To analyze the relationship between the number of miles driven and the number of gallons left in the tank, we can use a linear regression model.
Given the data points:
| Miles Driven | Gallons in Tank |
|--------------|-----------------|
| 27 | 13 |
| 65 | 12 |
| 83 | 11 |
| 109 | 10 |
| 142 | 9 |
| 175 | 8 |
### Step 1: Determining the Linear Regression Model
The linear regression model can be represented by the equation:
[tex]\[ y = mx + b \][/tex]
where [tex]\( y \)[/tex] is the miles driven, [tex]\( x \)[/tex] is the gallons in the tank, [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the intercept.
For this specific case:
- The slope ([tex]\( m \)[/tex]) is [tex]\(-28.49\)[/tex]
- The intercept ([tex]\( b \)[/tex]) is [tex]\(399.27\)[/tex]
Thus, the linear regression equation is:
[tex]\[ y = -28.49x + 399.27 \][/tex]
### Step 2: Finding the Correlation Coefficient
The correlation coefficient ([tex]\( r \)[/tex]) measures the strength and direction of a linear relationship between two variables.
For this data, the correlation coefficient is:
[tex]\[ r = -0.996 \][/tex]
This value indicates a very strong negative linear relationship between the miles driven and the gallons in the tank.
### Step 3: Determining the Strength of the Model
To determine the strength of the model, we can use the absolute value of the correlation coefficient ([tex]\( |r| \)[/tex]).
- If [tex]\( |r| > 0.8 \)[/tex], the model is considered strong.
- If [tex]\( 0.5 < |r| \leq 0.8 \)[/tex], the model is considered moderate.
- If [tex]\( |r| \leq 0.5 \)[/tex], the model is considered weak.
Given that [tex]\( |r| = 0.996 \)[/tex], which is greater than 0.8, the strength of the model is:
[tex]\[ \text{strong} \][/tex]
### Summary
- Linear regression model: [tex]\( y = -28.49x + 399.27 \)[/tex]
- Correlation coefficient: [tex]\( r = -0.996 \)[/tex]
- Strength of the model: strong
