Chapter 12, Problem R12.5RE

(a)

To determine

To explain did the transformation achieve linearity.

Answer to Problem R12.5RE

Yes, it achieve.

Explanation of Solution

A study was conducted on the intensity of the light bulb at various distance from the light source. The scatterplot for the same is given in the question. And the computer output of the residual is also given. Thus, physics textbooks suggest that the relationship between the intensity and the distance should follow an inverse square law. Therefore, they transformed the distance measurement by squaring them and then taking their reciprocals. Thus, this transformation achieve linearity because the residuals in the residual plot are centered about zero and there is no obvious pattern in the residual plot.

(b)

To determine

To find out what is the equation of the least square regression line.

Answer to Problem R12.5RE

  $\widehat{y}=-0.000595+0.299624\frac{1}{x^2}$ .

Explanation of Solution

A study was conducted on the intensity of the light bulb at various distance from the light source. The scatterplot for the same is given in the question. And the computer output of the residual is also given. Thus, physics textbooks suggest that the relationship between the intensity and the distance should follow an inverse square law. Therefore, they transformed the distance measurement by squaring them and then taking their reciprocals. Now, the general equation of the least square regression line is:

  $\widehat{y}=a+bx$

The coefficient of a and bare given in the column of “Coef” in the given computer output as:

  $\begin{array}{l}a=-0.000595\\ b=0.299624\end{array}$

We also note that the explanatory variable is “Distance ${}^{-2}$ ”. Then we have the equation of least square regression line as:

  $\begin{array}{l}\widehat{y}=a+bx\\ =-0.000595+0.299624\frac{1}{x^2}\end{array}$

Where x is the distance and y the light intensity.

(c)

To determine

To predict the intensity of a $100-$ watt bulb at a distance of $2.1$ meters.

Answer to Problem R12.5RE

The predicted intensity is $0.0673$ candelas.

Explanation of Solution

A study was conducted on the intensity of the light bulb at various distance from the light source. The scatterplot for the same is given in the question. And the computer output of the residual is also given. Thus, physics textbooks suggest that the relationship between the intensity and the distance should follow an inverse square law. Therefore, they transformed the distance measurement by squaring them and then taking their reciprocals. Thus, form part (b) we have that,

  $\widehat{y}=-0.000595+0.299624\frac{1}{x^2}$

Now we have to predict the intensity of a $100-$ watt bulb at a distance of $2.1$ meters. Then, by replacing the value of x, we have,

  $\begin{array}{l}\widehat{y}=-0.000595+0.299624\frac{1}{x^2}\\ =-0.000595+0.299624\frac{1}{{2.1}^2}\\ =0.0673\end{array}$

This then implies that the predicted intensity is $0.0673$ candelas.