Given a set of points (x,y) the line of best fit is given by the least squares method. According to this method the equation of this line is:
[tex]y=a+bx[/tex]
Where a and b are given by:
There are a few quantities to note here. First we have the mean values of x and y:
[tex]\begin{gathered} \bar{X}=\frac{\sum ^n_{i\mathop=1}X_i}{n} \\ \bar{Y}=\frac{\sum ^n_{i\mathop{=}1}Y_i}{n} \end{gathered}[/tex]
Where Xi and Yi are the x values and y values given by the table and n is the total number of points (x,y). So first of all let's find these two values. We just need to sum all the x values and divide them by 11 (the total amount of points) and then do the same for the y values. Then we get:
[tex]\begin{gathered} \bar{X}=\frac{2+3+\cdots+6+8}{11}=\frac{50}{11} \\ \bar{Y}=\frac{3.4+2.5+\cdots+1.2+12}{11}=\frac{59.9}{11} \end{gathered}[/tex]
Now let's calculate the denominator of b. We need to find this expression for each value of x:
[tex](x-\bar{X})^2=(x-\frac{50}{11})^2[/tex]
And then add all the results. If we do this for each x value we get the following set of values:
The denominator is given by their sum And this is equal to 24.7273.
For the numerator of b we first need to find:
[tex](x-\bar{X})\text{ and }(y-\bar{Y})[/tex]
For each x and y. Remember that:
[tex]\begin{gathered} \bar{X}=\frac{50}{11} \\ \bar{Y}=\frac{59.9}{11} \end{gathered}[/tex]
Then we have the following table of values:
The sum of all this values is the numerator of b and it's equal to 30.1273. Then b is equal to:
[tex]b=\frac{30.1273}{24.7273}=1.2184[/tex]
Then a is:
[tex]a=\bar{Y}-b\bar{X}=\frac{59.9}{11}-1.2184\cdot\frac{50}{11}=-0.0927[/tex]
So the slope of the line of best fit is given by b and is equal to 1.2184.
The slope tells us how much does the the y value increases when the x value increases in 1 unit. In this case since x represents the number of people in a household and y represents the pounds wasted the slope represents the food waste per additional person i.e. the amount of food wasted by a person on average. So this is basically saying than a person on average wastes 1.2184 pounds of food per day which seems to be a little high.
