Question

The scatter plot shows the number of hours worked, x, and the amount of money spent on entertainment, y, by each of 25 students.Use the equation of the line of best fit, =y+1.82x11.36, to answer the questions below.Give exact answers, not rounded approximations. (a) For an increase of one hour in time worked, what is the predicted increase in the amount of money spent on entertainment?$(b) What is the predicted amount of money spent on entertainment for a student who doesn't work any hours?$(c) What is the predicted amount of money spent on entertainment for a student who works 8 hours?$

Answer 1

Given the scatterplot below:

where the equation of the line of best fit is expressed as

`y=1.82x+11.36`

A) Predicted increase in the amount of money spent on entertainment, for an increase of one hour in time worked.

Recall that the line equation is expressed as

`\begin{gathered} y=mx+c \\ where \\ m=slope \\ slope=\frac{increase\text{ in y}}{increase\text{ in x}} \end{gathered}`

By comparison with the equation of line of best fit, we see that

`\begin{gathered} slope=1.82 \\ where \\ slope=\frac{increase\text{ in amout of money spent}}{increase\text{ in the number of hours worked}} \end{gathered}`

Thus, we have

`\begin{gathered} 1.82=\frac{increase\text{ in amount of money spent}}{1} \\ \Rightarrow predicted\text{ increase in amount of money spent on entertainment = \$1.82} \end{gathered}`

B) Predicted amount of money spent on entertainment for a student with no number of hours worked

This implies that from the equation of the line of best fit, the value of x is zero.

By substitution, we have

`\begin{gathered} y=1.82(0)+11.36 \\ =0+11.36 \\ \Rightarrow y=\$11.36 \end{gathered}`

C) Predicted amount of money spent on entertainment for a student with8 hours of work.

Thus, we have the value of x to be 8 from the equation of the line of best fit.

By substitution, we have

`\begin{gathered} y=1.82\left(8\right)+11.36 \\ =14.56+11.36 \\ \Rightarrow y=\$25.92 \end{gathered}`