Question

R12.5 Light intensity In a physics class, the intensity of a 100-watt light bulb was measured by a sensor at various distances from the light source. Here is a scatterplot of the data. Note that a candela is a unit of luminous intensity in the International System of Units.

Physics textbooks suggest that the relationship between light intensity y and distance x should follow an “inverse square law,” that is, a power law model of the form $y=a{x}^{-2}=a\frac{1}{x^2}$. We transformed the distance measurements by squaring them and then taking their reciprocals. Here is some computer output and a residual plot from a least-squares regression analysis of the transformed data. Note that the horizontal axis on the residual plot displays predicted light intensity.

a. Did this transformation achieve linearity? Give appropriate evidence to justify your answer.
b. What is the equation of the least-squares regression line? Define any variables you use.
c. Predict the intensity of a 100-watt bulb at a distance of 2.1 meters.

Short answer

(a) Yes, transformation achieve linearity.

(b) The equation of the least-squares regression line is$\hat{y}=-0.000595+0.299624\frac{1}{x^2}$.

(c) The predicted intensity is $0.0673$ candelas.

Step-by-step solution

Part (a) Step 1: Given information

To justify with appropriate evidence to the transformation achieve linearity.

Part (a) Step 2: Explanation

At different ranges from the light source, the intensity of the light bulb was evaluated. The question includes a scatterplot for the same.
The residual's computer output is also provided.

As a result, physics textbooks advise that the intensity-to-distance connection should follow an inverse square law. So, they squared the distance measurements and then calculated their reciprocals.
Because the residuals in the residual plot are centered around zero and there is no clear pattern in the residual plot, this transformation achieves linearity.
As a result, yes, it achieve linearity .

Part (b) Step 1: Given information

To determine the equation of the least-squares regression line.

Part (b) Step 2: Explanation

At different ranges from the light source, the intensity of the light bulb was evaluated. The question includes a scatterplot for the same.
The residual's computer output is also provided.
Hence, physics textbooks advise that the intensity-to-distance connection should follow an inverse square law.
As a result, they squared the distance measurements and then calculated their reciprocals. The least square regression line's general equation is now:
$\hat{y}=a+bx$
In the given computer output, the coefficients of a and bare are presented in the column "Coef" as:
$a=-0.000595$
$b=0.299624$
The explanatory variable is expressed as "$Dis\tan c{e}^{-2}$".
The least square regression line is calculated using the following equation:
$\hat{y}=a+bx$
$=-0.000595+0.299624\frac{1}{x^2}$
Where $y$indicates the light intensity and $x$indicates the distance.

As a result, the equation is $\hat{y}=-0.000595+0.299624\frac{1}{x^2}$.

Part (c) Step 1: Given information

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

Part (c) Step 2: Explanation

At different ranges from the light source, the intensity of the light bulb was evaluated. The question includes a scatterplot for the same.
The residual's computer output is also provided. So, physics textbooks advise that the intensity-to-distance connection should follow an inverse square law. As a result, they squared the distance measurements and then calculated their reciprocals. Hence from part (b):
$\hat{y}=-0.000595+0.299624\frac{1}{x^2}$
Now, at a distance of $2.1$meters, predict the intensity of a $100$-watt lamp.
Then, by substituting $x$for the value of $x$,
$\hat{y}=-0.000595+0.299624\frac{1}{x^2}$
$=-0.000595+0.299624\frac{1}{2.1^2}$
$=0.0673$
As a result, the predicted intensity is $0.0673$candelas.