To determine the type of relationship Brett's teacher discovered between the minutes Brett stayed on task in class and the grade he earned on the homework, we will analyze the provided correlation coefficient [tex]\(r = 0.97\)[/tex].
### Step-by-Step Analysis:
1. Understanding the Correlation Coefficient:
- The correlation coefficient [tex]\(r\)[/tex] is a measure of the strength and direction of the linear relationship between two variables.
- The value of [tex]\(r\)[/tex] ranges between -1 and 1.
- If [tex]\(r > 0\)[/tex], it indicates a positive correlation.
- If [tex]\(r < 0\)[/tex], it indicates a negative correlation.
- If [tex]\(r = 1\)[/tex] or [tex]\(r = -1\)[/tex], it indicates a perfect linear relationship.
- The closer [tex]\(r\)[/tex] is to 1 or -1, the stronger the correlation.
2. Value of [tex]\(r\)[/tex]:
- In this case, [tex]\(r = 0.97\)[/tex]. This is very close to 1, which suggests a strong positive correlation.
3. Interpretation of the Correlation Coefficient:
- A correlation of [tex]\(r = 0.97\)[/tex] indicates that there is a strong positive relationship between the minutes Brett stayed on task in class and the grades he earned on the homework.
- This means that as the minutes Brett stayed on task in class increased, his homework grades also tended to increase.
4. Causality Consideration:
- A correlation does not imply causation by itself. However, in this context, the observation relates to behavior (staying on task in class) and its effect on performance (homework grade).
- Given the nature of the variables (behavior in class and resulting grades), it is reasonable to infer that staying on task in class could likely have a causal effect on better homework grades. There is a logical basis to suspect causality here.
### Conclusion:
Based on the value of the correlation coefficient and the context of the variables observed, the teacher has discovered a positive correlation, likely causal relationship between Brett's behavior in class and his homework grades.
Thus, the type of relationship discovered is:
Positive correlation, likely causal.
