Question

There are many regulations for catching lobsters off the coast of New England including required permits, allowable gear, and size prohibitions. The Massachusetts Division of Marine Fisheries requires a minimum carapace length measured from a rear eye socket to the center line of the body shell. Any lobster measuring less than 3.25 inches must be returned to the ocean. The mean carapace length of the lobsters is 4.125 inches with a standard deviation of 1.05 inches. A random sample of 125 lobsters is obtained.

Note that if you do not round your z score to two decimal places, your answers may be marked wrong.

(0.5 pts.) a) Assuming that the sample mean carapace length is greater than 3.97 inches, what is the probability that the sample mean carapace length is more than 4.25 inches? Please use four decimal places.

b) If the sample mean carapace length is less than 3.97 inches, a lobsterman will look for other places to set his traps. What is the probability that a lobsterman will be looking for a different location? Please use four decimal places.

BONUS: Do you think that they will be moving to a new location? Please explain your answer based on your result of the calculated probability above.

Answer 1

Given that mean= 4.125 inches

standard deviation = 1.05

.sample n=  125

a) the sample mean carapace length is greater than 3.97 inches ,probability that the sample mean carapace length is more than 4.25 inches

P( X > 4.25 / X > 3.97) =

P( X > 4.25) = P( Z >  x - μ/ σ / √n)

= 4.25- 4.125 / 1.05 /sqrt(125)
= 0.125/0.0939
= 1.331

P( Z >1.331) =1−P ( Z<1.331 )

=1−0.9082=0.0918.

P( X > 3.97) = P( Z >  x - μ/ σ / √n)

=   3.97- 4.125 / 1.05 /sqrt(125)

= -1.65

P ( Z>1.65 )=0.9505

P( X > 4.25 / X > 3.97) = 0.0918/ 0.9505 =0.0966

b) If the sample mean carapace length is less than 3.97 inches

P( X < 3.97) = P( Z < x - μ/ σ / √n)

=   3.97- 4.125 / 1.05 /sqrt(125)

= -1.65

P ( Z<−1.65 )=1−P ( Z<1.65 )=1−0.9505=0.0495

No, they wont be moving to new location because this is so unlikely to happen