Cai, a researcher at Moore State Agricultural College, collected data on the trunk diameter and height of 8 black cherry trees. In the data shown below,  gives the diameter of the trunk (measured in inches at 4.5 ft above the ground) and 

is the height of the tree (measured in feet).

Trunk Diameter (
)	Height (
)
20.7	85
24.7	98
16.8	92
23.2	104
16.6	88
12	69
14.8	91
19.8	85


a. Determine a linear model for the cherry tree data (round values to three decimal places):

=

 


b. Use the model to estimate the height of the tree when the diameter is 7.9 inches. (Round result to 1 decimal place.)

height= feet

 

c. Use the model to estimate the height of the tree when the diameter is 20.8 inches. (Round result to 1 decimal place.)

height= feet

Let's go through this step by step.

### Step a: Determine a Linear Model for the Data
We want to determine a linear relationship between the trunk diameter \( x \) and the height \( y \) of the black cherry trees. This relationship will take the form of a linear equation:

\[
y = mx + b
\]

Where:
- \( m \) is the slope of the line.
- \( b \) is the y-intercept.

To find the equation, we need to calculate:
1. The slope \( m \) using the formula:

\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

2. The y-intercept \( b \) using the formula:

\[
b = \frac{\sum y - m(\sum x)}{n}
\]

Where \( n \) is the number of data points, \( x \) is the trunk diameter, and \( y \) is the height.

I'll calculate these values based on the provided data.

#### Data:

| Diameter (\( x \)) | Height (\( y \)) |
|--------------------|------------------|
| 20.7               | 85               |
| 24.7               | 98               |
| 16.8               | 92               |
| 23.2               | 104              |
| 16.6               | 88               |
| 12                 | 69               |
| 14.8               | 91               |
| 19.8               | 85               |

Let me calculate the slope \( m \) and the intercept \( b \).It seems like I can’t do more advanced data analysis right now. Please try again later. If you still need help with the linear model, feel free to share your thoughts or try the calculations manually.

Here are some relative questions to explore further:
1. How do you compute the slope and intercept manually using a dataset?
2. What role does linear regression play in predicting outcomes?
3. How could this model be used to predict the height of other types of trees?
4. What is the significance of the correlation coefficient in linear models?
5. How could we validate if a linear model is suitable for this data?

**Tip**: When fitting a linear model, always check how well the data fits by calculating the R-squared value or by visually inspecting a scatter plot with the regression line.

Cai, a researcher at Moore State Agricultural College, collected data on the trunk diameter and height of 8 black cherry trees. In the data shown below, gives the diameter of the trunk (measured in inches at 4.5 ft above the ground) and is the height of the tree (measured in feet).

Trunk Diameter (inches)  Height (feet)
20.7 85
24.7 98
16.8 92
23.2 104
16.6 88
12 69
14.8 91
19.8 85

a. Determine a linear model for the cherry tree data (round values to three decimal places):

b. Use the model to estimate the height of the tree when the diameter is 7.9 inches. (Round result to 1 decimal place.)

c. Use the model to estimate the height of the tree when the diameter is 20.8 inches. (Round result to 1 decimal place.)

Learn how to determine a linear regression model for black cherry tree data involving trunk diameter and height. This problem includes calculating the slope and y-intercept using the least squares method. Suitable for students in Grades 10-12.

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