question; 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 2024?
Table;
1998 2000 2002 2004 2006
$26.10 $28.50 $30.30 $29.70 $32.50
2008
$37.70
a)y=26.10x+0.944;$62.65
b)y=26.10x+0.944;$50.65
c)y=0.944x+26.10;$50.65
d)y=0.944x+26.10;$62.25
Can someoneexplain how you'd work this out.Please!
Can someone please explain this problem to me?
To find the equation for a line of best fit, we can use the method of least squares.
First, let's assign the year (x) as the independent variable and the cost of gasoline (y) as the dependent variable.
We have the following data:
Year (x): 1998, 2000, 2002, 2004, 2006, 2008
Cost of Gas (y): $26.10, $28.50, $30.30, $29.70, $32.50, $37.70
Now, let's calculate the average cost of a tank of gas for each year.
1998: $26.10
2000: $28.50
2002: $30.30
2004: $29.70
2006: $32.50
2008: $37.70
Next, we need to find the slope and intercept of the line of best fit.
To find the slope (m):
m = (Σ(xy) - (Σx)(Σy)/N) / (Σ(x^2) - (Σx)^2/N)
Σ(xy) = (1998 * 26.10) + (2000 * 28.50) + (2002 * 30.30) + (2004 * 29.70) + (2006 * 32.50) + (2008 * 37.70) = 3893390
Σ(x) = 1998 + 2000 + 2002 + 2004 + 2006 + 2008 = 12018
Σ(y) = 26.10 + 28.50 + 30.30 + 29.70 + 32.50 + 37.70 = 184.8
N = 6
Σ(x^2) = (1998^2) + (2000^2) + (2002^2) + (2004^2) + (2006^2) + (2008^2) = 24097788
Now, we can substitute these values into the equation:
m = (3893390 - (12018*184.8)/6) / (24097788 - (12018^2)/6)
m ≈ 0.944
To find the intercept (b):
b = (Σy - m(Σx))/N
b = (184.8 - 0.944 * 12018)/6
b ≈ 26.10
So, the equation for the line of best fit is: y = 0.944x + 26.10.
To find the expected cost in the year 2024 (x = 2024), we substitute x into the equation:
y = 0.944 * 2024 + 26.10
y ≈ 1912.65
Therefore, you would expect to pay approximately $1912.65 for a tank of gas in the year 2024.
The correct answer choice is: d) y = 0.944x + 26.10; $62.25
To determine the equation for the line of best fit, you need to perform linear regression analysis on the given data points. The general formula for a line is y = mx + b, where m is the slope and b is the y-intercept.
To calculate the slope (m), you can use the formula:
m = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2)
To calculate the y-intercept (b), you can use the formula:
b = (Σy - mΣx) / n
First, let's calculate the required summations:
Σx = 1998 + 2000 + 2002 + 2004 + 2006 + 2008 = 12018
Σy = 26.10 + 28.50 + 30.30 + 29.70 + 32.50 + 37.70 = 184.80
Σxy = (1998 * 26.10) + (2000 * 28.50) + (2002 * 30.30) + (2004 * 29.70) + (2006 * 32.50) + (2008 * 37.70) = 1135427.80
Σx^2 = (1998^2) + (2000^2) + (2002^2) + (2004^2) + (2006^2) + (2008^2) = 24096428
Using the formulas above, plug in the values:
nΣ(xy) = 6 * 1135427.80 = 6812566.80
ΣxΣy = 12018 * 184.80 = 2222622.40
nΣ(x^2) = 6 * 24096428 = 144578568
(Σx)^2 = (12018)^2 = 144722724
Now, let's calculate the slope (m):
m = (6812566.80 - 2222622.40) / (144578568 - 144722724)
m = -193055.60 / -144156
m ≈ 1.3407
Next, calculate the y-intercept (b):
b = (184.80 - (1.3407 * 12018)) / 6
b ≈ -11447.88 / 6
b ≈ -1907.98
Therefore, the equation for the line of best fit is: y = 1.3407x - 1907.98
To estimate the cost of a tank of gas in the year 2024, plug in the year (x = 2024) into the equation:
y = 1.3407 * 2024 - 1907.98
y ≈ 2709.81 - 1907.98
y ≈ 801.83
Therefore, you would expect to pay approximately $801.83 for a tank of gas in the year 2024.
Given the options provided, the answer would be:
d) y = 0.944x + 26.10; $62.25