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 175 lobsters is obtained.

What is the probability that the sample mean carapace length is more than 4.25 inches? Please use four decimal places.

Answer 1

The probability that the sample mean carapace length is more than 4.25 inches is 0.0764.

Step-by-step explanation:

To find the probability that the sample mean carapace length is more than 4.25 inches, we need to use the properties of the normal distribution. First, we need to calculate the z-score for the sample mean using the formula:
z = (x - μ) / (σ / sqrt(n))
Where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Plugging in the values:
z = (4.25 - 4.125) / (1.05 / sqrt(175))

Simplifying:
z = 1.428571

Next, we need to find the cumulative probability from the z-table. The table will give us the probability of getting a z-score less than or equal to a given value. Since we want the probability that the sample mean is more than 4.25 inches, we need to subtract the cumulative probability from 1:
Probability = 1 - cumulative probability

Looking up the cumulative probability in the z-table, we find that it is approximately 0.9236. Therefore, the probability that the sample mean carapace length is more than 4.25 inches is:
Probability = 1 - 0.9236 = 0.0764