Suppose that people's heights (in centimeters) are normally distributed, with a mean of 170 and a standard deviation of 5. We find the heights of 80 people. (You may need to use the standard normal distribution table. Round your answers to the nearest whole number.)
(a) How many would you expect to be between 170 and 180 cm tall?
people
(b) How many would you expect to be taller than 178 cm?
people
The use of this page is more efficient and more accurate than using a standard normal distribution table,
http://davidmlane.com/hyperstat/z_table.html
for the prob(between 170 and 180) is gives .4772
for the prob( > 178) it gives .0548
multiply each of these by the 80 people and round off to the nearest whole number ( can't have partial people)
Verify my answers by you entering the data.
0.56
To answer these questions, we need to use the properties of the normal distribution and the standard normal distribution table.
(a) To find the number of people expected to be between 170 and 180 cm tall, we can calculate the z-scores for the given heights and use the standard normal distribution table.
First, we need to standardize the values of 170 cm and 180 cm using the formula:
z = (x - μ) / σ
where x is the given height, μ is the mean, and σ is the standard deviation.
For 170 cm:
z1 = (170 - 170) / 5 = 0
For 180 cm:
z2 = (180 - 170) / 5 = 2
Next, we need to find the area under the normal distribution curve between z1 and z2 using the standard normal distribution table. The area represents the probability.
From the standard normal distribution table, the area corresponding to z = 2 is 0.9772. The area corresponding to z = 0 is 0.5000.
So, the probability of heights being between 170 and 180 cm is:
P(170 ≤ x ≤ 180) = P(0 ≤ z ≤ 2) = 0.9772 - 0.5000 = 0.4772
To find the number of people, we multiply the probability by the total number of people:
Number of people = probability * total number of people
Number of people = 0.4772 * 80 ≈ 38
Therefore, we would expect approximately 38 people to be between 170 and 180 cm tall.
(b) To find the number of people expected to be taller than 178 cm, we can calculate the z-score for the given height and use the standard normal distribution table.
First, we need to standardize the value of 178 cm using the formula:
z = (x - μ) / σ
where x is the given height, μ is the mean, and σ is the standard deviation.
For 178 cm:
z = (178 - 170) / 5 = 1.6
Next, we need to find the area under the normal distribution curve to the right of z = 1.6 using the standard normal distribution table.
The area corresponding to z = 1.6 is 0.9452.
So, the probability of heights being taller than 178 cm is:
P(x > 178) = P(z > 1.6) = 1 - 0.9452 = 0.0548
To find the number of people, we multiply the probability by the total number of people:
Number of people = probability * total number of people
Number of people = 0.0548 * 80 ≈ 4
Therefore, we would expect approximately 4 people to be taller than 178 cm.