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.

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(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.

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(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!