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 ____

Please help

To find the number of men weighing more than 165 pounds, we need to calculate the Z-score for 165 pounds using the formula:

Z = (X - μ) / σ

Where:
X = 165 pounds (value we want to find the number of men above)
μ = 150 pounds (mean weight)
σ = 15 pounds (standard deviation)

Z = (165 - 150) / 15
Z = 1

Using a standard normal distribution table or calculator, we can find the proportion of values above Z = 1, which is approximately 0.1587.

To find the actual number of men weighing more than 165 pounds, we need to multiply the proportion by the total number of men (1,000):

Number of men weighing more than 165 pounds = 0.1587 * 1,000
Number of men weighing more than 165 pounds ≈ 158.7

Therefore, the number of men weighing more than 165 pounds is about 159.

Similarly, to find the number of men weighing less than 135 pounds, we follow the same steps:

Z = (X - μ) / σ
Z = (135 - 150) / 15
Z = -1

Using the standard normal distribution table or calculator, we find the proportion of values below Z = -1, which is also approximately 0.1587.

Number of men weighing less than 135 pounds = 0.1587 * 1,000
Number of men weighing less than 135 pounds ≈ 158.7

Therefore, the number of men weighing less than 135 pounds is about 159.

To find the number of men weighing more than 165 pounds and less than 135 pounds, we need to calculate the z-scores and use the standard normal distribution table.

Step 1: Calculate the z-score for a weight of 165 pounds.
The z-score formula is:
z = (x - μ) / σ

Where:
x = the value from the data (165 pounds)
μ = the mean (150 pounds)
σ = the standard deviation (15 pounds)

Calculating the z-score:
z = (165 - 150) / 15
z = 1

Step 2: Find the area to the right of the z-score using the standard normal distribution table.
The standard normal distribution table provides the cumulative probability to the left of a given z-score. Since we want the area to the right of the z-score, we subtract the cumulative probability from 1.

From the standard normal distribution table, the cumulative probability for a z-score of 1 is approximately 0.8413.

Area to the right = 1 - 0.8413 = 0.1587

Step 3: Calculate the number of men weighing more than 165 pounds by multiplying the area from Step 2 by the total number of men (1,000).
Number of men weighing more than 165 pounds = 0.1587 * 1000 = 158.7 (approximately)

So, the number of men weighing more than 165 pounds is about 159.

Now let's calculate the number of men weighing less than 135 pounds using the same steps:

Step 1: Calculate the z-score for a weight of 135 pounds.
z = (135 - 150) / 15
z = -1

Step 2: Find the area to the left of the z-score using the standard normal distribution table.
From the standard normal distribution table, the cumulative probability for a z-score of -1 is approximately 0.1587.

Area to the left = 0.1587

Step 3: Calculate the number of men weighing less than 135 pounds by multiplying the area from Step 2 by the total number of men (1,000).
Number of men weighing less than 135 pounds = 0.1587 * 1000 = 158.7 (approximately)

So, the number of men weighing less than 135 pounds is about 159.

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