# 4.3 Fitting linear models to data  (Page 4/14)

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## Using a regression line to make predictions

Gasoline consumption in the United States has been steadily increasing. Consumption data from 1994 to 2004 is shown in [link] . http://www.bts.gov/publications/national_transportation_statistics/2005/html/table_04_10.html Determine whether the trend is linear, and if so, find a model for the data. Use the model to predict the consumption in 2008.

 Year '94 '95 '96 '97 '98 '99 '00 '01 '02 '03 '04 Consumption (billions of gallons) 113 116 118 119 123 125 126 128 131 133 136

The scatter plot of the data, including the least squares regression line, is shown in [link] .

We can introduce new input variable, $\text{\hspace{0.17em}}t,$ representing years since 1994.

The least squares regression equation is:

$C\left(t\right)=113.318+2.209t$

Using technology, the correlation coefficient was calculated to be 0.9965, suggesting a very strong increasing linear trend.

Using this to predict consumption in 2008 $\text{\hspace{0.17em}}\left(t=14\right),$

The model predicts 144.244 billion gallons of gasoline consumption in 2008.

Use the model we created using technology in [link] to predict the gas consumption in 2011. Is this an interpolation or an extrapolation?

150.871 billion gallons; extrapolation

Access these online resources for additional instruction and practice with fitting linear models to data.

## Key concepts

• Scatter plots show the relationship between two sets of data. See [link] .
• Scatter plots may represent linear or non-linear models.
• The line of best fit may be estimated or calculated, using a calculator or statistical software. See [link] .
• Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. See [link] .
• The correlation coefficient, $\text{\hspace{0.17em}}r,$ indicates the degree of linear relationship between data. See [link] .
• A regression line best fits the data. See [link] .
• The least squares regression line is found by minimizing the squares of the distances of points from a line passing through the data and may be used to make predictions regarding either of the variables. See [link] .

## Verbal

Describe what it means if there is a model breakdown when using a linear model.

When our model no longer applies, after some value in the domain, the model itself doesn’t hold.

What is interpolation when using a linear model?

What is extrapolation when using a linear model?

We predict a value outside the domain and range of the data.

Explain the difference between a positive and a negative correlation coefficient.

Explain how to interpret the absolute value of a correlation coefficient.

The closer the number is to 1, the less scattered the data, the closer the number is to 0, the more scattered the data.

## Algebraic

A regression was run to determine whether there is a relationship between hours of TV watched per day $\text{\hspace{0.17em}}\left(x\right)\text{\hspace{0.17em}}$ and number of sit-ups a person can do $\text{\hspace{0.17em}}\left(y\right).\text{\hspace{0.17em}}$ The results of the regression are given below. Use this to predict the number of sit-ups a person who watches 11 hours of TV can do.

what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice