Finding the Best Model
In Exercises 5–16, construct a sc...

Question

Finding the Best Model

In Exercises 5–16, construct a scatterplot and identify the mathematical model that best fits the given data. Assume that the model is to be used only for the scope of the given data, and consider only linear, quadratic, logarithmic, exponential, and power models.

Landing on the Moon When the Apollo spacecraft landed on the Moon, the rocket engine would typically cut off at about 1.3 meters above the surface so that hot gases and dust and other surface materials would not cause damage. The landing module was in freefall starting at about 1 meter above the surface. The table below lists the time t (seconds) after being dropped and the distance d (meters) travelled by an object dropped near the surface of the Moon.

Table showing time in seconds and distance in meters for an object dropped near the Moon's surface.

Accepted Answer

Hello. In this video, we are told that the following data lists the monthly active users of a social media app over several months after its launch. We want to construct a scatter plot to determine the best mathematical model for the data. Now in order to create a scattered plot, we are going to be using an external tool. In this video, I'm going to be using a tool called Google Sheets. Now in order to create the scatter plot, the first thing we want to do is we want to go ahead and take the given data and separate it into 2 columns in Google Sheets. So let's go ahead and do that. So what I've done here is I've created two columns for the given data. The first column represents the amount of months that have passed after the initial launch of the app, and the second column represents the amount of users per month. Now, in order to create a scatter plot, we are going to select both of the columns. Then what we are going to do is we're going to go to insert, and we are going to insert a chart. Now, currently, this is not in the form of a scatter plot, but we can change this to a scatter plot if we were to scroll to chart type, and underneath chart type, we can select the option of a scatter chart. So what we have now is a scatter plot of the given data. Now, in order to determine the best fitting model, we are going to use a tool called the trend line. Now, in order to enable the trend line, we are going to click on at least one of our data points, and then we are going to scroll down until we see a trend line option. In order to enable it, we will click the trend line option. Now underneath the trend line option, we have a few different options for selecting the different models. Now, we are going to be looking at each of the models, so linear, exponential, logarithmic, and polynomial. And one thing to note here is that polynomial is equivalent to quadratic. So what we wanna do is we want to select the option that best fits the data set. Taking a look at the linear trend line, the linear trend line doesn't fit the data set that well, since some of the points are either above or below the trend line. So linear won't be the correct option here. Looking at exponential, exponential fits a little bit better, but there are some points that are slightly above the trend line. So let's take a look at our different options here. If we take a look at logarithmic, logarithmic doesn't follow the pattern of the trend of the data points at all, so this won't be an option either. But if we take a look at polynomial, polynomial seems to fit the data set exceedingly well. One thing to note, by the way, is that when you select polynomial, you want to make sure that your polynomial degree is 2, so that you're working with an accurate coefficient of determination. But the polynomial model passes through the middle of many of the points at once. So what this means is that the polynomial option is going to be the best fit model for the scatter plot. And so what that means is that going back to the problem, the correct solution to this problem is going to be option A. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

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