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
answers:
y=1.02x+25.32;$46.74
y=26.40x+1.006:$58.92
y=26.40x+1.006:$47.52
1.006x+26.40:$47.52
To find the equation for a line of best fit, we need to calculate the slope and y-intercept using the given data points.
Let's calculate the slope (m) using the formula:
m = (Σxy - (Σx)(Σy) / n(Σx²) - (Σx)²)
Σxy = (0)(26.40) + (2)(27.20) + (4)(29.70) + (6)(29.30) + (8)(32.20) + (10)(37.80) = 691.60
Σx = 0 + 2 + 4 + 6 + 8 + 10 = 30
Σy = 26.40 + 27.20 + 29.70 + 29.30 + 32.20 + 37.80 = 182.60
Σx² = (0²) + (2²) + (4²) + (6²) + (8²) + (10²) = 220
n = 6
m = (691.60 - (30)(182.60) / (6)(220) - (30)²
m = (691.60 - 5478) / (1320 - 900)
m = -4786.40 / 420
m = -11.39
Now, let's calculate the y-intercept (b) using the formula:
b = (Σy - m(Σx)) / n
b = (182.60 - (-11.39)(30)) / 6
b = (182.60 + 341.70) / 6
b = 524.30 / 6
b = 87.38
Therefore, the equation for a line of best fit is:
y = -11.39x + 87.38
To find the expected cost for a tank of gas in 2019, we need to substitute x = 2019 - 1998 = 21 into the equation:
y = -11.39(21) + 87.38
y = -239.19 + 87.38
y = -151.81
Since a negative cost does not make sense, we can assume that the cost for a tank of gas in the year 2019 would be $0.
To find the equation for the line of best fit, we need to use linear regression to determine the slope and y-intercept.
Using the given data, we have the following values:
x = [0, 2, 4, 6, 8, 10] (representing the number of years after 1998)
y = [26.40, 27.20, 29.70, 29.30, 32.20, 37.80] (representing the average cost for one tank)
Using these values, we can calculate the equation for the line of best fit:
1. Calculate the average of x and y:
avg(x) = (0 + 2 + 4 + 6 + 8 + 10) / 6 = 30/6 = 5
avg(y) = (26.40 + 27.20 + 29.70 + 29.30 + 32.20 + 37.80) / 6 = 182.60 / 6 = 30.43
2. Calculate the sum of the products of (x - avg(x)) and (y - avg(y)):
Σ[(x - avg(x)) * (y - avg(y))] = [(0-5) * (26.40-30.43)] + [(2-5) * (27.20-30.43)] + [(4-5) * (29.70-30.43)] + [(6-5) * (29.30-30.43)] + [(8-5) * (32.20-30.43)] + [(10-5) * (37.80-30.43)]
= (-5 * -4.03) + (-3 * -3.23) + (-1 * -0.73) + (1 * -1.13) + (3 * 1.77) + (5 * 7.37)
= 20.15 + 9.69 + 0.73 + 1.13 + 5.31 + 36.85
= 73.86
3. Calculate the sum of the squares of (x - avg(x)):
Σ[(x - avg(x))^2] = [(0-5)^2] + [(2-5)^2] + [(4-5)^2] + [(6-5)^2] + [(8-5)^2] + [(10-5)^2]
= 25 + 9 + 1 + 1 + 9 + 25
= 70
4. Calculate the slope (m) of the line of best fit:
m = Σ[(x - avg(x)) * (y - avg(y))] / Σ[(x - avg(x))^2]
= 73.86 / 70
≈ 1.05
5. Calculate the y-intercept (c) of the line of best fit:
c = avg(y) - m * avg(x)
= 30.43 - 1.05 * 5
= 30.43 - 5.25
= 25.18
Therefore, the equation for the line of best fit is y = 1.05x + 25.18.
To estimate the average cost for a tank of gas in the year 2019 (x = 2019 - 1998 = 21), substitute x = 21 into the equation:
y = 1.05 * 21 + 25.18
= 22.05 + 25.18
≈ 47.23
So, you would expect to pay approximately $47.23 for a tank of gas in the year 2019.
To find the equation for a line of best fit, we need to perform linear regression on the given data points. This regression will help us determine the relationship between the number of years after 1998 (x) and the average cost for one tank of gas (y).
Once we have the equation for the line of best fit, we can use it to predict the cost of a tank of gas in the year 2019.
Here are the steps to find the equation for the line of best fit:
1. Convert the years into the number of years after 1998 (x). For example, 2000 becomes 2, 2002 becomes 4, and so on.
Year: 1998, 2000, 2002, 2004, 2006, 2008
x-value: 0, 2, 4, 6, 8, 10
2. Perform linear regression on the data points (x, y) using any statistical software or online calculator. This will give us the equation for the line of best fit.
After performing the regression using the given data points, the equation for the line of best fit is:
y = 1.02x + 25.32
So, this is the equation for the line of best fit for the given data.
Now, to predict the cost of a tank of gas in the year 2019, we need to substitute the corresponding x-value into the equation.
Here, the number of years after 1998 for the year 2019 would be:
2019 - 1998 = 21
Substituting x = 21 into the equation:
y = 1.02 * 21 + 25.32
y = 46.74
Therefore, we would expect to pay $46.74 for a tank of gas in the year 2019, according to the line of best fit equation.