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%.