Question

What is the slope of the line of best fit? What does it mean in this situation? Is this realistic?

Answer 1

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:

`y=a+bx`

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:

`\begin{gathered} \bar{X}=(\sum ^n_(i\mathop=1)X_i)/(n) \\ \bar{Y}=\frac{\sum ^n_{i\mathop{=}1}Y_i}{n} \end{gathered}`

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:

`\begin{gathered} \bar{X}=(2+3+\cdots+6+8)/(11)=(50)/(11) \\ \bar{Y}=(3.4+2.5+\cdots+1.2+12)/(11)=(59.9)/(11) \end{gathered}`

Now let's calculate the denominator of b. We need to find this expression for each value of x:

`(x-\bar{X})^2=(x-(50)/(11))^2`

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:

`(x-\bar{X})\text{ and }(y-\bar{Y})`

For each x and y. Remember that:

`\begin{gathered} \bar{X}=(50)/(11) \\ \bar{Y}=(59.9)/(11) \end{gathered}`

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:

`b=(30.1273)/(24.7273)=1.2184`

Then a is:

`a=\bar{Y}-b\bar{X}=(59.9)/(11)-1.2184\cdot(50)/(11)=-0.0927`

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.