Question

payton collected data to show the relationship between the number of hours he practices and the number of errors he makes when playing a new piece of music. the table shows his data
practice makes better

number of hours	1	2	3	4	5	6	7	8
number of errors	36	34	30	31	23	16	11	6

which is the approximate slope of the line of best fit for the data?
-5.5
-4.5
-2.0
-1.0

Explanation:

Step1: Recall slope formula

The formula for the slope $m$ of the line of best - fit using two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. We can also use a more comprehensive formula for the slope of the line of best - fit $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$, but for a quick estimate, we can choose two points. Let's choose the first point $(x_1,y_1)=(1,36)$ and the last point $(x_2,y_2)=(8,6)$.

Step2: Calculate the slope

$m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{6 - 36}{8 - 1}=\frac{- 30}{7}\approx - 4.3$. Another way is to use a more visual or intuitive approach. As the number of hours $x$ increases from $1$ to $8$, the number of errors $y$ decreases from $36$ to $6$. The change in $y$ is $\Delta y=6 - 36=-30$ and the change in $x$ is $\Delta x=8 - 1 = 7$. The slope $m=\frac{\Delta y}{\Delta x}\approx - 4.3$. The closest value to our estimate among the options is $-4.5$.

Answer:

-4.5