A student believes there is a correlation between the number of texts sent during class and GPA. The student collected data and found that the line of fit can be modeled by the equation [tex]$\hat{\varphi}=3.9-0.1 x$[/tex].
Identify and interpret the slope in this scenario.
A. The slope is 3.9. Starting at 0.1, the GPA will increase by 3.9 for every text sent in class.
B. The slope is 3.9. Starting at 0.1, the GPA will decrease by 3.9 for every text sent in class.
C. The slope is -0.1. Starting at 3.9, the GPA will increase by 0.1 for every text sent in class.
D. The slope is -0.1. Starting at 3.9, the GPA will decrease by 0.1 for every text sent in class.
Answer (1)
The equation is φ ^ = 3.9 − 0.1 x .
Identify the slope as the coefficient of x , which is -0.1.
Interpret the slope: For each additional text sent, the GPA decreases by 0.1.
The slope is -0.1, indicating a decrease of 0.1 GPA for each text sent. The slope is -0.1 . Starting at 3.9 , the GPA will decrease by 0.1 for every text sent in class.
Explanation
Understanding the Problem The problem provides a linear equation that models the relationship between the number of texts sent during class and a student's GPA. The equation is given as φ ^ = 3.9 − 0.1 x , where φ ^ represents the predicted GPA and x represents the number of texts sent during class. We need to identify the slope of this equation and interpret its meaning in the context of the problem.
Identifying the Slope The given equation is in the form of a linear equation, which can be generally represented as y = m x + b , where m is the slope and b is the y-intercept. In our equation, φ ^ = 3.9 − 0.1 x , we can rewrite it as φ ^ = − 0.1 x + 3.9 . Comparing this with the general form y = m x + b , we can identify the slope m as -0.1 and the y-intercept b as 3.9.
Interpreting the Slope The slope represents the rate of change of the GPA with respect to the number of texts sent during class. In this case, the slope is -0.1. This means that for each additional text sent during class, the GPA is predicted to decrease by 0.1. The negative sign indicates a negative correlation between the number of texts sent and the GPA.
Final Answer Therefore, the slope is -0.1, and it indicates that for every additional text sent during class, the GPA is predicted to decrease by 0.1.
Examples
Understanding the relationship between texting in class and GPA can help students make informed decisions about their study habits. For example, if a student knows that their GPA decreases by 0.1 for every text they send in class, they might choose to limit their texting to improve their academic performance. This concept extends to other areas, such as understanding the impact of social media usage on mental health or the relationship between exercise and physical well-being. By analyzing the slope of a linear equation, we can quantify the relationship between two variables and make predictions about how one variable will change in response to changes in the other.
