Question

the weight of adult men is approximately normally distributed with a mean of 190 pounds and a standard deviation of 30 pounds. a) if you randomly selected one adult man, what is the probability that his weight exceeds 200 pounds? b) in random samples of 3 men, what are the mean and standard deviation of the sum of their weights? c) an elevator in a small apartment building has a maximum weight capacity of 600 pounds. if 3 randomly selected adult men get on the elevator, what is the probability that they exceed the maximum capacity? d) suppose that the weight of the elevator car is 2300 pounds. what is the mean and standard deviation of the total weight of the elevator car and three randomly selected adult males?

Answer 1

a) The probability that a randomly selected adult man's weight exceeds 200 pounds is 1/3.

b) The mean of the sum of their weights will be 3 times the mean weight of an individual man, or 570 pounds.

The standard deviation of the sum will be 52.36 pounds.

c) The mean weight of the 3 men will be 570 pounds, and the standard deviation will be 52.36 pounds.

d) The mean weight of the total weight of the elevator car and 3 men will be 2870 pounds. The standard deviation will be is 52.36 pounds.

Probability can be calculated using various methods, depending on the nature of the event and the information available. For example, if an event has two equally likely outcomes, such as flipping a coin, the probability of either outcome occurring is 0.5, or 50%. If an event is influenced by random variables, such as rolling a die, the probability of a particular outcome occurring can be calculated using probability distributions and statistical methods.

a) To find the probability that a randomly selected adult man's weight exceeds 200 pounds, we need to use the standard normal distribution and the z-score formula:

z = (x - mean) / standard deviation

Plugging in the values, we get:

z = (200 - 190) / 30 = 10/30 = 1/3

b) In random samples of 3 men, the mean of the sum of their weights will be 3 times the mean weight of an individual man, or 3 * 190 = 570 pounds.

The standard deviation of the sum will be the square root of the number of men, multiplied by the standard deviation of an individual man's weight. In this case, the standard deviation of the sum will be the square root of 3, multiplied by 30 pounds, or 30 * sqrt(3) = 52.36 pounds.

c) To find the probability that 3 randomly selected men will exceed the maximum weight capacity of the elevator, we need to use the normal distribution again. The mean weight of the 3 men will be:

3 * 190 = 570 pounds, and the standard deviation will be 52.36 pounds as calculated above.

We can use the z-score formula again to find the probability that the total weight of the 3 men exceeds 600 pounds:

z = (600 - 570) / 52.36 = 30/52.36 = 5/8.875 = 0.56

d) The mean weight of the total weight of the elevator car and 3 men will be 2300 + 570 = 2870 pounds. The standard deviation will be the same as the standard deviation of the sum of the weights of the 3 men, which is 52.36 pounds.

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