Answer:
The length of the longest 15% of Atlantic cod in this area is 53.79cm, roundeed to 2 decimal places.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X.
In this problem, we have that:
A study conducted in 2012 found the length of Atlantic cod caught in nets in Karlskrona to have a mean of 49.9 cm and a standard deviation of 3.74 cm, so
.
What is the length in cm of the longest 15% of Atlantic cod in this area?
We have to find the value of X for the value of Z that has a pvalue of 0.85.
Looking at the zscore table, we have that Z = 1.04 has a pvalue of 0.8508. So, we have to find the value of X when
.
So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![1.04 = \frac{X - 49.9}{3.74}](https://tex.z-dn.net/?f=1.04%20%3D%20%5Cfrac%7BX%20-%2049.9%7D%7B3.74%7D)
![X - 49.9 = 3.8896](https://tex.z-dn.net/?f=X%20-%2049.9%20%3D%203.8896)
![X = 53.7896](https://tex.z-dn.net/?f=X%20%3D%2053.7896)
The length of the longest 15% of Atlantic cod in this area is 53.79cm, roundeed to 2 decimal places.