Answer
**3. Equation of the Linear Model:**
\[
y = 10x + 50
\]
**4. Predicted Test Score for 6 Hours:**
110
**5. Relationship Description:**
There is a positive linear relationship between study hours and test scores, meaning more study time leads to higher test scores.
Solution
Sure, let's tackle each of your questions step-by-step. We'll assume that the **y-intercept** of the linear model is provided or can be determined from the scatter plot. For illustrative purposes, let's assume the y-intercept is **50**. If your specific problem has a different y-intercept, you can adjust the calculations accordingly.
---
### **3. Equation of the Linear Model**
**Given:**
- **Y-intercept (b):** 50 (i.e., the point (0, 50))
- **Another point on the line:** (4, 90)
**Objective:**
Find the equation of the line in the form \( y = mx + b \), where:
- \( m \) is the **slope** of the line
- \( b \) is the **y-intercept**
**Steps:**
1. **Calculate the Slope (\( m \)):**
The slope \( m \) represents the change in \( y \) for a unit change in \( x \).
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points (0, 50) and (4, 90):
\[
m = \frac{90 - 50}{4 - 0} = \frac{40}{4} = 10
\]
2. **Write the Equation:**
Now that we have both \( m \) and \( b \), plug them into the linear equation formula:
\[
y = mx + b
\]
\[
y = 10x + 50
\]
**Answer:**
\[
\boxed{y = 10x + 50}
\]
---
### **4. Predicting Adam's Test Score When He Studies for 6 Hours**
**Given:**
- **Linear Model:** \( y = 10x + 50 \)
- **Study Hours (\( x \)):** 6
**Objective:**
Find Adam's predicted test score (\( y \)) when he studies for 6 hours.
**Calculation:**
\[
y = 10(6) + 50 = 60 + 50 = 110
\]
**Interpretation:**
A score of 110 might exceed the maximum possible score (commonly 100). This suggests that the linear model is extrapolating beyond the range of the original data. It's important to consider the context and practical limitations when making such predictions.
**Answer:**
\[
\boxed{110}
\]
*(Note: If the maximum test score is 100, Adam's predicted score should be adjusted accordingly, indicating that studying 6 hours might be sufficient to achieve a perfect score.)*
---
### **5. Describing the Relationship Between the Data in the Scatter Plot**
**Observation Based on the Linear Model:**
- The relationship between study hours and test scores is **linear**.
- **Positive Slope:** As study hours increase, test scores also increase.
- **Consistency:** Each additional hour of study is associated with an average increase of 10 points in the test score.
**Interpretation:**
There is a **direct proportional relationship** between the number of hours Adam studies and his test score. This means that more study time generally leads to better performance on the test.
**Graphical Representation:**
- The scatter plot would show points that roughly align along an upward-sloping straight line.
- The **y-intercept (50)** suggests that with 0 hours of study, the expected score is 50.
- The **slope (10)** indicates the rate at which the test score improves with each additional hour of study.
**Answer:**
The data shows a **positive linear relationship** where increasing the number of study hours is associated with higher test scores. Specifically, each additional hour of studying tends to raise the test score by about 10 points.
---
**Summary:**
1. **Equation of the Linear Model:** \( y = 10x + 50 \)
2. **Predicted Test Score for 6 Hours of Study:** 110 (subject to contextual constraints)
3. **Relationship Description:** A positive linear relationship where more study hours lead to higher test scores.
If your specific problem has different values for the y-intercept or other parameters, please adjust the calculations accordingly or provide additional details for a more tailored solution.
Reviewed and approved by the UpStudy tutoring team
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