Question

the lengths of mature trout in a local lake are approximately normally distributed with a mean of $mu$ 12.5 inches, and a standard deviation of $sigma$ = 1.7 inches. fill in the indicated boxes. find the z - score corresponding to a fish that is 13.7 inches long. round your answer to the nearest hundredth as needed. $z=$ how long is a fish that has a z - score of 0.4? round your answer to the nearest tenth as needed. inches question help: video message instructor submit question

Explanation:

Step1: Calculate the z - score for \(x = 13.7\)

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\).
Given \(\mu = 12.5\), \(\sigma=1.7\), and \(x = 13.7\).
Substitute the values into the formula: \(z=\frac{13.7 - 12.5}{1.7}\)

$$z=\frac{1.2}{1.7}\approx0.71$$

Step2: Calculate the value of \(x\) for \(z = 0.4\)

The formula \(z=\frac{x-\mu}{\sigma}\) can be rewritten as \(x=\mu+z\sigma\).
Given \(\mu = 12.5\), \(\sigma = 1.7\), and \(z = 0.4\).
Substitute the values into the formula: \(x=12.5+0.4\times1.7\)

$$x=12.5 + 0.68=13.2$$

Answer:

The z - score for a 13.7 - inch fish is approximately \(0.71\).
The length of a fish with a z - score of \(0.4\) is \(13.2\) inches.