Question

melania is tracking her running training program. the table gives her 5k run time at the end of each month.

month	1	2	3	4	5	6

what is the equation for the line of best fit, where x represents the month and y-hat represents the predicted time?
options:
$hat{y} = 33.6 + 2.77x$
$hat{y} = 48.5 + 2.77x$
$hat{y} = 33.6 - 2.77x$
$hat{y} = 48.5 - 2.77x$

Explanation:

Step1: Calculate $\bar{x}$ (mean of x)

$x$ values: 1,2,3,4,5,6
$\bar{x} = \frac{1+2+3+4+5+6}{6} = 3.5$

Step2: Calculate $\bar{y}$ (mean of y)

$y$ values: 47,43,39,36,35,33
$\bar{y} = \frac{47+43+39+36+35+33}{6} = 38.833$

Step3: Calculate slope $m$

First compute numerator $\sum(x_i-\bar{x})(y_i-\bar{y})$:
$(1-3.5)(47-38.833)+(2-3.5)(43-38.833)+(3-3.5)(39-38.833)+(4-3.5)(36-38.833)+(5-3.5)(35-38.833)+(6-3.5)(33-38.833)$
$=(-2.5)(8.167)+(-1.5)(4.167)+(-0.5)(0.167)+(0.5)(-2.833)+(1.5)(-3.833)+(2.5)(-5.833)$
$=-20.4175-6.2505-0.0835-1.4165-5.7495-14.5825=-48.5$

Denominator $\sum(x_i-\bar{x})^2$:
$(1-3.5)^2+(2-3.5)^2+(3-3.5)^2+(4-3.5)^2+(5-3.5)^2+(6-3.5)^2$
$=6.25+2.25+0.25+0.25+2.25+6.25=17.5$

$m = \frac{-48.5}{17.5} \approx -2.77$

Step4: Calculate y-intercept $b$

$b = \bar{y} - m\bar{x} = 38.833 - (-2.77)(3.5) = 38.833 + 9.695 = 48.528 \approx 48.5$

Step5: Form the regression equation

$\hat{y} = b + mx$

Answer:

$\hat{y}=48.5-2.77x$