Question

practice problem #1
background: the thickness of the annual rings indicates what type of environmental situation was occurring the time of the trees development. a thin ring usually indicates a rough period of development such as lack of water, forest fires, or insect infestation. on the other hand, a thick ring means a prosperous period of development. use the information from the data table below to create a proper scientific graph and to answer the corresponding questions.

age of trees (in years)	average thickness of annual rings in forest a (millimeters)	average thickness of annual rings in forest b (millimeters)
20	24	28
30	30	35
35	34	38
50	41	45
60	45	51

1. what is the dependent variable?
2. what is the independent variable?
3. what was the average thickness of annual rings for 40 - year old trees in forest a?
4. what is it called when you make predictions within given data, such as made in question #3?
5. what was the mean thickness of annual rings for all trees found in forest b?
6. based on the data shown, what can be concluded about the comparative health of forest a & b?
7. what type of relationship (constant, direct, or indirect) exists between the age of trees and the average thickness of the trees rings? explain.

use pencil to make graph!

Explanation:

Step1: Identify dependent variable

The variable that depends on another is the average thickness of annual rings. It varies based on the age of trees.

Step2: Identify independent variable

The independent variable is the age of trees as it is the variable being manipulated or changed.

Step3: Find thickness for 40 - year - old trees in Forest A

There is no 40 - year data point. But we can assume a linear trend between 35 and 50 years. The difference in age is \(50 - 35=15\) years and the difference in thickness is \(41 - 34 = 7\) mm. The rate of change is \(\frac{7}{15}\) mm per year. From 35 years to 40 years (a 5 - year increase), the increase in thickness is \(\frac{7}{15}\times5=\frac{7}{3}\approx2.33\) mm. So the thickness for 40 - year - old trees in Forest A is approximately \(34+\frac{7}{3}=34 + 2.33 = 36.33\) mm.

Step4: Define prediction within data

This is called interpolation.

Step5: Calculate mean for Forest B

The data points for Forest B are 24, 28, 35, 38, 45, 51. The sum is \(24 + 28+35 + 38+45+51=221\). There are 6 data points. The mean is \(\frac{221}{6}\approx36.83\) mm.

Step6: Conclude forest health

On average, the annual rings in Forest B are thicker than those in Forest A at each age interval, indicating that, on average, trees in Forest B have had a more prosperous development period.

Step7: Determine relationship

There is a direct relationship. As the age of the trees increases, the average thickness of the tree's rings also generally increases.

Answer:

1. Average thickness of annual rings
2. Age of trees
3. Approximately 36.33 mm
4. Interpolation
5. Approximately 36.83 mm
6. Trees in Forest B generally had a more prosperous development period as their annual - ring thickness is on average greater than that of Forest A.
7. Direct relationship. As the age of trees increases, the average thickness of the tree's rings generally increases.