Answer :
Answer:
Probability that the sample mean carapace length is more than 4.25 inches = 0.9993
Step-by-step explanation:
Given - 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. For a particular local
municipality, any lobster measuring less than 3.37 inches must
be returned to the ocean. The mean carapace length of the
lobsters is 4.01 inches with a standard deviation of 2.13 inches.
A random sample of 60 lobsters is obtained.
To find - What is the probability that the sample mean carapace length
is more than 4.25 inches?
Proof -
Given that, μ = 4.01, σ = 2.13 , n = 60
Now,
μₓ⁻ = σ / √n
= [tex]\frac{2.13}{\sqrt{60} }[/tex]
= [tex]\frac{2.13}{7.746}[/tex] = 0.275
⇒μₓ⁻ = 0.275
Now,
P(X⁻ > 3.37) = 1 - P( X⁻ < 3.37)
= 1 - P(z < [tex]\frac{3.37 - 4.25}{0.275}[/tex] )
= 1 - P( z < -3.2 )
= 1 - 0.0007
= 0.9993
⇒P(X⁻ > 3.37) = 0.9993
∴ we get
Probability that the sample mean carapace length is more than 4.25 inches = 0.9993