7.03 Linear, quadratic, and exponential models

Determine whether a linear or exponential model best fits the relationship between the years and the population of fish.

Use technology to plot the data on a coordinate plane, then examine the shape of the data.

To do this, enter the x- and y-values in two separate columns, then highlight the data and select Two Variable Regression Analysis:

To find the line of best fit, choose Linear under the Regression Model drop down menu:

To find the exponential curve of best fit, choose Growth under the Regression Model drop down menu:

The exponential model is a better model of the data because most of the points are tightly clustered around the curve. The linear model does not represent the smallest and largest x-values well.

An exponential model would best fit the data after examining the plot.

We can also use the correlation coefficient and coefficient of determination to examine the fit of the linear and exponential models.

To examine the fit of the exponential model, click the \Sigma x button to show the statistics and find the coefficient of determination \left(R^2\right). For this curve, R^2=0.9844 which means this curve closely models the actual data.

To examine the fit of the linear model, find the correlation coefficient \left(r\right). For this line, r=-0.9538 which shows a strong correlation, but it is not as strong as the exponential model.

Calculate the regression model for the data and use it to predict the population of fish in the lake after 4 years.

Use technology to calculate the exponential regression, then use the equation to determine the population when x=4.

The function that fits the model best is y=1005.4784\left(0.5161\right)^{x}

If x=4, then \begin{aligned}y&=1005.4784\left(0.5161\right)^{4}\\&=71.3359\end{aligned} According to the regression model, we can expect there to be about 71 fish remaining in the lake after 4 years.

Remember that the coefficients in the equation for the curve of best fit have been rounded. For that reason, the answer is not 100\% accurate. If we had used technology to make this prediction, we would have gotten a slightly different answer.

The calculator's answer is more accurate because it includes more decimal values in the coefficients and does not round them to only four place values. However, the differences between these values is small because we used four decimals in the coefficients of the equation.

If we had rounded the coefficients to even fewer decimal places, the answer would have been less accurate.

Formulate a statistical question that Jocelyn can use to investigate the relationship between the shots she makes and the distance she is from the hoop.

We can assume that Jocelyn is investigating the relationship between the shots she makes and the distance she is from the hoop because she wants to improve her game. There are many statistical questions we can ask, but we should focus the question around the purpose of the investigation.

One possible statistical question is, "At what distances does Jocelyn make less than half of her shots?"

Other possible questions are:

• How does the percentage of shots Jocelyn makes change with the distance from the hoop?

• If Jocelyn is at the three point line, what percent of shots can we expect her to make?

• How far is Jocelyn from the hoop when she makes most of her shots?

Describe a method Jocelyn can use to collect the data.

Jocelyn needs to collect data on two variables: the distance she is from the hoop and the number or percent of shots she makes from that distance.

Bivariate data can be collected by:

• acquiring data through research

• collecting data using surveys, observations, scientific experiments, polls, or questionnaires

Jocelyn will need to collect the data herself, rather than acquiring it through research. She cannot collect the data through a survey, poll, or questionnaire either. The only variable Jocelyn is interested in controling is the distance from the hoop, so she does not need to use a scientific experiment.

Jocelyn can use a systematic observation to collect this data. One way to do this would be to draw arcs that are various distances from the hoop (such as 2\text{ ft} from the hoop, 3\text{ ft} from the hoop, etc.). Then, she can shoot a large number of shots from each of the distances and keep track of how many she made from each distance.

To get a random sample, she should not take consecutive shots from the same distance or same side of the hoop. For example, rather than taking 10 consecutive shots from the free throw line, she should take a few shots from the left or right side of the hoop that is the same distance as the free throw line. This will help the data be more representative of how she shoots in general.

Jocelyn collected data on the percentage of shots she made from various distances. Her data is shown in the table.

Organize the data into a scatterplot.

We can use technology to organize this data into a scatterplot. The distance from the hoop is the independent variable, and the percentage of shots she made is the dependent variable.

Enter the x-values into one column and the y-values in a second column, then highlight the data and select Two Variable Regression Analysis.

Determine the type of model that fits the data best, and find the equation of the curve of best fit.

We can use either the table or the scatterplot to analyze how the y-values change as x increases. Then, we can use technology to find the equation of the curve of best fit.

As the x-values increase, the y-values increase, reach a maximum point, then decrease. This shows the data would be best represented by a quadratic equation. Recall that quadratic functions are polynomial functions of degree 2.

Using technology, we can choose a polynomial regression model, and the degree is set to 2 by default.

The quadratic curve of best fit is y=-0.7048x^2+16.1349x-3.2206.

The coefficient of determination, R^2=0.9784, shows that the model is a good representation of the data.

Use the data to answer the statistical question from part (a).

Our statistical question was, "At what distances does Jocelyn make less than half of her shots?" We can use the scatterplot to find the x-values for which the y-values are less than 50.

To help read the graph better, we can use the settings to adjust the axis scale and add grid lines.

The y-values are less than 50 when x is less than 4 or greater than 19. This means Jocelyn makes less than half of her shots when she is less than 4 feet from the hoop or more than 19 feet from the hoop.

Because the coefficient of determination is high, Jocelyn can be fairly confident in this analysis. Notice that the domain of the curve is \left[2,21\right], so Jocelyn should not use this model to make predictions outside of this domain.

Is the relationship between the days and the number of views best approximated by a line, quadratic curve, exponential curve, or a combination of these functions? Explain.

Use the scatterplot to determine how the number of views changes over time. If the change is not consistent, consider which functions would best model the data over different subsets of the domain.

Over the first 8 days, the number of views increases at an increasing rate. From 0 to 8 days, the data is best modeled by an exponential function.

After 8 days, the number of views increases at a relatively constant rate. From 8 to 20 days, the data is best modeled by a linear function.

To check our answer, we can try to fit a linear, exponential, and quadratic model to the data. However, none of the best fit curves model the data well over the entire domain.

Use your answer from part (a) to find the regression model for the data.

In part (a), we found that the data from 0 to 8 days should be modeled by an exponential curve of best fit, and the data from 8 to 20 days should be modeled by a line of best fit.

Using technology, we can enter each subset of data separately to find the two curves of best fit for our piecewise regression model.

To find the exponential curve of best fit for the first subset of the domain, enter the data points on the interval 0\leq x\leq 8 into the calculator. Then, find the growth regression model.

To find the line of best fit for the second subset of the domain, enter the data points on the interval 8\leq x\leq 20 into the calculator. Then, find the linear regression model.

Notice that when x=8, the exponential model represents the data better. When creating the piecewise function, we will include x=8 in the domain for the exponential piece, but exclude x=8 from the domain for the linear piece.

The piecewise regression model for the data shown in the scatterplot isy=\begin{cases}32.1599\left(2.0894\right)^x, & 0\leq x\leq 8 \\1\,553.7473x-808.5385, &x\gt 8 \end{cases}

If we graph the piecewise model, we can see that it represents the data much better than using a single function to model the data over the entire domain.