Researchers studying how a car’s Fuel Efficiency varies with...

Researchers studying how a car’s Fuel Efficiency varies with its Speed drove a compact car 200 miles at various speeds on a test track. Their data are shown in the table. $$\begin{array}{lll}{\text { Speed (mph) }} & {35} & {40} & 45 & 50 & 55 & 60 & 65 & 70 & 75\\ {\text { Fuel Eff. }(\mathrm{mpg})} & {25.9} & {27.7} & 28.5 & 29.5 & 29.2 & 27.4 & 26.4 & 24.2 & 22.8\end{array}$$ Create a linear model for this relationship and report any concerns you may have about the model.

Researchers studying how a car’s Fuel Efficiency varies with its Speed drove a compact car
200 miles at various speeds on a test track. Their data are shown in the table.
$$\begin{array}{lll}{\text { Speed (mph) }} & {35} & {40} & 45 & 50 & 55 & 60 & 65 & 70 & 75\\ {\text { Fuel Eff. }(\mathrm{mpg})} & {25.9} & {27.7} & 28.5 & 29.5 & 29.2 & 27.4 & 26.4 & 24.2 & 22.8\end{array}$$
Create a linear model for this relationship and report any concerns you may have about the model.

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to the observed data. In this context, it involves using a straight-line equation to approximate how fuel efficiency changes with speed.

Linearity Assumption

A fundamental assumption of linear regression is that the relationship between the dependent and independent variables is linear. This means that changes in the independent variable (speed) are assumed to produce proportional changes in the dependent variable (fuel efficiency). If the true relationship is non-linear, the linear model may not provide an accurate fit.

Extrapolation Caution

Extrapolation refers to making predictions using a model outside the range of the observed data. When using a linear model for fuel efficiency, caution must be exercised because predictions at speeds far beyond the tested range may not be reliable if the underlying relationship deviates from linearity.

Model Fit and Residual Analysis

Evaluating how well a linear model fits the data is crucial. Residual analysis involves examining the differences between the observed values and the values predicted by the model. This process helps identify patterns, non-linearities, or other issues (such as heteroscedasticity) that might indicate that a linear model is not the best choice for the data.

Transcript

0:00 All right.

00:01 So for number 27, we're trying to compare speed and fuel efficiency on a track.

00:09 And we're trying to see if we graph this information, what kind of model are we going to get? it's going to be linear.

00:16 And if we have any concerns about this.

00:19 So i'm going to desmos .com, d -e -s -m -o -s -com, and you're going to click start graphing.

00:27 And this is your home screen of the graphing calculator.

00:31 So you click the plus button, and we're going to add a table.

00:35 And so to add a table, we're going to input all of the information that they gave us.

00:40 So the 35 was our first value.

00:43 That's going to be our x.

00:45 The 25 .9 is going to be our ys, and so on and so forth.

00:50 And you want to use the tab button to help you get your information put in.

00:55 It just helps with your spacing and less likely to make a mistake.

00:59 So this does end up being the most tedious part of the whole problem.

01:05 It's just making sure we're putting everything in correctly.

01:07 But beyond that, it's smooth sailing because then we're able to actually look at the data and look at what we're trying to answer this question on.

01:19 And luckily, there's not as many pieces to plug in, so that always is nice for a change...

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