You research the average cost of a tank of gasoline for a certain type of car for several recent years to look for trends. The table shows your data. What is the equation for a line of best fit? How much would you expect to pay for a tank of gas in the year 2019? Let x be the number of years after 1998.
Year:1998,2000,2002,2004,2006,2008
Average cost for one tank:$26.40,$27.20,$29.70,$29.30,$32.20,$37.80
To find the equation for a line of best fit, we can use linear regression. Using the given data, let's calculate the equation:
Year (x): 0, 2, 4, 6, 8, 10
Cost ($): 26.40, 27.20, 29.70, 29.30, 32.20, 37.80
Using a calculator or statistical software, we find that the equation for the line of best fit is:
Cost ($) = 0.5600x + 26.560
To find how much you would expect to pay for a tank of gas in the year 2019, we need to find the corresponding value of x. Since 2019 is 21 years after 1998, the value of x would be:
x = 21
Substitute x = 21 into the equation for the line of best fit:
Cost (2019) = 0.5600(21) + 26.560
Cost (2019) = 11.760 + 26.560
Cost (2019) ≈ 38.32
Therefore, you would expect to pay approximately $38.32 for a tank of gas in the year 2019.
To find the equation for a line of best fit and estimate the cost of a tank of gas in 2019, we can use linear regression.
Step 1: Convert the given years to the number of years after 1998 (x).
Year: 1998, 2000, 2002, 2004, 2006, 2008
x: 0, 2, 4, 6, 8, 10
Step 2: Calculate the average cost of a tank of gas for each year.
Average cost for one tank: $26.40, $27.20, $29.70, $29.30, $32.20, $37.80
Step 3: Calculate the average of both x and the average cost values.
x̅ = (0 + 2 + 4 + 6 + 8 + 10) / 6 = 5
y̅ = ($26.40 + $27.20 + $29.70 + $29.30 + $32.20 + $37.80) / 6 ≈ $30.23
Step 4: Calculate the sum of the differences between each x value and x̅, and the differences between each y value and y̅.
Σ(x - x̅) = (0 - 5) + (2 - 5) + (4 - 5) + (6 - 5) + (8 - 5) + (10 - 5) = -5
Σ(y - y̅) = ($26.40 - $30.23) + ($27.20 - $30.23) + ($29.70 - $30.23) + ($29.30 - $30.23) + ($32.20 - $30.23) + ($37.80 - $30.23) ≈ $5.71
Step 5: Calculate the sum of the squared differences between each x value and x̅.
Σ(x - x̅)² = (-5)² + (-3)² + (-1)² + (1)² + (3)² + (5)² = 55
Step 6: Calculate the sum of the product of the differences between each x value and x̅, and the differences between each y value and y̅.
Σ((x - x̅)(y - y̅)) = (-5)($26.40 - $30.23) + (-3)($27.20 - $30.23) + (-1)($29.70 - $30.23) + (1)($29.30 - $30.23) + (3)($32.20 - $30.23) + (5)($37.80 - $30.23) ≈ $23.86
Step 7: Calculate the slope (m) of the line of best fit.
m = Σ((x - x̅)(y - y̅)) / Σ(x - x̅)² ≈ $23.86 / 55 ≈ $0.43
Step 8: Calculate the y-intercept (b) of the line of best fit.
b = y̅ - m * x̅ ≈ $30.23 - $0.43 * 5 ≈ $27.02
Therefore, the equation for the line of best fit is y ≈ $0.43x + $27.02.
To estimate the cost of a tank of gas in 2019 (x = 2019 - 1998 = 21 years after 1998), substitute x = 21 into the equation:
y ≈ $0.43 * 21 + $27.02
y ≈ $9.03
Therefore, you would expect to pay approximately $9.03 for a tank of gas in the year 2019.
To find the equation for a line of best fit, we can use linear regression analysis. This technique helps us determine the relationship between two variables, in this case, the year (x) and the average cost of a tank of gasoline (y).
Step 1: Convert the years to the number of years after 1998.
Year: 1998, 2000, 2002, 2004, 2006, 2008
x: 0, 2, 4, 6, 8, 10
Step 2: Calculate the mean values for x and y.
Mean of x = (0 + 2 + 4 + 6 + 8 + 10)/6 = 4
Mean of y = ($26.40 + $27.20 + $29.70 + $29.30 + $32.20 + $37.80)/6 = $30.00
Step 3: Calculate the deviations from the mean for x and y.
Deviation of x = x - mean of x
Deviation of y = y - mean of y
Deviations of x: -4, -2, 0, 2, 4, 6
Deviations of y: -$3.60, -$2.80, -$0.30, -$0.70, $2.20, $7.80
Step 4: Calculate the product of the deviations.
Product of deviations = (deviation of x) * (deviation of y)
Products of deviations: 14.4, 5.6, 0, -1.4, 8.8, 46.8
Step 5: Calculate the squared deviations.
Squared deviation of x = (deviation of x)^2
Squared deviation of y = (deviation of y)^2
Squared deviations of x: 16, 4, 0, 4, 16, 36
Squared deviations of y: 12.96, 7.84, 0.09, 0.49, 4.84, 60.84
Step 6: Calculate the sum of squared deviations.
Sum of squared deviations of x = 16 + 4 + 0 + 4 + 16 + 36 = 76
Sum of squared deviations of y = 12.96 + 7.84 + 0.09 + 0.49 + 4.84 + 60.84 = 87.96
Step 7: Calculate the sum of the product of deviations.
Sum of products of deviations = 14.4 + 5.6 + 0 - 1.4 + 8.8 + 46.8 = 75.2
Step 8: Calculate the slope (m) using the formula:
m = (sum of products of deviations) / (sum of squared deviations of x)
m = 75.2 / 76 ≈ 0.989
Step 9: Calculate the y-intercept (b) using the formula:
b = mean of y - (m * mean of x)
b = $30.00 - (0.989 * 4) ≈ $25.04
Therefore, the equation for the line of best fit is:
y = 0.989x + $25.04
To find the expected cost for a tank of gas in the year 2019 (x = 21):
y = 0.989 * 21 + $25.04
y ≈ $46.04
Based on the line of best fit equation, you would expect to pay approximately $46.04 for a tank of gas in the year 2019.