Light Intensity Velma and Reggie gathered the data in Table...

Light Intensity Velma and Reggie gathered the data in Table 2.13 using a 100 -watt light bulb and a Calculator-Based Laboratory $\left(\mathrm{CBL}^{\text { Th }}\right)$ with a light-intensity probe. (a) Draw a scatter plot of the data in Table 2.13 (b) Find the power regression model. Is the power close to the theoretical value of $a=-2 ?$ (c) Superimpose the regression curve on the scatter plot. (d) Use the regression model to predict the light intensity at distances of 1.7 $\mathrm{m}$ and 3.4 $\mathrm{m}$ . $$ \begin{array}{ll}{\text { Distance }} & {\text { Intensity }} \\ {(\mathrm{m})} & {\left(\mathrm{W} / \mathrm{m}^{2}\right)} \\ {1.0} & {7.95} \\ {1.5} & {2.01} \\ {2.5} & {1.27} \\ {2.5} & {1.27} \\ {3.0} & {0.90}\end{array} $$

Light Intensity Velma and Reggie gathered the data in Table 2.13 using a 100 -watt light bulb and a Calculator-Based Laboratory $\left(\mathrm{CBL}^{\text { Th }}\right)$ with a light-intensity probe.
(a) Draw a scatter plot of the data in Table 2.13
(b) Find the power regression model. Is the power close to the theoretical value of $a=-2 ?$
(c) Superimpose the regression curve on the scatter plot.
(d) Use the regression model to predict the light intensity at distances of 1.7 $\mathrm{m}$ and 3.4 $\mathrm{m}$ .
$$
\begin{array}{ll}{\text { Distance }} & {\text { Intensity }} \\ {(\mathrm{m})} & {\left(\mathrm{W} / \mathrm{m}^{2}\right)} \\ {1.0} & {7.95} \\ {1.5} & {2.01} \\ {2.5} & {1.27} \\ {2.5} & {1.27} \\ {3.0} & {0.90}\end{array}
$$

Key Concepts

Scatter Plot

A scatter plot is a graphical representation that displays individual data points on a two-dimensional graph with two variables. It is used to visualize the relationship between the variables and identify any patterns, clusters, or outliers that may be present.

Regression Analysis

Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It helps in understanding how the typical value of the dependent variable changes when any one of the independent variables is varied.

Power Regression Model

A power regression model is a form of nonlinear regression that fits data to an equation of the form y = bx^a, where b and a are constants. This model is particularly useful when the dependent variable varies as a power of the independent variable, and its parameters can be estimated using transformations such as logarithms.

Inverse Square Law

The inverse square law is a principle in physics stating that a specified physical quantity, like light intensity, is inversely proportional to the square of the distance from the source. This law provides a theoretical basis for expecting an exponent close to -2 in the power regression model for light intensity versus distance.

Model Prediction

Model prediction involves using the fitted regression model to estimate the value of the dependent variable for given inputs of the independent variable. By substituting specific values into the regression equation, one can predict outcomes and assess the model's applicability to new data.

Transcript

00:04 This is problem number 57, and we're doing an application that has to do with light intensity as a function of distance for light bulbs, for 100 watt light bulbs.

00:15 And so most of this problem will solve using the graphing calculator.

00:19 I've just started by setting up an equation that relates light intensity in watts per meters squared to distance in meters.

00:26 And we're given a table of values.

00:29 So the first thing we want to do is use a graphing calculator or other kind of grapher.

00:33 Go to statistics, go to edit, and type your data from list 1 and list 2 into the calculator.

00:43 I've already done that.

00:45 Now we want to look at a scatter plot.

00:48 So if i go to y equals and you take a look and see that plot 1 is highlighted, and that indicates that i have already turned on my scatter plot, but if you haven't turned on a scatter plot yet, you press second and then stat plot.

01:05 And that will take you into the stat plot menu.

01:08 You can go into the menu for plot one and turn it on.

01:12 Besides turning on the scatter plot, you also want the calculator to establish a good window for you to view the scatter plot.

01:20 So what i like to do is press zoom.

01:22 And then inside the zoom menu, if you go to zoom number 9, which is zoom stat, the calculator will select the window dimension based on the numbers from the lists.

01:33 So zoom 9.

01:34 And we can see our full scatter plot.

01:36 So that takes care of part a of the problem.

01:40 Part b is to find the power regression model.

01:44 So we can go back to stat, over to calculate, and go through the list of all the different types of regression models that the calculator can compute until you find power regression.

01:56 So we select power regression, and we go through this list here and press enter several times.

02:04 Now notice here i have an option to store my regression equations, somewhere and knowing that when i get to part c i want to superimpose my regression curve on the scatter plot this at this point i'm going to stop and tell my calculator to store the regression equation into y1 so the way i do that is i press vars and go over to y vars vars for variables select function number one and select y one there are also shortcuts you can find for that some if you know your calculator shortcuts.

02:41 Okay, i'm going to press enter and calculate.

02:44 And there we have the values that go into our power regression model.

02:49 So for the sake of writing it down, we might just write down 7 .93 for the constant and negative 1 .99 for the power.

02:58 So i'm going to go back and do that.

03:00 Light intensity is equal to approximately 7 .93 times distance raised to the negative 1.

03:10 That's our model based on the data that we were given.

03:13 The theoretical value of the power is negative 2...