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dezoksy [38]
2 years ago
15

There are many regulations for catching lobsters off the coast of New England including required permits, allowable gear, and si

ze 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.
Required:
What is the probability that the sample mean carapace length is more than 4.25 inches?
Mathematics
1 answer:
Georgia [21]2 years ago
6 0

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

    = \frac{2.13}{\sqrt{60} }

    = \frac{2.13}{7.746} = 0.275

⇒μₓ⁻ = 0.275

Now,

P(X⁻ > 3.37) = 1 - P( X⁻ <  3.37)

                   = 1 - P(z <  \frac{3.37 - 4.25}{0.275} )

                   = 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

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