You research the cost of a gallon of gasoline over several years to look for a trend.
The table below shows your data. The line of best fit uses years since 1970 as the
input value. What is the line of best fit for the data provided? Based upon these
prices, how much would you expect to pay in 2024?
Year
1970
1974
1978
1982
1986
Price per Gallon
$1.20
$1.22
$1.03
$1.24
$1.40
To find the line of best fit, we need to find a linear equation that represents the relationship between the years and the price per gallon.
Using the data provided and using years since 1970 as the input value, we can create the following table:
Input (x) - Years since 1970
0
4
8
12
16
Output (y) - Price per Gallon
$1.20
$1.22
$1.03
$1.24
$1.40
Now, let's calculate the slope (m) and y-intercept (b) using the formula for a line:
y = mx + b
First, find the average of the input values (x) and the output values (y):
x̄ = (0 + 4 + 8 + 12 + 16) / 5 = 8
ȳ = ($1.20 + $1.22 + $1.03 + $1.24 + $1.40) / 5 = $1.238
Next, calculate the sum of the products of the input and output values (xy) and the sum of the squared input values (x^2):
xy = (0*1.20) + (4*1.22) + (8*1.03) + (12*1.24) + (16*1.40) = 96.20
x^2 = (0^2) + (4^2) + (8^2) + (12^2) + (16^2) = 496
Now, calculate the slope (m) using the formula:
m = (xy - n*x̄*ȳ) / (x^2 - n*x̄^2) = (96.20 - 5*8*$1.238) / (496 - 5*8^2) = 0.0032 (rounded to four decimal places)
Finally, calculate the y-intercept (b) using the formula:
b = ȳ - m*x̄ = $1.238 - 0.0032*8 = $1.2116 (rounded to four decimal places)
Therefore, the line of best fit for the data provided is: y = 0.0032x + $1.2116
To find an estimate for the price per gallon in 2024, we can substitute x = (2024 - 1970) = 54 into the equation:
y = 0.0032*54 + $1.2116 = $1.3874 (rounded to four decimal places)
Based on these prices and the line of best fit, we would expect to pay approximately $1.39 per gallon in 2024.