Car Performance A V8 car engine is coupled to a dynamometer,...

Car Performance $A$ V8 car engine is coupled to a dynamometer, and the horsepower $y$ is measured at different engine speeds $x$ (in thousands of revolutions per minute). The results are shown in the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline y & {40} & {85} & {140} & {200} & {225} & {245} \\ \hline\end{array} $$ (a) Use the regression capabilities of a graphing utility to find a cubic model for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use the model to approximate the horsepower when the engine is running at 4500 revolutions per minute.

Car Performance $A$ V8 car engine is coupled to a dynamometer, and the horsepower $y$ is measured at different engine speeds $x$ (in thousands of revolutions per minute). The results are shown in the table.
$$
\begin{array}{|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline y & {40} & {85} & {140} & {200} & {225} & {245} \\ \hline\end{array}
$$
(a) Use the regression capabilities of a graphing utility to find a cubic model for the data.
(b) Use a graphing utility to plot the data and graph the model.
(c) Use the model to approximate the horsepower when the engine is running at 4500 revolutions per minute.

Key Concepts

Graphical Visualization

Visualizing both the data points and the regression curve on a graph helps assess how well the model fits the observed data. It enables users to identify trends, detect anomalies, and understand the overall pattern, making it easier to validate the effectiveness of the predictive model.

Cubic Regression Model

A cubic regression model involves fitting a third-degree polynomial to the data to establish a relationship between the independent and dependent variables. This model is particularly useful when the trend in the data shows non-linear behavior that can be better captured with a polynomial that includes cubic terms.

Prediction Using Regression Models

Once a regression model has been established, it can be used to predict or estimate values of the dependent variable for given values of the independent variable. This process, known as interpolation (or extrapolation when outside the observed range), is fundamental in many practical applications for forecasting and decision-making.

Transcript

00:01 They want us to use this data here to find a cubic model.

00:10 So if you're doing this with a calculator, you might get a little bit different of an answer since i'll be using google sheets to be doing this, and they might use slightly different algorithms to actually find these models.

00:25 But whatever you get should be fairly close to what we have if you were to just kind of round enough.

00:32 So if you get something slightly different, then it's probably still correct.

00:37 Just google sheets is rounding a little bit different or might be using a slightly different algorithm than what your calculator does.

00:43 But we're going to be using google sheets to do this.

00:46 So what we will first do is once we get the data into here, we're just going to highlight this.

00:51 Then we'll come up here to more and then insert chart.

00:56 So now this is going to give us the chart of the data.

00:59 And since in part b they want us to have this as a scatter plot so let's just go ahead and do that so you would come up here hit setup and then if it was a line like we had before just change it to scatter plot so we have that then we are going to come up here to customize and we'll scroll down hit series if that's not open and then we're going to hit trend line and so we want a cubic so we're going to going to come down here to polynomial, and we're going to change this to a degree three polynomial.

01:36 And so if we want the regression, we'll hit label and use equation, and then it'll pop up the equation up top here.

01:44 So this is what google sheets is saying should be our regression.

01:48 And again, you might have something really close to this...

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