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18. The lengths of Atlantic croaker fish are normally distributed, with a mean of 10 inches and a standard deviation

of 2 inches. An Atlantic croaker fish is randomly selected.
a) Find the probability that the length of the croaker fish is less than 7 inches.
b) Find the probability that the length of the fish is between 7 and 15 inches

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Answer:

a) To find the probability that the length of the croaker fish is less than 7 inches, we need to calculate the z-score of 7 inches using the formula:

z = (x - mu) / sigma

where x is the length of the fish we are interested in, mu is the mean length of the population, and sigma is the standard deviation of the population.

In this case, x = 7, mu = 10, and sigma = 2. So the z-score is:

z = (7 - 10) / 2 = -1.5

We can look up the probability of a z-score less than -1.5 in a standard normal distribution table or use a calculator. The probability is approximately 0.0668 or 6.68%.

Therefore, the probability that the length of the croaker fish is less than 7 inches is 0.0668 or 6.68%.

b) To find the probability that the length of the fish is between 7 and 15 inches, we need to calculate the z-scores of 7 inches and 15 inches using the same formula as above:

z1 = (7 - 10) / 2 = -1.5

z2 = (15 - 10) / 2 = 2.5

We can look up the probability of a z-score between -1.5 and 2.5 in a standard normal distribution table or use a calculator. The probability is approximately 0.9332 or 93.32%.

Therefore, the probability that the length of the fish is between 7 and 15 inches is 0.9332 or 93.32%.

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