Final answer:
The probability that the total weight of 64 randomly loaded crates will exceed 7,100 pounds is approximately 6.68%. This is calculated using the Central Limit Theorem, where the mean and standard deviation of the sum of crate weights are used to find a z-score, which yields the probability.
Step-by-step explanation:
To determine the probability that the total weight of 64 crates exceeds the specified truck capacity of 7,100 pounds, we can use the Central Limit Theorem. Since we're dealing with a sample of crates, the distribution of the sample mean can be approximated by a normal distribution given the sample size is large enough (in this case, n=64 is sufficiently large).
First, we calculate the mean weight of 64 crates by multiplying the mean weight of one crate by the number of crates: 110 pounds * 64 = 7040 pounds.
Next, we calculate the standard deviation of the total weight of 64 crates. The standard deviation of the sum is the standard deviation of one crate times the square root of the number of crates: 5 pounds * sqrt(64) = 5 pounds * 8 = 40 pounds.
To find the probability that the sum weight is greater than 7,100 pounds, we need to find the z-score for 7,100 pounds. The z-score is calculated by the formula (X - mean) / standard deviation. For this problem, it is (7100 - 7040) / 40 = 1.5. We can then look up the z-score in a standard normal distribution table, or use a calculator to find that the probability of a z-score being more than 1.5 is approximately 0.0668. Therefore, the probability the total weight exceeds 7,100 pounds is roughly 0.0668, or 6.68%.