Questions: Jacob distributed a survey to his fellow students asking them how many hours they'd spent playing sports in the past day. He also asked them to rate their mood on a scale from 0 to 10, with 10 being the happiest. A line was fit to the data to model the relationship. Which of these linear equations best describes the given model? Choose 1 answer: (A) ŷ = 5x + 1.5 (B) ŷ = 1.5x + 5 (C) ŷ = -1.5x + 5

Jacob distributed a survey to his fellow students asking them how many hours they'd spent playing sports in the past day. He also asked them to rate their mood on a scale from 0 to 10, with 10 being the happiest. A line was fit to the data to model the relationship.

Which of these linear equations best describes the given model?

Choose 1 answer:
(A) ŷ = 5x + 1.5
(B) ŷ = 1.5x + 5
(C) ŷ = -1.5x + 5

Transcript text: Jacob distributed a survey to his fellow students asking them how many hours they'd spent playing sports in the past day. He also asked them to rate their mood on a scale from 0 to 10 , with 10 being the happiest. A line was fit to the data to model the relationship. Which of these linear equations best describes the given model? Choose 1 answer: (A) $\hat{y}=5 x+1.5$ (B) $\hat{y}=1.5 x+5$ (c) $\hat{y}=-1.5 x+5$

Solution

Solution Steps

Step 1: Identify the Slope of the Line

To determine the slope of the line, we need to find the change in the mood rating (y-axis) over the change in hours playing sports (x-axis). From the graph, we can see that as the hours playing sports increase from 0.5 to 1.5 (a change of 1 hour), the mood rating increases from 4 to 5.5 (a change of 1.5).

Slope (m) = Change in y / Change in x = 1.5 / 1 = 1.5

Step 2: Identify the Y-Intercept

The y-intercept is the point where the line crosses the y-axis (when x = 0). From the graph, we can see that the line crosses the y-axis at approximately 5.

Step 3: Form the Equation of the Line

Using the slope-intercept form of the equation of a line, y = mx + b, where m is the slope and b is the y-intercept, we can form the equation:

y = 1.5x + 5