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.

Global Warming Listed below are mean annual temperatures (°C) of the earth for each decade, beginning with the decade of the 1880s. Find the best model and then predict the value for 2090–2099. Comment on the result.

Table showing mean annual Earth temperatures (°C) by decade from 1880s to 1970s for model fitting.

Accepted Answer

Hello. In this video, we are told that the following data lists the number of bacteria in a culture over several hours. We want to construct a scatter plot to determine the best mathematical model for the data set. So in order to approach this problem, we are going to need to use an external tool to help us create a scatter plot using the given data. So in this video I'm going to be using Google Sheets, but the first thing we want to do is we want to go ahead and take the given data set and separate it into two separate 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. One column represents the hours that have passed by for the culture, and the second column represents the amount of bacteria per hour. Now, in order to create the scatter plot, we are going to want to select both of the columns. Then we are going to go up to insert and create a chart. Now, we're automatically going to get a connected data set like this, but we want to use a scatter plot. So what we're going to do is we're going to go under chart type during the setup, and we're going to click on line chart, and what we're going to do is we're going to select the scatter chart option. So this is what our scatter chart should look like. Now, in order to determine which model best fits, we are going to be using a tool called the trend line. In order to access the trend line, we will go ahead and select one of our data points, and we will scroll down until we see a trend line option. We will select it Now, what this is going to do is this is going to create a trend line for different options of models. So currently, we are looking at the linear model, but if we click underneath the type, we get a list of different models to choose from as well. Now there are 4 options that we want to go ahead and explore. We have linear, exponential, logarithmic, and polynomial is equivalent to quadratic. So, let's go ahead and first take a look at observing the linear one. As we look at the linear trend line, many of the data points tend to be either slightly above or slightly below the trend line, so this model won't fit the data set. If we explore logarithmic, logarithmic doesn't follow the pattern of the data points at all, so this won't be an option either. Exponential is slightly better, but it still has some data points that tend away from the trend line, so this won't be our correct option. And if we take a look at polynomial, polynomial fits the trend line that is the best fitting trend line. Now, one thing to note is that when you're looking at the polynomial trend line, you wanna make sure that your polynomial degree is 2. That way you are working with an accurate coefficient of determination. But comparing all the options together, polynomial is going to be the best fitting model. So going back to the original problem, that means that our correct solution is going to be option C. The quadratic model is the best fitting model by using a scattered plot. 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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