Question

The chart below contains a portion of the fuel consumption information for a 2002 Toyota Echo that I (Jeff) used to own. The first row is the cumulative number of gallons of gasoline that I had used and the second row is the odometer reading when I refilled the gas tank. So, for example, the fourth entry is the point (28.25, 1051) which says that I had used a total of 28.25 gallons of gasoline when the odometer read 1051 miles. Gasoline Used (Gallons) 0 9.26 19.03 28.25 36.45 44.64 53.57 62.62 71.93 81.69 90.43 Odometer (Miles) 11 356 731 1051 1347 1631 1966 2310 2670 3030 3371 Find the least squares line for this data. Is it a good fit? What does the slope of the line represent? Do you and your classmates believe this model would have held for ten years hacd I not crashed the car on the Turnpike a few years ago? (I'm keeping a fuel log for my 2006 Scion xa fer future College Algebra books so I hope not to crash it, too.)

Answer 1

Gasoline usedOdometer (Gallons) (X)(miles) (Y) 2 XY 0 9.26 19.03 28.25 36.45 44.64 53.57 62.62 71.93 81.69 90.43 41 356 731 1051 1347 1631 1966 2310 2670 3030 3371 0 3296.56 13910.93 29690.7!5 49098.15 72807.84 105318.62 144652.2 192053.1 247520.7 304839.53 18504 5xY-1163188.385x-31383.0583 0 85.7476 362.1409 798.0625 1328.6025 1992.7296 2869.7449 3921.2644 5173.9249 6673.2561 8177.5849 X= 497.87 Y:

* Least Squares line is also called Line of Regression.

We need to find the average of all these points to find the best fit.

The equation for the Least Square line is :   $y=mx+c$

where m= slope of the line

c= y-intercept

Equation to find the slope and y-intercept are:

$m=\frac{n(\sum XY)-(\sum X)(\sum y)}{n\sum X^{2}-(\sum X)^{2} }$

$c=\frac{(\sum Y)-m(\sum X)}{n}$

where n=number of data points

$m=\frac{11(1163188.38)-(497.87)(18504)}{11*31383.0583-497.87^{2}}$

$=36.8042$

$c=\frac{(\sum Y)-m(\sum X)}{n}$

  $=\frac{18504-(36.8042*497.87)}{11}$

$=16.39$

Plugging these values to the equation of line we get:

$y=mx+c$

$Y=36.8042*X+16.39$--------------------------> Required least squares line.

Since it gives the average of all the data points it going to be the best fit.

* Slope of a line or Gradient of a line always represents the direction and steepness of a line.In equation form

  $m=\frac{\Delta y}{\Delta x}$

  $\Delta y =$ change in y-value

  $\Delta x =$ change in x-value.

* If you take any data points from the table and plug in the value of X, you will get a y-value close to that indicated in the table.

ex: Take X=44.64

Y= 36.8042* 44.64 + 16.39

= 1659 ( which is a value near to that given 1631. This point (44.64,1659) has the least perpendicular distance to the regression line we just found out).

Thus this model can be held for years to get the best fit.

Thanks!!!