\) is the standard deviation.

Given 
- Mean length (\(\mu\)): unknown (you might have missed providing the mean; I'll assume it based on context to complete the calculations, or you can specify it).
- Standard deviation (\(\sigma\)): 41 millimeters.

### (a) What proportion of six-year-old rainbow trout are less than 451 millimeters long?

1. **Find the z-score for 451 millimeters**:

\[
z = \frac{451 - \mu}{41}
\]

2. **Look up the proportion associated with this z-score** from a standard normal distribution table or use a calculator to find the cumulative probability.

---

### (b) What proportion of six-year-old rainbow trout are between 420 and 518 millimeters long?

1. **Find the z-scores for 420 and 518 millimeters**:

\[
z_1 = \frac{420 - \mu}{41}
\]
\[
z_2 = \frac{518 - \mu}{41}
\]

2. **Find the cumulative probabilities for both z-scores** and subtract to get the proportion in between.

---

### (c) Is it unusual for a six-year-old rainbow trout to be less than 412 millimeters long?

1. **Find the z-score for 412 millimeters**:

\[
z = \frac{412 - \mu}{41}
\]

2. **Determine if the proportion is very small** (typically, if the probability is less than 0.05, it is considered unusual).

Please provide the missing mean, and I will complete the calculations for you!

According to a report by the U.S. Fish and Wildlife Service, the mean length of six-year-old rainbow trout in the Arolik River in Alaska is millimeters with a standard deviation of 41 millimeters. Assume these lengths are normally distributed.

(a) What proportion of six-year-old rainbow trout are less than 451 millimeters long?

(b) What proportion of six-year-old rainbow trout are between 420 and 518 millimeters long?

(c) Is it unusual for a six-year-old rainbow trout to be less than 412 millimeters long?

Round the answers to at least four decimal places.

Discover how to solve problems related to the normal distribution of six-year-old rainbow trout lengths in Alaska. Learn to calculate the proportion of fish below a specific length and the probability of lengths within a range using Z-scores. Detailed steps provided with key formulas and solutions.

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