assume the mean height of soldiers to be 68.22 inches with a variance of 10.8 inches. how many soldiers in a regiment of 1000 would you expect to be over 6 feet tall

assume the mean height of soldiers to be 68.22 inches with a variance of 10.8 inches. how many soldiers in a regiment of 1000 would you expect to be over 6 feet tall

To find the number of soldiers you would expect to be over 6 feet tall in a regiment of 1000, first, we need to convert 6 feet to inches.

Since 1 foot is equal to 12 inches, 6 feet is equal to 6 * 12 = 72 inches.

Next, we need to calculate the standard deviation from the variance. The standard deviation (σ) is the square root of the variance (σ^2). So, in this case, the standard deviation is the square root of 10.8, which is √10.8 ≈ 3.29 inches.

To find the number of soldiers over 6 feet tall, we can use the normal distribution to determine the probability of a soldier being taller than 72 inches, assuming a normal distribution with a mean of 68.22 inches and a standard deviation of 3.29 inches.

Using a standard normal distribution table or a calculator, we can find the probability of a soldier being taller than 72 inches. Let's denote this probability as P(Z > (72-68.22)/3.29).

P(Z > (72-68.22)/3.29) ≈ P(Z > 1.15)

From the standard normal distribution table or a calculator, we find that P(Z > 1.15) ≈ 0.1251.

Since the probability mentioned above is the probability of a soldier being taller than 72 inches, we can multiply it by the total number of soldiers (1000) to estimate the number of soldiers over 6 feet tall.

Number of soldiers over 6 feet tall = 0.1251 * 1000 ≈ 125.1

However, since the number of soldiers must be a whole number, we round down the fractional part.

Therefore, in a regiment of 1000 soldiers, you would expect approximately 125 soldiers to be over 6 feet tall.

To solve this question, we need to convert the height of 6 feet into inches as the given mean height is in inches. One foot is equivalent to 12 inches, so 6 feet would be equal to 6 * 12 = 72 inches.

Next, we need to find the standard deviation, which is the square root of the variance. The standard deviation is √10.8 ≈ 3.29 inches.

Now we can use the concept of the normal distribution to estimate the number of soldiers over 6 feet tall. The mean height is 68.22 inches with a standard deviation of 3.29 inches.

Since the question doesn't specify whether the height follows a normal distribution, we will assume it does. We can use a z-score to find the proportion of soldiers above 6 feet tall.

The z-score formula is: z = (x - μ) / σ
Where:
z = the z-score
x = the given value (72 inches in this case)
μ = the mean height (68.22 inches)
σ = the standard deviation (3.29 inches)

Using the formula, we can calculate the z-score:
z = (72 - 68.22) / 3.29 ≈ 1.15

To find the proportion of soldiers above 6 feet tall, we need to find the area under the normal curve to the right of the z-score. We can consult a standard normal distribution table or use a statistical calculator to find the corresponding probability.

Looking up the z-score of 1.15 in a standard normal distribution table, we find that the area to the right is 0.8749.

This means that approximately 87.49% of soldiers have a height below 6 feet, so the remaining 100% - 87.49% = 12.51% would be above 6 feet tall.

To calculate the expected number of soldiers over 6 feet tall in a regiment of 1000, we can multiply the proportion (12.51%) by the total number of soldiers:
Expected number = 0.1251 * 1000 = 125.1

Therefore, we would expect around 125 soldiers in a regiment of 1000 to be over 6 feet tall.