Question

The number of murders and robberies per 100,000 population for a random selection of states is shown.

Answer 1

The equation of the regression line has the following shape:

$y=mx+b$

Where m is calculated through the following equation:

$m=\frac{N\sum^{}_{}xy-\sum^{}_{}x\sum^{}_{}y}{N\sum^{}_{}x^2-(\sum^{}_{}x)^2}$

And b is calculated through the following equation:

$b=\frac{\sum^{}_{}y-m\sum^{}_{}x}{N}$

N is the number of samples. 8 for this case.

The values of all the sums present in the above equation are reported in the last row of the table:

$\begin{gathered} \sum ^{}_{}x=31 \\ \sum ^{}_{}y=680.1 \\ \sum ^{}_{}xy=3202.71 \\ \sum ^{}_{}x^2=142.52 \\ \sum ^{}_{}y^2=80033.99 \end{gathered}$

Now, we can begin calculating m by replacing the values:

$\begin{gathered} m=\frac{8\cdot3202.71-31\cdot680.1}{8\cdot142.52-31^2} \\ m=25.333 \end{gathered}$

The slope of the equation is m = 25.333.

Now, we can calculate b:

$\begin{gathered} b=\frac{680.1-25.333\cdot31}{8} \\ b=-13.153 \end{gathered}$

Now that we know the parameters m and b for the linear regression, we can build the equation:

$\begin{gathered} y=mx+b \\ y=25.333x-13.153 \end{gathered}$

Where x represents the murders and y the robberies per 100,000 population.

Then, (a): the equation of the regression line is y = 25.333x - 13.153.

To predict the robberies per 100,000 population when x = 4.5 murders, we just need to replace that 4.5 in the equation that we just found:

$y=25.333\cdot4.5-13.153=100,85$

Finally, (b): according to the linear regression, the number of robberies per 100,000 population when x = 4.5 murders is approximately 100,85.