Respuesta :
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu = 49.9, \sigma = 3.74[/tex].
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 [tex]Z = 1.04, \mu = 49.9, \sigma = 3.74[/tex].
So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.04 = \frac{X - 49.9}{3.74}[/tex]
[tex]X - 49.9 = 3.8896[/tex]
[tex]X = 53.7896[/tex]
The length of the longest 15% of Atlantic cod in this area is 53.79cm, roundeed to 2 decimal places.
