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.

Earthquakes Listed below are earthquake depths (km) and magnitudes (Richter scale) of different earthquakes. Find the best model and then predict the magnitude for the last earthquake with a depth of 3.78 km. Is the predicted value close to the actual magnitude of 7.1?

Table showing earthquake depths in km and corresponding magnitudes for model fitting and prediction analysis.

Accepted Answer

Hello. In this video, we are told that the following data lists the number of subscribers to a streaming service over several years. We want to construct a scatter plot to determine the best mathematical model for the data set. So in order to approach this problem and in order to construct a scatter plot, what we are going to do is we are going to use Google Sheets. Now, the first thing we want to do is we want to take this data set and separate it into two different columns for Google Sheets. So let's go ahead and do that. So what I've done here is I've taken in the data set that was given to us in the table and I separate it into two separate columns. One column has the years and the other column has the respective revenue. Now, in order to create the scatter plot, the first thing we want to do is you want to select both of the columns. Then we want to go ahead and insert a chart. Now, this is not a scatter plot, but when you create the chart, what we are going to do is we are going to go to chart type, and we are going to scroll down until we see a scatter plot. So we will select a scatter chart, and this is going to be our data set with respect to the scatter chart. Now, in order to determine what kind of model best fits this data set, we are going to click on at least one of the data points and scroll down until we find the trend line option. We then want to select the trend line option to create a trend line and determine which model best fits. Now, the four options that was given to us is exponential, logarithmic, quadratic, and linear. So in order to determine the different types, we can select the option under type and scroll through each of our options. Now, one thing to note here is that polynomial is the same as quadratic. So let's take a look at each of our options. So if we see linear, as you can see, linear is a trend line where most of the data points don't line up with the model, so linear is not going to be the solution here. If we take a look at exponential, exponential fits a little bit better, but most of the data points tend to tend away from the trend line, so this won't be a good pick either. If we take a look at logarithmic. Logarithmic has, is very similar to linear, but has the same problem where many of the points don't line up with the trend line, so this won't be the solution either. Now if we pick polynomial, the first thing we want to check and make sure is that a polynomial is of degree 2 in order to have an accurate coefficient of determination. But as we can see, polynomial is the best fit because it passes through all of the given points for the data set. So what this means is that the polynomial option is going to be the correct solution. And going back to the original problem. That means that the correct option is going to be option C. Quadratic is going to be the best fitting model by using a scattered plot. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Scatterplot

Mathematical Models (Linear, Quadratic, Logarithmic, Exponential, Power)

Prediction and Model Evaluation

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