Question

jacob distributed a survey to his fellow students asking them how many hours they spent on the internet 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:
$hat { y } = x + 7.5$
$hat { y } = - x + 7.5$
$hat { y } = - \frac { 1 } { 2 } x + 7.5$
based on this equation, estimate the mood rating for a student that spent 5.5 hours online.

Explanation:

Step1: Analyze the slope of the line

The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope. Looking at the scatter - plot and the line of best fit, we can see that the line has a positive slope. For the equation \(y=-\frac{1}{2}x + 7.5\), the slope \(m =-\frac{1}{2}\) (negative), so it can be eliminated. For the equation \(y=-x + 7.5\), the slope \(m=- 1\) (negative), so it can be eliminated.

Step2: Estimate the slope of the line of best fit

We can use two points on the line of best fit. Let's assume two points \((x_1,y_1)\) and \((x_2,y_2)\). If we take a point close to \((5,2)\) and \((7,3.5)\) (approximate points on the line). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3.5 - 2}{7 - 5}=\frac{1.5}{2}=0.75\approx1\) (approximate visual estimate). The equation \(y=x + 7.5\) has a slope \(m = 1\).

Step3: Estimate the mood rating for \(x = 5.5\)

Substitute \(x = 5.5\) into the equation \(y=x + 7.5\). Then \(y=5.5+7.5\)

Answer:

The linear equation that best describes the model is \(y=x + 7.5\). When \(x = 5.5\), \(y=13\)