Question

First scatterplot y1~ mx1 +b
r^2= 0.7955. e1
r = 0.8919
m = -0.12654 b = 23837.8best line of fit


Scatterplot 2
Y1~ mx1+b
r^2= 0.8209. e1
r = -0.906
m = -1451.95
b = 23746.4
Line of best fit
The sale price of the car that Kara owns ______by ________ each year as it ages
Describe the strength and direction of the linear association between mileage (scatterplot 1) and sale price of a used vehicle

Answer 1

Best line of fit for each scatterplot

Answer 2

To analyze the information provided about the scatterplots and the relationship between mileage and sale price of a used vehicle, let's summarize the key points and respond to your questions.

### Sale Price of the Car
From the equations of the lines of best fit, we see:

- For Scatterplot 1:
- Slope \(m = -0.12654\)
- Intercept \(b = 23837.8\)

- For Scatterplot 2:
- Slope \(m = -1451.95\)
- Intercept \(b = 23746.4\)

Given that the slope is negative in both scatterplots, we can conclude that the sale price of the car that Kara owns:

- **Decreases by approximately 0.12654 dollars (in Scatterplot 1) or 1451.95 dollars (in Scatterplot 2) for each unit of measurement in the x-variable (mileage, in this case) each year as it ages.**

### Strength and Direction of Linear Association
Now let's describe the strength and direction of the linear association between mileage (x) and sale price (y):

1. **Direction**:
- Both scatterplots indicate a **negative relationship** between mileage and sale price. This means as the mileage of the vehicle increases, the sale price tends to decrease.

2. **Strength**:
- The strength of the linear relationship can be assessed using the coefficient of determination \(r^2\):
- Scatterplot 1: \(r^2 = 0.7955\) indicates a strong linear relationship.
- Scatterplot 2: \(r^2 = 0.8209\) indicates an even stronger linear relationship.

Both values are quite high, suggesting that a large proportion of the variance in sale price can be explained by mileage.

### Overall Conclusion
In summary:
- The sale price of the car that Kara owns decreases as the car ages and accumulates mileage, by approximately \(0.12654\) or \(1451.95\) for each increase in mileage.
- The linear association shows a strong negative correlation, meaning that as mileage increases, the sale price tends to decrease significantly. The statistics from both scatterplots reinforce this trend, indicating a reliable predictive model.

Answer 3

The best line of fit for each scatterplot can be expressed using the slope-intercept form of a linear equation, which is given by the formula:

\[ y = mx + b \]

where:
- \( y \) is the dependent variable (sale price in this case),
- \( m \) is the slope of the line,
- \( x \) is the independent variable (mileage), and
- \( b \) is the y-intercept.

From your previous information, we can summarize the lines of best fit for each scatterplot as follows:

### Scatterplot 1
- Slope (\( m \)): \(-0.12654\)
- Intercept (\( b \)): \(23837.8\)

**Equation of the line of best fit:**
\[
y = -0.12654x + 23837.8
\]

### Scatterplot 2
- Slope (\( m \)): \(-1451.95\)
- Intercept (\( b \)): \(23746.4\)

**Equation of the line of best fit:**
\[
y = -1451.95x + 23746.4
\]

### Summary
1. **Scatterplot 1:**
- Equation: \( y = -0.12654x + 23837.8 \)

2. **Scatterplot 2:**
- Equation: \( y = -1451.95x + 23746.4 \)

These equations represent the best lines of fit for each of the scatterplots, modeling the relationship between car mileage and sale price.