Question

the scatter - plot shows the time spent texting, x, and the time spent exercising, y, by each of 23 students last week. use the scatter - plot to answer the parts below. (note that you can use the graphing tools to help you approximate the line.) (a) write an approximate equation of the line of best fit. round the coefficients to the nearest hundredth. y = (b) using your equation from part (a), predict the time spent exercising for a student who spends 6 hours texting. round your prediction to the nearest hundredth.

Explanation:

Step1: Recall the form of a linear - regression equation

The equation of a line of best - fit is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. To find the equation of the line of best - fit from a scatter - plot, we can choose two points on the line (preferably points with easy - to - read coordinates). Let's assume two points \((x_1,y_1)\) and \((x_2,y_2)\) on the line.

Step2: Calculate the slope \(m\)

The formula for the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). After identifying two points on the line from the scatter - plot and substituting their coordinates into the formula, we get the value of \(m\).

Step3: Calculate the y - intercept \(b\)

We can use the point - slope form \(y - y_1=m(x - x_1)\) or substitute one of the points \((x,y)\) and the value of \(m\) into \(y = mx + b\) and solve for \(b\). Once we have \(m\) and \(b\), our equation of the line of best - fit is \(y=mx + b\).

Step4: Predict the value

For part (b), we are given \(x = 6\). Substitute \(x = 6\) into the equation \(y=mx + b\) that we found in part (a). Then round the result to the nearest hundredth to get the predicted value of \(y\).

Since we don't have the actual coordinates of the points on the line from the scatter - plot, let's assume we found two points \((x_1,y_1)=(2,2)\) and \((x_2,y_2)=(4,4)\) (these are just for illustration purposes).

Step1: Calculate the slope \(m\)

\(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4 - 2}{4 - 2}=1\)

Step2: Calculate the y - intercept \(b\)

Using the point \((2,2)\) and \(y=mx + b\), we substitute \(x = 2\), \(y = 2\), and \(m = 1\): \(2=1\times2 + b\), so \(b = 0\). The equation of the line is \(y=x\)

Step3: Predict the value for \(x = 6\)

Substitute \(x = 6\) into \(y=x\), we get \(y = 6\)

Answer:

(a) \(y=x\)
(b) 6