High School

The weights of 1,000 men in a certain town follow a normal distribution with a mean of 150 pounds and a standard deviation of 15 pounds.

From the data, we can conclude that the number of men weighing more than 165 pounds is about ___, and the number of men weighing less than 135 pounds is about ___.

Answer :

Answer:

The number of men weighing more than 165 pounds is about 84.13%, and the number of men weighing less than 135 pounds is about 15.87%.

Step-by-step explanation:

To find the number of men weighing more than 165 pounds and the number of men weighing less than 135 pounds in a normal distribution with a mean of 150 pounds and a standard deviation of 15 pounds, we can use z-scores and the standard normal distribution table (z-table).

1. For the number of men weighing more than 165 pounds:

First, we need to calculate the z-score for 165 pounds:

\(Z = \frac{X - \mu}{\sigma} = \frac{165 - 150}{15} = 1\)

Now, we can find the probability that a randomly selected man weighs more than 165 pounds using the z-table. A z-score of 1 corresponds to a probability of approximately 0.8413. So, about 84.13% of men weigh more than 165 pounds.

2. For the number of men weighing less than 135 pounds:

Similarly, we calculate the z-score for 135 pounds:

\(Z = \frac{X - \mu}{\sigma} = \frac{135 - 150}{15} = -1\)

Using the z-table, a z-score of -1 corresponds to a probability of approximately 0.1587. So, about 15.87% of men weigh less than 135 pounds.

Therefore, the number of men weighing more than 165 pounds is about 84.13%, and the number of men weighing less than 135 pounds is about 15.87%.