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.

Sunspot Numbers Listed below in order by row are annual sunspot numbers beginning with 1980. Is the best model a good model? Carefully examine the scatterplot and identify the pattern of the points. Which of the models fits that pattern?

[IMAGE]

Accepted Answer

Hello, everyone. Let's take a look at this question together. The following data set represents the population in thousands of a city recorded at five-year intervals from 1980 to 2020. Construct a scatter plot and determine which mathematical model best fits the data. Assume that the model is to be used only for the scope of the given data. And consider only linear, quadratic, logarithmic, exponential, or power models. And here we have our data set that represents the year and the population in thousands, and we have to determine which mathematical model best fits the data. Is it answer choice A, the linear model, answer choice B, the quadratic model, answer choice C, the exponential model, answer choice D, the logarithmic model, or answer choice E, the power model. And so in order to solve this question, we have to use our TI 84 + calculator to generate a scatter plot where we first enter the data in the lists using the following steps, and then we can plot the scatter plot using the following steps, giving us the following scatter plot. And the next step is to test different models using the data where we fit the different models to this data and calculate the goodness of fit values or the R squared values, starting with the linear model, which the linear model gives us an R2d value of 0.946. Then we have the quadratic model, which the quadratic model gives us an R2d value of 0.999. And then we have the exponential model, which the exponential model gives an R2d value of 0.994, and then we have the power model, which the power model gives us an R2d value of 0.930. And the last model being the logarithmic model, the logarithmic model describes situations where growth starts rapidly. But slows down over time and based on the given data, it shows accelerating growth rather than slowing growth, making a logarithmic model inappropriate. Therefore, the only models that we are considering are the linear model, the quadratic model, the exponential model, and the power model with the following R squared values and the mat. Thematical model that best fits the data is the model with the highest R2 value, and looking at the R2d values that we calculated, we can identify that the quadratic model provides the highest R2d value, making it the mathematical model that best fits the data. So answer choice be quadratic model is the correct answer, and all other models are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Scatterplot

Mathematical Models

Model Fit

Watch next