Answer:
14
Step-by-step explanation:
In the given histogram, the vertical axis shows frequency density. The horizontal axis has a continuous scale, and there are no gaps between the columns. The left-hand edge of a bar corresponds to the lower class boundary, and the right-hand edge of a bar corresponds to the upper class boundary.
Therefore, the classes of the given histogram, where h is the height in cm of each tree, are:
- 100 < h ≤ 200
- 200 < h ≤ 250
- 250 < h ≤ 300
- 300 < h ≤ 400
- 400 < h ≤ 450
- 450 < h ≤ 600
- 600 < h ≤ 800
- 800 < h
To estimate the number of trees with a height greater than 500 cm, we first need to determine the scale of the frequency density axis.
We can use this formula to find frequency:
[tex]\large\boxed{\textsf{Frequency}=\textsf{Class width}\times \textsf{Frequency density}}[/tex]
The frequency in a class is proportional to the area of its bar. In other words, frequency = k × area of bar (where k is a number).
We are told that the number of trees for which 300 < h ≤ 400 is 8 fewer than the number of trees for which 400 < h ≤ 500, so:
- Let x be the number of trees for which 400 < h ≤ 500.
- Let (x - 8) be the number of trees for which 300 < h ≤ 400.
Class 400 < h ≤ 500 is divided into two sub-groups: 400 < h ≤ 450 and 450 < h ≤ 500.
The width of class 400 < h ≤ 450 is 50, and its bar has a height of 37.5 small squares. Hence, its frequency density (height) is 37.5k.
The width of class 450 < h ≤ 500 is 50, and its bar has a height of 12.5 small squares. Hence, its frequency density (height) is 12.5k.
Therefore:
[tex]x = 50 \times 37.5k+50 \times 12.5k\\\\x=1875k+625k\\\\x=2500k[/tex]
The width of class 300 < h ≤ 400 is 100, and its bar has a height of 15 small squares. Hence, its frequency density (height) is 15k.
Therefore:
[tex]x - 8 = 15k \times 100\\\\x-8=1500k[/tex]
Substitute in x = 2500k and solve for k:
[tex]2500k-8=1500k\\\\2500k-1500k=8\\\\1000k=8\\\\k=0.008[/tex]
Therefore, the scale of the frequency density axis is 0.008 per increment or 0.04 per 5 increments.
To find an estimate for the number of trees (frequency) that have a height greater than 500 cm, we simply multiply the class width by the frequency density for classes 500 < h ≤ 600 and 600 < h ≤ 800.
The width of class 500 < h ≤ 600 is 100, and the frequency density of this bar is 0.1, so the frequency of this class is:
[tex]\textsf{Frequency} = 100 \times 0.1 = 10[/tex]
The width of class 600 < h ≤ 800 is 200, and the frequency density of this bar is 0.02, so the frequency of this class is:
[tex]\textsf{Frequency} = 200 \times 0.02 = 4[/tex]
Therefore, an estimate for the number of trees that have a height greater than 500 cm is:
[tex]h > 500=10 + 4 \\\\h > 500= 14[/tex]
