Answer :
The Central Limit Theorem allows us to assume a normal distribution for the total weight of a large number of crabs. Calculating the z-score for a total weight of 150 kg gives a probability of about 5.3% that the actual total weight is less than this.
Since we know the expected value and standard deviation of the weight of one crab, we can calculate an estimated total weight for all the crabs in one net. It's important to remember that the Central Limit Theorem applies here: as the sample size (in this case, number of crabs) gets larger, the distribution of the sum tends towards a normal distribution.
The expected value for the total weight of 112 crabs is simply 112 * 1.4 kg = 156.8 kg. The standard deviation for the total weight of 112 crabs is sqrt(112) * 0.4 kg = 4.2 kg. Now we're interested in finding the probability that the total weight is less than 150 kg. We convert this to a z-score using the formula Z = (X - μ) / σ, giving us Z = (150 kg - 156.8 kg) / 4.2 kg = -1.62. Consulting a Z-table, we find this score corresponds to a cumulative probability of about 0.053. Therefore, assuming our distribution is approximately normal, the probability that the total weight of the crabs in the net is less than
the probability that the total weight of the crabs in the net is less than 150 kg is approximately 5.3%, assuming the Central Limit Theorem applies and the distribution tends towards normal.
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