Suppose that people's heights (in centimeters) are normally distributed, with a mean of 165 and a standard deviation of 6. We find the heights of 80 people.
(a) How many would you expect to be between 159 and 171 cm tall
(b) How many would you expect to be taller than 160 cm
To find the number of people that would be between a certain height range or taller than a certain height, you can use the properties of the normal distribution and the z-score.
(a) To find the number of people between 159 and 171 cm tall, we first need to standardize these values using the z-score formula:
z = (x - μ) / σ
For 159 cm:
z1 = (159 - 165) / 6 = -1
For 171 cm:
z2 = (171 - 165) / 6 = 1
Next, we need to find the area under the curve between these two z-scores. We can use a standard normal distribution table or a calculator to find the probability associated with each z-score.
Using a standard normal distribution table, the probability associated with z = -1 is approximately 0.1587, and the probability associated with z = 1 is also approximately 0.1587.
To find the number of people between 159 and 171 cm, we subtract the probability associated with -1 from the probability associated with 1:
P(-1 < z < 1) = 0.1587 - 0.1587 = 0
Since the probability is 0, we would expect no people between 159 and 171 cm tall.
(b) To find the number of people taller than 160 cm, we need to find the probability of the z-score being greater than 160 cm.
First, we standardize the value of 160 cm using the z-score formula:
z = (x - μ) / σ
z = (160 - 165) / 6
z = -5 / 6
Using a standard normal distribution table or a calculator, we find the probability associated with z = -5 / 6 is approximately 0.6615.
To find the number of people taller than 160 cm, we subtract the probability associated with z = -5 / 6 from 1 (since we want the probability of being above 160 cm):
P(z > -5 / 6) = 1 - 0.6615
P(z > -5 / 6) ≈ 0.3385
Since the probability is 0.3385, we would expect approximately 0.3385 * 80 = 27 people to be taller than 160 cm.