Question

determining the trend line of best fit
use the points to describe the data and determine the line of best fit.
what type of correlation do the data points represent?
what is the equation that best represents the data?
( y = - 400x + 3,900 )
( y = - 500x + 4,500 )
( y = - 100x + 3,100 )
( y = 350x + 3,100 )

Explanation:

Step1: Determine the correlation type

Looking at the scatter - plot, as the number of rainy days (x - value) increases, the total ticket sales (y - value) decrease. So, it is a negative correlation.

Step2: Check the equation of the line of best fit

We can use the slope - intercept form \(y = mx + b\).
Let's assume two points \((x_1,y_1)\) and \((x_2,y_2)\). If we take \(x = 0\), for \(y=-400x + 3900\), \(y = 3900\); for \(y=-500x + 4500\), \(y = 4500\); for \(y=-100x+3100\), \(y = 3100\); for \(y = 350x+3100\), it is a positive - slope line (rejected as we have a negative correlation).
Let's check the slope. If we assume two points \((1,2700)\) and \((3,1300)\) (approximate from the scatter - plot). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{1300 - 2700}{3 - 1}=\frac{-1400}{2}=-700\) (approximate).
If we substitute \(x = 1\) into \(y=-400x + 3900\), \(y=-400\times1+3900 = 3500\) (not a good fit).
If we substitute \(x = 1\) into \(y=-500x + 4500\), \(y=-500\times1 + 4500=4000\) (not a good fit).
If we substitute \(x = 1\) into \(y=-100x + 3100\), \(y=-100\times1+3100 = 3000\) (not a good fit).
Let's use another approach. The general trend: when \(x = 0\), we can assume a higher \(y\) - value. For \(y=-400x + 3900\), when \(x = 0\), \(y = 3900\); when \(x=9\), \(y=-400\times9 + 3900=-3600 + 3900 = 300\).
For \(y=-500x + 4500\), when \(x = 0\), \(y = 4500\); when \(x = 9\), \(y=-500\times9+4500=0\).
If we check the spread of points. The line \(y=-400x + 3900\) seems to be a better fit as it is in the middle of the data points' general trend.

Answer:

Negative correlation, \(y=-400x + 3900\)