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

 Page 4 / 14

## 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.

#### Questions & Answers

(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
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Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
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master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
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Miranda
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
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Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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Propessor
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