Question

Long jumps Here are the 40-yard-dash times (in seconds) and long-jump distances (in inches) for a small class of 12 students:

a. Sketch a scatterplot of the data using dash time as the explanatory variable.

b. Use technology to calculate the equation of the least-squares regression line for predicting the long-jump distance based on the dash time. Add the line to the scatterplot from part (a).

c. Explain why the line calculated in part (b) is called the “least-squares” regression line.

Short answer

Part (b) $\stackrel{⏜}{y}=414.7862-45.7433x$

Part (c) A residual is a difference between the actual $y-$values and the predicted $y-$values.

Part (a)

Step-by-step solution

Part (a) Step 1: Given information

Part (a) Step 2: Concept

The least-squares regression line reduces the sum of squares of vertical distances between the observed points and the line to zero.

Part (a) Step 3: Explanation

In the question for the students, the data for the sprint time and the long-jump distance is provided. As a result, the scatterplot for the same is as follows:

On the horizontal axis is the dash time in seconds, while on the vertical axis is the long jump distance in inches.

Part (b) Step 1: Calculation

In the question for the students, the data for the sprint time and the long-jump distance is provided. We must now determine the least-squares regression line's equation. As a result,

$Linreg(a+bx)L1,L2$

Select 1: Edit after pressing STAT for the first time. Then, in list $L1$enter the data, and in list $L2$enter the rest of the information. Next, press $STAT$then pick $CALC$, then $LinReg(a+bx)$After that, we must complete the command by entering the list.

Finally pressing on ENTER ten gives us the following results:

$$\begin{gathered} y=a+bx \\ a=414.7862 \\ b=-45.7433 \\ {r}^2=0.7023 \\ r=-0.8380 \end{gathered}$$

Thus the equation of the least square regression line will be as:

$\stackrel{⏜}{y}=414.7862-45.7433x$

And the line added to the scatterplot in part (a) will be as:

Part (c) Step 1: Explanation

In the question for the students, the data for the sprint time and the long-jump distance is provided. Because it is the straight line that minimizes the sum of the squared residuals, the regression line from component (b) is termed the least-squares regression line. The disparity between the actual and expected y-values is known as a residual.