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Chapter 12: Q. 5 (page 794)

In physics class, the intensity of a 100-watt light bulb was measured by a sensor at various distances from the light source. A scatterplot of the data is shown below. Note that a candela (cd) is a unit of luminous intensity in the International System of Unit

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=ax-2. We transformed the distance measurements by squaring them and then taking their reciprocals. Some computer output and a residual plot from a least-squares regression analysis on the transformed data are shown below. 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 would you predict for the intensity of a 100-watt bulb at a distance of 2.1meters? Show your work.

Short Answer

Expert verified

(a) Yes, because the residuals in the residual plot are centered about 0 and there is no obvious pattern in the residual plot.

(b) The predicted light intensity is 0.06735candelas.

Step by step solution

01

Part (a) Step 1: Given information

The given data is

02

Part (a) Step 2: Explanation

A transformation to obtain linearity in regression is a special type of nonlinear transformation. It's a nonlinear transformation that makes two variables' linear relationship stronger.

03

Part (b) Step 1: Given information

The given data is

04

Part (b) Step 2: Explanation

General least-squares regression equation

y^=a+bx

The coefficient a and b are given in the column of "Coef":

a=-0.000595

b=0.299624

The least-squares regression equation then becomes:

y^=-0.000595+0.2996241x2

with x the distance and y the light intensity.

Replace x with 2.1

localid="1650646784430" y^=-0.000595+0.2996241(2.1)2≈0.06735.

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