Maleia is tracking her running training program. The table gives her 5K run time at the end of each month.
Month Time (minutes)
1 47
2 43
3 39
4 36
5 35
6 33
What is the equation for the line of best fit where x represents the month and y represents the time?
y = −2.78x + 33.6
y = −2.78x + 48.5
y = 2.78x + 33.6
y = 2.78x + 48.5
To find the equation for the line of best fit, we need to determine the slope and y-intercept.
To find the slope, we can use the formula:
m = (Σ(xy) - (Σx)(Σy) / n(Σ(x²) - (Σx)²)
where Σ represents the sum, Σ(xy) represents the sum of the products of x and y, Σx represents the sum of x values, Σy represents the sum of y values, Σ(x²) represents the sum of the squares of x values, n represents the number of data points.
Calculating the necessary values:
Σxy = (1*47) + (2*43) + (3*39) + (4*36) + (5*35) + (6*33) = 597
Σx = 1 + 2 + 3 + 4 + 5 + 6 = 21
Σy = 47 + 43 + 39 + 36 + 35 + 33 = 233
Σ(x²) = (1²) + (2²) + (3²) + (4²) + (5²) + (6²) = 91
n = 6
Plugging these values into the formula:
m = (597 - (21*233) / (6*91 - (21)²)
m = (597 - 4803) / (546 - 441)
m = -4206 / 105
m = -40.06
The slope (m) is approximately -40.06.
To find the y-intercept, we can use the formula:
b = (Σy - m(Σx)) / n
Plugging in the values:
b = (233 - (-40.06)(21)) / 6
b = (233 + 841.26) / 6
b = 1074.26 / 6
b = 179.04
The y-intercept (b) is approximately 179.04.
Putting it all together, the equation for the line of best fit is:
y = -40.06x + 179.04.