Central Limit Theorem Probability question

Question

The weight of people in a small town in Missouri is known to be normally distributed with a mean of 198 pounds and a standard deviation of 31 pounds. On a raft that takes people across the river, a sign states, “Maximum capacity 3,582 pounds or 18 persons.” What is the probability that a random sample of 18 persons will exceed the weight limit of 3,582 pounds?

Answer 1

The weights of people are normally distributed with a mean of 198 pounds and a standard deviation of 31 pounds. We want to find out the probability that 18 randomly chosen persons exceed the weight limit of 3,582 pounds on the raft.

First, calculate the mean and standard deviation for a sample of 18 persons:

1. Sample mean = Population mean = 198 pounds
2. Sample standard deviation = Population standard deviation / sqrt(18) = 31 / sqrt(18) = 7.3 pounds

Next, the total mean weight for 18 persons would be 18 x 198 = 3564 pounds, and the total standard deviation would be 18 x 7.3 = 131.4 pounds.

Now we find the Z-score for a total weight of 3,582 pounds: Z = (3582 - 3564) / 131.4 = 18 / 131.4 = 0.137

Using a standard normal distribution table or calculator, the probability of having a Z-score less than 0.137 is about 0.555. Since we want the other tail (weights greater than 3,582 pounds), we do 1 - 0.555 = 0.445.

So, the probability that a random sample of 18 persons will exceed the weight limit of 3,582 pounds is approximately 0.445 or 44.5%.