Respuesta :
Answer:
The length in cm of the longest 10% of Atlantic cod in this area is at least 53.92 cm.
Step-by-step explanation:
When the distribution is normal, we use 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]
The Z-score measures how many standard deviations the measure is from the mean. 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, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this question:
[tex]\mu = 47.8, \sigma = 3.72[/tex]
What is the length in cm of the longest 10% of Atlantic cod in this area
At least x cm.
x is the 100 - 10 = 90th percentile, which is X when Z has a pvalue of 0.9. So X when Z = 1.645. Then
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.645 = \frac{X - 47.8}{3.72}[/tex]
[tex]X - 47.8 = 1.645*3.72[/tex]
[tex]X = 53.92[/tex]
The length in cm of the longest 10% of Atlantic cod in this area is at least 53.92 cm.
