# 2.4 Fitting linear models to data

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In this section, you will:
• Draw and interpret scatter plots.
• Find the line of best fit.
• Distinguish between linear and nonlinear relations.
• Use a linear model to make predictions.

A professor is attempting to identify trends among final exam scores. His class has a mixture of students, so he wonders if there is any relationship between age and final exam scores. One way for him to analyze the scores is by creating a diagram that relates the age of each student to the exam score received. In this section, we will examine one such diagram known as a scatter plot.

## Drawing and interpreting scatter plots

A scatter plot is a graph of plotted points that may show a relationship between two sets of data. If the relationship is from a linear model , or a model that is nearly linear, the professor can draw conclusions using his knowledge of linear functions. [link] shows a sample scatter plot.

Notice this scatter plot does not indicate a linear relationship . The points do not appear to follow a trend. In other words, there does not appear to be a relationship between the age of the student and the score on the final exam.

## Using a scatter plot to investigate cricket chirps

[link] shows the number of cricket chirps in 15 seconds, for several different air temperatures, in degrees Fahrenheit Selected data from http://classic.globe.gov/fsl/scientistsblog/2007/10/. Retrieved Aug 3, 2010 . Plot this data, and determine whether the data appears to be linearly related.

 Chirps 44 35 20.4 33 31 35 18.5 37 26 Temperature 80.5 70.5 57 66 68 72 52 73.5 53

Plotting this data, as depicted in [link] suggests that there may be a trend. We can see from the trend in the data that the number of chirps increases as the temperature increases. The trend appears to be roughly linear, though certainly not perfectly so.

## Finding the line of best fit

Once we recognize a need for a linear function to model that data, the natural follow-up question is “what is that linear function?” One way to approximate our linear function is to sketch the line that seems to best fit the data. Then we can extend the line until we can verify the y -intercept. We can approximate the slope of the line by extending it until we can estimate the $\frac{\text{rise}}{\text{run}}.$

## Finding a line of best fit

Find a linear function that fits the data in [link] by “eyeballing” a line that seems to fit.

On a graph, we could try sketching a line.

Using the starting and ending points of our hand drawn line, points (0, 30) and (50, 90), this graph has a slope of

$m=\frac{60}{50}=1.2$

and a y -intercept at 30. This gives an equation of

$T\left(c\right)=1.2c+30$

where $c$ is the number of chirps in 15 seconds, and $T\left(c\right)$ is the temperature in degrees Fahrenheit. The resulting equation is represented in [link] .

## Recognizing interpolation or extrapolation

While the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data. We use a process known as interpolation when we predict a value inside the domain and range of the data. The process of extrapolation is used when we predict a value outside the domain and range of the data.

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this