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 60 people.
(a) How many would you expect to be between 167 and 183 cm tall?
(b) How many would you expect to be taller than 170 cm?
http://davidmlane.com/hyperstat/z_table.html is an excellent calculator. Enter mean, std deviation, then look at the shaded areas. You can multipy those by 60 to get numbers.
To answer these questions, we will use the standard normal distribution and the Z-score formula.
The Z-score is calculated as:
Z = (X - μ) / σ
Where:
X is the value you want to find the probability for,
μ is the mean of the distribution,
and σ is the standard deviation of the distribution.
(a) To find the number of people expected to be between 167 and 183 cm tall:
1. Calculate the Z-score for the lower value (167 cm):
Z1 = (167 - 175) / 6 = -1.33
2. Calculate the Z-score for the upper value (183 cm):
Z2 = (183 - 175) / 6 = 1.33
3. Use a Z-score table or a statistical calculator to find the probability corresponding to each Z-score.
Using the Z-score table, you can find that the probability corresponding to Z1 is approximately 0.0918, and the probability corresponding to Z2 is approximately 0.9082.
4. Subtract the probability corresponding to Z1 from the probability corresponding to Z2:
0.9082 - 0.0918 = 0.8164
This means that approximately 81.64% of the 60 people would be expected to have heights between 167 and 183 cm.
(b) To find the number of people expected to be taller than 170 cm:
1. Calculate the Z-score for 170 cm:
Z = (170 - 175) / 6 = -0.83
2. Use the Z-score table or a statistical calculator to find the probability corresponding to the Z-score.
Looking up the Z-score of -0.83 in the table, you can find that the corresponding probability is approximately 0.2033.
3. Subtract the probability from 1 since we want to find the probability of being taller than 170 cm:
1 - 0.2033 ≈ 0.7967
0.7967 is the probability that a randomly selected person's height would be greater than 170 cm. To find the number of people, we multiply this probability by the total number of individuals, which is 60:
0.7967 × 60 = 47.8
Therefore, approximately 47.8 people would be expected to be taller than 170 cm. Since the number of people must be whole, we can round this number up to 48.
So, (a) Approximately 81.64% or 49 people would be expected to have heights between 167 and 183 cm.
(b) Approximately 48 people would be expected to be taller than 170 cm.