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Answer :
There is a 3.51% probability that a group of 10 persons will exceed the load limit of the elevator.
To find the probability that a group of 10 persons will exceed the load limit of the elevator, we need to calculate the probability that the total weight of the group exceeds 2000 lb.
Since the weights of the people are normally distributed with a mean of 185 lb and a standard deviation of 22 lb, we can use the properties of the normal distribution to solve this problem.
First, we need to calculate the mean and standard deviation of the total weight of the group. The mean of the total weight is simply the mean weight of a person multiplied by the number of persons, which is 185 lb * 10 = 1850 lb.
The standard deviation of the total weight is calculated by taking the square root of the sum of the variances of each person's weight. Since the weights are independent, the variance of the total weight is equal to the sum of the variances of each person's weight. So, the standard deviation of the total weight is sqrt(10) * 22 lb ≈ 69.296 lb.
Next, we can use the normal distribution to calculate the probability that the total weight exceeds 2000 lb. We standardize the value using the formula z = (x - mean) / standard deviation, where x is the threshold value of 2000 lb.
Calculating the z-score: z = (2000 - 1850) / 69.296 ≈ 1.81
Finally, we look up the probability corresponding to this z-score in the standard normal distribution table or use a calculator to find the area under the curve to the right of the z-score. The probability that the group of 10 persons will exceed the load limit of the elevator is approximately 0.0351, or 3.51%.
Therefore, there is a 3.51% probability that a group of 10 persons will exceed the load limit of the elevator.
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