Question

1. using the line of best fit, which equation most closely represents the set of data? {(-0.09, -0.91), (2.75, -2.31), (-4.76, 0.51), (-0.81, -0.59), (6.07, -3.35), (0.13, -3.11)} \\( y = -0.36x - 1.07 \\) \\( y = -0.81x - 1.62 \\) \\( y = -0.15x - 0.51 \\) \\( y = -0.09x - 0.32 \\)

Explanation:

Step1: Calculate mean of x-values

The x - values are \(-0.09\), \(2.75\), \(-4.76\), \(-0.81\), \(6.07\), \(0.13\).
The sum of x - values: \(\sum x=- 0.09 + 2.75-4.76 - 0.81+6.07 + 0.13=3.29\)
The mean of x - values, \(\bar{x}=\frac{\sum x}{n}=\frac{3.29}{6}\approx0.55\)

Step2: Calculate mean of y - values

The y - values are \(-0.91\), \(-2.31\), \(0.51\), \(-0.59\), \(-3.35\), \(-3.11\)
The sum of y - values: \(\sum y=-0.91-2.31 + 0.51-0.59-3.35-3.11=-9.76\)
The mean of y - values, \(\bar{y}=\frac{\sum y}{n}=\frac{-9.76}{6}\approx - 1.63\)

Step3: Check which line passes through \((\bar{x},\bar{y})\)

We check the equations:
- For \(y=-0.36x - 1.07\): Substitute \(x = 0.55\), \(y=-0.36\times0.55-1.07=-0.198 - 1.07=-1.268 eq - 1.63\)
- For \(y=-0.81x - 1.62\): Substitute \(x = 0.55\), \(y=-0.81\times0.55-1.62=-0.4455-1.62=-2.0655 eq - 1.63\)
- For \(y=-0.15x - 0.51\): Substitute \(x = 0.55\), \(y=-0.15\times0.55-0.51=-0.0825-0.51=-0.5925 eq - 1.63\)
- For \(y=-0.09x - 0.32\): Substitute \(x = 0.55\), \(y=-0.09\times0.55-0.32=-0.0495-0.32=-0.3695 eq - 1.63\) (Wait, maybe a better way is to use the concept of line of best fit. The line of best fit should have a slope and intercept such that it minimizes the sum of squared residuals. Alternatively, we can plug in one of the points. Let's take the point \((- 4.76,0.51)\)
- For \(y=-0.36x - 1.07\): \(y=-0.36\times(-4.76)-1.07 = 1.7136-1.07 = 0.6436\approx0.51\) (close)
- For \(y=-0.81x - 1.62\): \(y=-0.81\times(-4.76)-1.62=3.8556 - 1.62 = 2.2356 eq0.51\)
- For \(y=-0.15x - 0.51\): \(y=-0.15\times(-4.76)-0.51 = 0.714-0.51 = 0.204 eq0.51\)
- For \(y=-0.09x - 0.32\): \(y=-0.09\times(-4.76)-0.32 = 0.4284-0.32 = 0.1084 eq0.51\)

Another way: Calculate the slope between two points. Let's take \((-0.09,-0.91)\) and \((2.75,-2.31)\)
Slope \(m=\frac{-2.31 + 0.91}{2.75 + 0.09}=\frac{-1.4}{2.84}\approx - 0.49\) (close to - 0.36? Wait, maybe my first approach was wrong. Let's use the formula for the line of best fit \(y=mx + b\), where \(m=\frac{\sum(x_i-\bar{x})(y_i - \bar{y})}{\sum(x_i-\bar{x})^2}\)

Calculate \((x_i-\bar{x})\) and \((y_i-\bar{y})\) for each point:

1. For \((-0.09,-0.91)\): \(x_i-\bar{x}=-0.09 - 0.55=-0.64\), \(y_i-\bar{y}=-0.91+1.63 = 0.72\), \((x_i - \bar{x})(y_i-\bar{y})=-0.64\times0.72=-0.4608\), \((x_i - \bar{x})^2=0.4096\)
2. For \((2.75,-2.31)\): \(x_i-\bar{x}=2.75 - 0.55 = 2.2\), \(y_i-\bar{y}=-2.31 + 1.63=-0.68\), \((x_i - \bar{x})(y_i-\bar{y})=2.2\times(-0.68)=-1.496\), \((x_i - \bar{x})^2 = 4.84\)
3. For \((-4.76,0.51)\): \(x_i-\bar{x}=-4.76 - 0.55=-5.31\), \(y_i-\bar{y}=0.51 + 1.63 = 2.14\), \((x_i - \bar{x})(y_i-\bar{y})=-5.31\times2.14=-11.3634\), \((x_i - \bar{x})^2 = 28.1961\)
4. For \((-0.81,-0.59)\): \(x_i-\bar{x}=-0.81 - 0.55=-1.36\), \(y_i-\bar{y}=-0.59 + 1.63 = 1.04\), \((x_i - \bar{x})(y_i-\bar{y})=-1.36\times1.04=-1.4144\), \((x_i - \bar{x})^2 = 1.8496\)
5. For \((6.07,-3.35)\): \(x_i-\bar{x}=6.07 - 0.55 = 5.52\), \(y_i-\bar{y}=-3.35 + 1.63=-1.72\), \((x_i - \bar{x})(y_i-\bar{y})=5.52\times(-1.72)=-9.4944\), \((x_i - \bar{x})^2 = 30.4704\)
6. For \((0.13,-3.11)\): \(x_i-\bar{x}=0.13 - 0.55=-0.42\), \(y_i-\bar{y}=-3.11 + 1.63=-1.48\), \((x_i - \bar{x})(y_i-\bar{y})=-0.42\times(-1.48)=0.6216\), \((x_i - \bar{x})^2 = 0.1764\)

Now, \(\sum(x_i-\bar{x})(y_i - \bar{y})=-0.4608-1.496-11.3634-1.4144-9.4944 + 0.6216=-23.6074\)

\(\sum(x_i-\bar{x})^2=0.4096 + 4.84+28.1961+1.8496+30.4704+0.1764 = 65.9421\)

Slope \(m=\frac{-23.6074}{65.9421}\approx - 0.36\)

Now, \(b=\bar{y}-m\bar{x}=-1.63-(-0.36)\times0.55=-1.63 + 0.198=-1.432\) (close to - 1.07? Wait, maybe due to rounding errors. Let's check the equation \(y=-0.36x - 1.07\) with the point \((6.07,-3.35)\)

Substitute \(x = 6.07\) into \(y=-0.36x - 1.07\): \(y=-0.36\times6.07-1.07=-2.1852-1.07=-3.2552\approx - 3.35\) (close)

For \(y=-0.81x - 1.62\), \(x = 6.07\): \(y=-0.81\times6.07-1.62=-4.9167-1.62=-6.5367 eq - 3.35\)

For \(y=-0.15x - 0.51\), \(x = 6.07\): \(y=-0.15\times6.07-0.51=-0.9105-0.51=-1.4205 eq - 3.35\)

For \(y=-0.09x - 0.32\), \(x = 6.07\): \(y=-0.09\times6.07-0.32=-0.5463-0.32=-0.8663 eq - 3.35\)

So the equation \(y=-0.36x - 1.07\) is the closest.

Answer:

\(y = - 0.36x - 1.07\)