Which of the following linear models best represents the line of best fit for the following set of data?
x 2 4 5 6 8 10 11
y 26 38 45 49 64 77 82
The data set can be represented by the equation y = 5x + 16.
Explanation: We can find the equation of the line of best fit by finding the slope and y-intercept of the line.
Using the formula for the slope of a line, m = (Σ(xy) - ΣxΣy / nΣx^2 - (Σx)^2), where Σ denotes the sum, n is the number of data points, x and y are the variables, and xy denotes the product of x and y, we can calculate the slope:
Σ(xy) = (2*26) + (4*38) + (5*45) + (6*49) + (8*64) + (10*77) + (11*82) = 840.
ΣxΣy = (2+4+5+6+8+10+11)(26+38+45+49+64+77+82) = 8124.
Σx^2 = 2^2 + 4^2 + 5^2 + 6^2 + 8^2 + 10^2 + 11^2 = 312.
(Σx)^2 = (2+4+5+6+8+10+11)^2 = 289.
nΣx^2 - (Σx)^2 = 7(312) - 289 = 2175.
m = (840 - 8124) / 2175 ≈ -0.1825.
The y-intercept can be found using the formula b = (Σy - mΣx) / n:
Σy = 26 + 38 + 45 + 49 + 64 + 77 + 82 = 381.
b = (381 - (-0.1825)(2+4+5+6+8+10+11)) / 7 ≈ 15.2459.
Therefore, the equation of the line of best fit is y = -0.1825x + 15.2459.
However, none of the given models exactly match this equation.
A) y=6.3x+12.8
B) y=−6.3x+12.8
C) y=−0.6x+20.2
C) y=0.6x+20.2
C) y = -0.6x + 20.2
This linear model best represents the line of best fit for the given set of data.