When fishing off the shores of Florida, a spotted seatrout must be between 24 and 30 inches long before it can be kept; otherwise, it must be returned to the waters. In a region of the Gulf of Mexico, the lengths of the spotted seatrout that are caught, are normally distributed with a mean of 22 inches, and a standard deviation of 4 inches. What is the probability that a fisherman catches a spotted seatrout within the legal limits?

Answer:  0.2858Step-by-step explanation:Given : When fishing off the shores of Florida, a spotted seatrout must be between 24 and 30 inches long before it can be kept.In a region of the Gulf of Mexico, the lengths of the spotted seatrout that are caught, are normally distributed with [tex]\mu=22\text{ inches}[/tex] and [tex]\sigma=4\text{ inches}[/tex].Let x be the lengths of the spotted seatrout that are caught .Using formula [tex]z=\dfrac{x-\mu}{\sigma}[/tex]For x= 24 [tex]z=\dfrac{24-22}{4}=0.5[/tex]For x= 30 [tex]z=\dfrac{30-22}{4}=2[/tex]Then, by using the z-value table , the probability that a fisherman catches a spotted seatrout within the legal limits will be :-[tex]P(24<x<30)=P(0.5<z<2)\\\\=P(z<2)-P(z<0.5)\\\\=0.9772498-0.6914625\\\\=0.2857873\approx0.2858[/tex]Hence, the probability that a fisherman catches a spotted seatrout within the legal limits = 0.2858