Question

Age and height A random sample of 195 students was selected from the United Kingdom using the Census At School data selector. The age x (in years) and height y (in centimeters) were recorded for each student. Here is a scatterplot with the least-squares regression line y^=106.1+4.21x. For this model, s = 8.61 and r2 = 0.274.

a. Calculate and interpret the residual for the student who was 141 cm tall at age 10.

b. Interpret the slope of the least-squares regression line.

c. Interpret the value of s.

d. Interpret the value of r2.

Short answer

Part (a) The residual is $-7.2$

Part (b) The height increases by $4.21$cm per year on average.

Part (c) The predicted height of a pupil using the least square regression equation is $8.61$cm shorter than the actual height of the student.

Part (d) The least square regression line employing age as an explanatory variable may explain $27.4$percent of the variation in height.

Step-by-step solution

Part (a) Step 1: Given information

Part (a) Step 2: Calculation

The relationship between the height and age of the students is given in the question. And the regression line is as:

$\stackrel{⏜}{y}=106.1+4.21x$

And also given that the student was $141cm$tall at age $10$Thus, we have,

$$\begin{gathered} x=10 \\ y=141 \end{gathered}$$

Thus the predicted height of the student is as:

$\stackrel{⏜}{y}=106.1+4.21x$

$$\begin{gathered} =106.1+4.21\left(10\right) \\ =148.2 \end{gathered}$$

Thus, the residual will be calculated as:

$$\begin{gathered} Residual=y-\stackrel{⏜}{y} \\ =141-148.2 \\ =-7.2 \end{gathered}$$

This means that while using the regression line to make a prediction, we overestimated the height of the student at age $10$ by $7.2cm$

Part (b) Step 1: Calculation

The question specifies the relationship between the students' height and age. The regression line is as follows:

$\stackrel{⏜}{y}=106.1+4.21x$

Also, at the age of ten, the pupil was $141$ cm tall. As a result,

$$\begin{gathered} x=10 \\ y=141 \end{gathered}$$

The slope in the least square regression equation is the coefficient of $x$, and it indicates the average rise or decrease of $y$ per unit of $x$ Thus,

$b_1=4.21$

As a result, the height increases by $4.21$ cm per year on average.

Part (c) Step 1: Calculation

The question specifies the relationship between the students' height and age. The regression line is as follows:

$\stackrel{⏜}{y}=106.1+4.21x$

Also, at the age of ten, the pupil was 141 cm tall. As a result,

$$\begin{gathered} x=10 \\ y=141 \end{gathered}$$

$s=8.61$ is the value of s. The standard error of the estimations, as we all know, is the average error of forecasts, and thus the average difference between actual and predicted values. Thus, using the equation of least square regression line deviation, a student's anticipated height differs by $8.61cm$from his or her actual height.

Part (d) Step 1: Calculation

The question specifies the relationship between the students' height and age. The regression line is as follows:

$\stackrel{⏜}{y}=106.1+4.21x$

Also, we have, given that the pupil was $141$ cm tall at the age of ten,

$$\begin{gathered} x=10 \\ y=141 \end{gathered}$$

And the value of $r^2$is,

$$\begin{gathered} r^2=0.274 \\ =27.4\% \end{gathered}$$

The coefficient of determination, as we know, is a measurement of how much variation in the answers y variable is explained by the least square regression model with the explanatory variable. As a result, the least square regression line employing age as an explanatory variable may explain $27.4$percent of the variation in height.