Question

Use residuals to calculate the quality of fit for the line y=0.25+29

Answer 1

Given the question in the image, the following are the solution steps to complete the table

STEP 1: Write the equation of the line

`y=0.25x+29`

STEP 2: Calculating the residual when x=10

`\begin{gathered} y=0.25x+29 \\ x=10 \\ By\text{ substitution,} \\ y=0.25(10)+29=2.5+29=31.5 \\ \text{ To get Residual} \\ 31.3-31.5=-0.2 \end{gathered}`

STEP 3: Calculating the residual when x=20

`\begin{gathered} y=0.25x+29 \\ x=20 \\ By\text{ substitution,} \\ y=0.25(20)+29=5+29=34^(\square) \\ \text{ To get Residual} \\ 34-34=0 \end{gathered}`

STEP 4: Calculating the residual when x=30

`\begin{gathered} y=0.25x+29 \\ x=30 \\ By\text{ substitution,} \\ y=0.25(30)+29=7.5+29=36.5 \\ \text{ To get Residual} \\ 36.5-36.5=0 \end{gathered}`

STEP 5: Calculating the residual when x=40

`\begin{gathered} y=0.25x+29 \\ x=40 \\ By\text{ substitution,} \\ y=0.25(40)+29=10+29=39 \\ \text{ To get Residual} \\ 38.2-39=-0.8 \end{gathered}`

STEP 6: Plotting the quality of fit graph

A residual plot has the Residual Values on the vertical axis; the horizontal axis displays the independent variable. A residual plot is typically used to find problems with regression. A residual is a measure of how well a line fits an individual data point. This vertical distance is known as a residual. For data points above the line, the residual is positive, and for data points below the line, the residual is negative. The closer a data point's residual is to 0, the better the fit.

We have the table below:

This table for plotting the residuals gives the graph below: