Question

Sarah’s parents are concerned that she seems short for her age. Their doctor has kept the following record of Sarah’s height:

a. Make a scatterplot of these data using age as the explanatory variable. Describe what you see.

b. Using your calculator, find the equation of the least-squares regression line.

c. Calculate and interpret the residual for the point when Sarah was 48 months old.

d. Would you be confident using the equation from part (b) to predict Sarah’s height when she is 40 years old? Explain.

Short answer

Part (a) Because the leftmost point in the scatterplot lies far from the other points in the scatterplot.

Part (b)$\stackrel{⏜}{y}=71.95+0.3833x$

Part (c) Residual is$-0.3484$

Part (d) No, we are not.

Step-by-step solution

Part (a) Step 1: Given information

The following is a scatterplot of age on the horizontal axis and height on the vertical axis:

Because the pattern in the scatterplot slopes upwards, we know the direction is positive. Because there is no substantial curvature in the scatterplot, the shape is linear. Because the points only stray little from the main pattern in the points, the strength is high. The Because the leftmost point in the scatterplot is far from the other points in the scatterplot, the strange points appear to be one outlier.

Part (b) Step 1: Explanation

Using a calculator, press on $STAT$ and then select $1$: Edit. and then enter the data of sugar in the list $L_1$and enter the data of calories in the list $L_2$

Next, press on $STAT$ select $CALC$ and then select $Linreg(a+bx)$ Next we need to finish the command by entering $L_1$$L_2$

Finally, pressing on $ENTER$then gives us the following result:

$$\begin{gathered} y=a+bx \\ a=71.95 \\ b=0.3833 \\ {r}^2=0.9888 \\ r=0.9944 \end{gathered}$$

This then implies the regression line as:

$$\begin{gathered} \stackrel{⏜}{y}=a+bx \\ \Rightarrow \stackrel{⏜}{y}=71.95+0.3833x \end{gathered}$$

Where $x$is age and $y$be height.

Part (c) Step 1: Explanation

The regression line is:

$\stackrel{⏜}{y}=71.95+0.3833x$

As a result, Sarah's height will be when she was $48$ months old:

$$\begin{gathered} \stackrel{⏜}{y}=71.95+0.3833x \\ =71.95+0.3833\left(48\right) \\ =90.3484 \end{gathered}$$

The residual will be as:

$$\begin{gathered} Residual=y-\stackrel{⏜}{y} \\ =90-90.3484 \\ =-0.3484 \end{gathered}$$

This means that while using the regression line to make a prediction, we overestimated Sarah's height by $0.3484cm$

Part (d) Step 1: Explanation

Because there are twelve months in a year, $40$ years corresponds to one year.

$$\begin{gathered} 40\times 12 \\ =480 \end{gathered}$$

The data ranges in age from $36$ to $60$ months. We then notice that $480$ months is outside of the age range, implying that we wouldn't be confident in applying the equation from part (b) to forecast Sarah's height when she was 48 months old because we'd be extrapolating.