Question

darius is studying the relationship between mathematics and art. he asks friends to each draw a \typical\ rectangle. he measures the length and width in centimeters of each rectangle and plots the points on a graph, where x represents the width and y represents the length. the points representing the rectangles are (6.1, 12.0), (5.0, 8.1), (9.1, 15.2), (6.5, 10.2), (7.4, 11.3), and (10.9, 17.5). which equation could darius use to determine the length, in centimeters, of a \typical\ rectangle for a given width in centimeters?
o y = 0.605x + 0.004
o y = 0.959x + 0.041
o y = 1.518x + 0.995
o y = 1.967x + 0.984

Explanation:

Step1: Recall the form of a linear - regression equation

The general form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We can use a graphing calculator or software to perform linear regression on the data points $(x_1,y_1)=(6.1,12.0),(x_2,y_2)=(5.0,8.1),(x_3,y_3)=(9.1,15.2),(x_4,y_4)=(6.5,10.2),(x_5,y_5)=(7.4,11.3),(x_6,y_6)=(10.9,17.5)$. But we can also estimate by calculating the slope between two points.
Let's take two points $(x_1,y_1)=(6.1,12.0)$ and $(x_2,y_2)=(5.0,8.1)$.
The slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{8.1 - 12.0}{5.0 - 6.1}=\frac{- 3.9}{-1.1}\approx3.545$ (this is just a quick check for concept, the better way is to use all points for regression).
Another way is to substitute the $x$ - values of the given points into the equations and see which equation gives $y$ - values closest to the observed $y$ - values.
Let's start with the first point $(x = 6.1)$:
For $y = 0.605x+0.004$, when $x = 6.1$, $y=0.605\times6.1 + 0.004=3.6905+0.004 = 3.6945$.
For $y = 0.959x+0.041$, when $x = 6.1$, $y=0.959\times6.1+0.041 = 5.8499+0.041=5.8909$.
For $y = 1.518x+0.995$, when $x = 6.1$, $y=1.518\times6.1 + 0.995=9.2598+0.995 = 10.2548$.
For $y = 1.967x+0.984$, when $x = 6.1$, $y=1.967\times6.1+0.984=12.0987+0.984 = 13.0827$.
The closest value to $y = 12.0$ (from the point $(6.1,12.0)$) among these calculations is for $y = 1.518x + 0.995$.
We can also check other points:
For example, for the point $(x = 5.0)$:
For $y = 1.518x+0.995$, when $x = 5.0$, $y=1.518\times5.0+0.995=7.59+0.995 = 8.585$ which is close to $8.1$.

Step2: Confirm the best - fit equation

By substituting more points into the four equations, we find that the equation $y = 1.518x+0.995$ gives $y$ - values that are closest to the observed $y$ - values of the data points.

Answer:

$y = 1.518x + 0.995$