Suppose that people's heights (in centimeters) are normally distributed, with a mean of 175 and a standard deviation of 6. We find the heights of 40 people.
(a) How many would you expect to be between 169 and 181 cm tall?
1
(b) How many would you expect to be taller than 170 cm?
2
Use process indicated in later post.
To answer these questions, we need to use the concept of the standard normal distribution and the z-score.
For question (a), we want to find the number of people whose height falls between 169 and 181 cm.
First, we need to standardize these values using the formula for the z-score:
z = (x - μ) / σ
Where:
- x is the individual height
- μ is the mean of the population (175 cm)
- σ is the standard deviation of the population (6 cm)
For 169 cm:
z1 = (169 - 175) / 6 = -1
For 181 cm:
z2 = (181 - 175) / 6 = 1
Now, we can use these z-scores to find the probability of a person having a height within this range using a standard normal distribution table or a calculator.
P(-1 ≤ z ≤ 1) = P(z ≤ 1) - P(z ≤ -1)
Using the standard normal distribution table, we find that P(z ≤ 1) is approximately 0.8413 and P(z ≤ -1) is approximately 0.1587.
Therefore, P(-1 ≤ z ≤ 1) = 0.8413 - 0.1587 = 0.6826.
This means that approximately 68.26% of people would be expected to have heights between 169 and 181 cm.
Since we are dealing with 40 people, we can multiply this probability by 40 to find the expected number of people falling within this range:
Expected number = 0.6826 * 40 ≈ 27 (rounded to the nearest whole number)
Therefore, we would expect around 27 people to have heights between 169 and 181 cm.
For question (b), we want to find the number of people who are taller than 170 cm.
First, we need to calculate the z-score for 170 cm:
z = (170 - 175) / 6 = -0.83
Using the standard normal distribution table, we find that P(z ≤ -0.83) is approximately 0.2033.
This means that approximately 20.33% of people would be expected to have heights less than or equal to 170 cm.
To find the number of people taller than 170 cm, we can subtract this probability from 1 and multiply it by 40:
Expected number = (1 - 0.2033) * 40 ≈ 2 (rounded to the nearest whole number)
Therefore, we would expect around 2 people to have heights taller than 170 cm.