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

(a) Make a scatterplot of these data.

(b) Using your calculator, find the equation of the least squares regression line of height on age.

(c) Use your regression line to predict Sarah’s height at age $40$ years ($480$ months). Convert your prediction to inches

($2.54cm=1inch$).

(d) The prediction is impossibly large. Explain why this happened.

A regression line shows how an explanatory variable $x$ affects a response variable $y$ You can use a regression line to forecast the value of $y$ for any value of $x$ by plugging this $x$ into the equation of the line.

The scatter plot for the given data is as follows

We can see from the scatter plot that all points have a linear rising trend. As a result, we can conclude that there is a positive relationship between age and height. To put it another way, Age and Height are positively connected.

As a result, a scatterplot is created.

The explanatory variable is "age," while the response variable is "height." The output of the least-square regression line using MINITAB is as follows.

Analysis of Regression: Age versus Height

From the above output, the least-square regression line is given by

$Height=71.95+0.0383Age$

Hence, the least-square regression line is $Height=71.95+0.0383Age$

Part (b) yields the following regression line between Age and Height:

$Height=71.95+0.0383Age$

We can anticipate Sarah's height at $40$years old using the regression line shown above ($480$months) That means, Sarah's height at $40$years ($480$months) is expected to be, $\stackrel{⏜}{Y}=71.95+0.383\times 480$

$$\begin{gathered} =255.94984cm \\ =\frac{255.94984}{2.54}inches(\mathrm{\backslash 1}inch=2.54cm) \end{gathered}$$

$$\begin{gathered} =100.7677inches \\ =255.94984cm \\ =100.7677inches \end{gathered}$$

The age of $40$ years ($480$ months) is well outside the range of our data's $X-$ values. At such extreme levels, we can't say whether the relationship remains linear. Predicting Sarah's height at $40$ years old ($480$ months) is an extension of the connection beyond the data. As a result, we can conclude that the estimated height is inaccurate.