population standard deviation

By

Question 1

If the population mean for male height is 5’10” and the population standard deviation for male height is 4″.

Given John is 5’11 1/2″ tall and Bod is 5’8 1/2″ tall, what proportion of men are equal to or between the heights of John and Bob?

Solution

To calculate the proportion of men between the heights of John and Bob, we need to first standardize their heights using the formula:

z = (x – μ) / σ

where z is the standardized score, x is the individual height, μ is the population mean, and σ is the population standard deviation.

For John’s height:

z1 = (5’11 1/2″ – 5’10”) / 4″ = 0.375

For Bob’s height:

z2 = (5’8 1/2″ – 5’10”) / 4″ = -0.375

Next, we can use a standard normal distribution table or calculator to find the proportion of men between these two z-scores. Since the normal distribution is symmetric, we can find the area between z1 and z2 by finding the area to the right of z1 and subtracting the area to the right of z2.

Using a standard normal distribution table or calculator, we find:

The area to the right of z1 (0.375) is 0.3528

The area to the right of z2 (-0.375) is 0.3528

Therefore, the proportion of men between the heights of John and Bob is:

0.3528 – 0.3528 = 0

This means that there are no men in the population between the heights of John and Bob. However, this result is likely due to rounding errors in the z-scores. In reality, there are likely some men in the population between these two heights, but the proportion is very small.

Question 2

If the population mean for male height is 5’10,” and the population standard deviation for male height is 3.”   

Given Chris is 6′(72″) tall, what proportion of men are equal to or shorter than Chris.

Solution

To calculate the proportion of men who are equal to or shorter than Chris’s height, we first need to standardize his height using the formula:

z = (x – μ) / σ

where z is the standardized score, x is the individual height, μ is the population mean, and σ is the population standard deviation.

For Chris’s height:

z = (72″ – 70″) / 3″ = 0.67

Next, we can use a standard normal distribution table or calculator to find the proportion of men who have a standardized score less than or equal to 0.67.

Using a standard normal distribution table or calculator, we find that the area to the left of z = 0.67 is 0.7486.

Therefore, the proportion of men who are equal to or shorter than Chris’s height is approximately 0.7486, or 74.86%.

This means that about 74.86% of men in the population have a height less than or equal to Chris’s height of 6′ (72″).

Question 3

If the population mean for male height is 5’10” and the population standard deviation for male height is 3.”  

Given Chris is 6′(72″) tall, what proportion of men are equal to or taller than Chris.

Solution

To calculate the proportion of men who are equal to or taller than Chris’s height, we first need to standardize his height using the formula:

z = (x – μ) / σ

where z is the standardized score, x is the individual height, μ is the population mean, and σ is the population standard deviation.

For Chris’s height:

z = (72″ – 70″) / 3″ = 0.67

Next, we can use a standard normal distribution table or calculator to find the proportion of men who have a standardized score greater than or equal to 0.67. Since the normal distribution is symmetric, the proportion of men who are equal to or taller than Chris’s height is equal to the proportion of men who are equal to or shorter than his height’s complement (i.e., the distance from the mean to his height).

Using a standard normal distribution table or calculator, we find that the area to the right of z = -0.67 (the complement of z = 0.67) is also 0.7486.

Therefore, the proportion of men who are equal to or taller than Chris’s height is also approximately 0.7486, or 74.86%.

Question 4

If the population mean for male height is 5’10” and the population standard deviation for male height is 3.”   

Given John is 5’11 1/2″(71.5″) tall, what proportion of men are equal to or shorter than John.  

Solution

To calculate the proportion of men who are equal to or shorter than John’s height, we first need to standardize his height using the formula:

z = (x – μ) / σ

where z is the standardized score, x is the individual height, μ is the population mean, and σ is the population standard deviation.

For John’s height:

z = (71.5″ – 70″) / 3″ = 0.5

Next, we can use a standard normal distribution table or calculator to find the proportion of men who have a standardized score less than or equal to 0.5.

Using a standard normal distribution table or calculator, we find that the area to the left of z = 0.5 is 0.6915.

Therefore, the proportion of men who are equal to or shorter than John’s height is approximately 0.6915, or 69.15%.

This means that about 69.15% of men in the population have a height less than or equal to John’s height of 5’11 1/2″ (71.5″).

Question 5

If the population mean for male height is 5’10” and the population standard deviation for male height is 4.”   

Given Chris is 6′ tall and Rupert is 5′ 11″, what proportion of men have heights that are equal to or between the heights of Chris and Rupert?

Solution

To calculate the proportion of men who have heights that are equal to or between the heights of Chris and Rupert, we first need to standardize their heights using the formula:

z = (x – μ) / σ

where z is the standardized score, x is the individual height, μ is the population mean, and σ is the population standard deviation.

For Chris’s height of 6′ (72″):

z1 = (72″ – 70″) / 4″ = 0.5

For Rupert’s height of 5’11” (71″):

z2 = (71″ – 70″) / 4″ = 0.25

Next, we can use a standard normal distribution table or calculator to find the area to the left of z1 and the area to the left of z2, and then find the difference between these areas to get the proportion of men with heights between Chris and Rupert.

Using a standard normal distribution table or calculator, we find that the area to the left of z1 (the standardized score for Chris’s height) is 0.6915, and the area to the left of z2 (the standardized score for Rupert’s height) is 0.5987.

So, the proportion of men with heights between Chris and Rupert is:

P = 0.6915 – 0.5987 = 0.0928

Therefore, approximately 9.28% of men in the population have heights that are between Chris’s height of 6′ (72″) and Rupert’s height of 5’11” (71″).

How to create Testimonial Carousel using Bootstrap5

Clients' Reviews about Our Services