Question

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 \\(\hat{y} = 3.9 - 0.1x\\). identify and interpret the slope in this scenario.\
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the slope is 3.9. starting at 0.1, the gpa will increase by 3.9 for every text sent in class.\
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the slope is 3.9. starting at 0.1, the gpa will decrease by 3.9 for every text sent in class.\
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the slope is -0.1. starting at 3.9, the gpa will increase by 0.1 for every text sent in class.\
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the slope is -0.1. starting at 3.9, the gpa will decrease by 0.1 for every text sent in class.

Explanation:

Step1: Recall slope-intercept form

The slope - intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. In the given equation \(\hat{y}=3.9 - 0.1x\), we can rewrite it as \(\hat{y}=- 0.1x + 3.9\). So, the slope \(m=-0.1\) and the y - intercept \(b = 3.9\).

Step2: Interpret the slope

The slope of a linear regression equation \(y=mx + b\) represents the change in the dependent variable (\(y\), in this case, GPA) for a one - unit change in the independent variable (\(x\), in this case, number of texts sent in class). Since the slope \(m=-0.1\), this means that for each additional text sent in class (a one - unit increase in \(x\)), the GPA (\(y\)) will change by \(- 0.1\). The y - intercept \(b = 3.9\) represents the predicted GPA when the number of texts sent in class (\(x = 0\)). So, starting at a GPA of \(3.9\) (when \(x = 0\)), the GPA will decrease by \(0.1\) for every text sent in class.

Answer:

The slope is - 0.1. Starting at 3.9, the GPA will decrease by 0.1 for every text sent in class.