Question

1. using the line of best fit, which equation most closely represents the set of data? {(5.43, 7.68), (-9.58, -15.54), (2.67, 2.87), (4.98, 4.62), (-0.03, -1.41), (-5.5, -8.33)} options: y = 1.13x + 0.23; y = 0.54x + 0.20; y = 2.27x + 1.78; y = 1.44x - 1.2

Explanation:

Step1: Calculate the slope and intercept

To find the line of best fit, we can calculate the slope \( m \) and intercept \( b \) using the formula for the line \( y = mx + b \). First, we find the mean of \( x \) values (\( \bar{x} \)) and mean of \( y \) values (\( \bar{y} \)).

The \( x \) values are: \( 5.43, -9.58, 2.67, 4.98, -0.03, -5.5 \)
The \( y \) values are: \( 7.68, -15.54, 2.87, 4.62, -1.41, -8.33 \)

Calculating \( \bar{x} \):
\[ \bar{x} = \frac{5.43 - 9.58 + 2.67 + 4.98 - 0.03 - 5.5}{6} = \frac{(5.43 + 2.67 + 4.98) + (-9.58 - 0.03 - 5.5)}{6} = \frac{13.08 - 15.11}{6} = \frac{-2.03}{6} \approx -0.338 \]

Calculating \( \bar{y} \):
\[ \bar{y} = \frac{7.68 - 15.54 + 2.87 + 4.62 - 1.41 - 8.33}{6} = \frac{(7.68 + 2.87 + 4.62) + (-15.54 - 1.41 - 8.33)}{6} = \frac{15.17 - 25.28}{6} = \frac{-10.11}{6} \approx -1.685 \]

Now, we calculate the slope \( m \) using the formula:
\[ m = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2} \]

Calculating \( (x_i - \bar{x})(y_i - \bar{y}) \) for each point:
- For \( (5.43, 7.68) \): \( (5.43 + 0.338)(7.68 + 1.685) \approx 5.768 \times 9.365 \approx 53.92 \)
- For \( (-9.58, -15.54) \): \( (-9.58 + 0.338)(-15.54 + 1.685) \approx -9.242 \times -13.855 \approx 128.05 \)
- For \( (2.67, 2.87) \): \( (2.67 + 0.338)(2.87 + 1.685) \approx 3.008 \times 4.555 \approx 13.70 \)
- For \( (4.98, 4.62) \): \( (4.98 + 0.338)(4.62 + 1.685) \approx 5.318 \times 6.305 \approx 33.53 \)
- For \( (-0.03, -1.41) \): \( (-0.03 + 0.338)(-1.41 + 1.685) \approx 0.308 \times 0.275 \approx 0.0847 \)
- For \( (-5.5, -8.33) \): \( (-5.5 + 0.338)(-8.33 + 1.685) \approx -5.162 \times -6.645 \approx 34.30 \)

Summing these up: \( 53.92 + 128.05 + 13.70 + 33.53 + 0.0847 + 34.30 \approx 263.58 \)

Calculating \( (x_i - \bar{x})^2 \) for each point:
- For \( 5.43 \): \( (5.43 + 0.338)^2 \approx 5.768^2 \approx 33.27 \)
- For \( -9.58 \): \( (-9.58 + 0.338)^2 \approx (-9.242)^2 \approx 85.41 \)
- For \( 2.67 \): \( (2.67 + 0.338)^2 \approx 3.008^2 \approx 9.05 \)
- For \( 4.98 \): \( (4.98 + 0.338)^2 \approx 5.318^2 \approx 28.28 \)
- For \( -0.03 \): \( (-0.03 + 0.338)^2 \approx 0.308^2 \approx 0.0949 \)
- For \( -5.5 \): \( (-5.5 + 0.338)^2 \approx (-5.162)^2 \approx 26.65 \)

Summing these up: \( 33.27 + 85.41 + 9.05 + 28.28 + 0.0949 + 26.65 \approx 182.75 \)

Now, the slope \( m \approx \frac{263.58}{182.75} \approx 1.44 \)

Now, using the point-slope form \( y - \bar{y} = m(x - \bar{x}) \), we can find \( b \):
\[ y = 1.44x + b \]
Substituting \( \bar{x} \) and \( \bar{y} \):
\[ -1.685 = 1.44(-0.338) + b \]
\[ -1.685 = -0.487 + b \]
\[ b = -1.685 + 0.487 \approx -1.198 \approx -1.2 \]

So the equation of the line of best fit is \( y = 1.44x - 1.2 \)

Step2: Verify with the given options

We can also verify by plugging in some of the data points into each equation.

For example, take the point \( (5.43, 7.68) \):

- For \( y = 1.13x + 0.23 \): \( 1.13(5.43) + 0.23 \approx 6.14 + 0.23 = 6.37 \) (not close to 7.68)
- For \( y = 0.54x + 0.20 \): \( 0.54(5.43) + 0.20 \approx 2.93 + 0.20 = 3.13 \) (not close)
- For \( y = 2.27x + 1.78 \): \( 2.27(5.43) + 1.78 \approx 12.33 + 1.78 = 14.11 \) (not close)
- For \( y = 1.44x - 1.2 \): \( 1.44(5.43) - 1.2 \approx 7.82 - 1.2 = 6.62 \)? Wait, maybe I made a mistake in calculation earlier. Wait, let's check another point. Take \( (-9.58, -15.54) \):

- \( y = 1.44(-9.58) - 1.2 \approx -13.79 - 1.2 = -14.99 \), which is close to -15.54 (considering rounding errors).

Take \( (2.67, 2.87) \):

- \( y = 1.44(2.67) - 1.2 \approx 3.84 - 1.2 = 2.64 \), close to 2.87.

Take \( (4.98, 4.62) \):

- \( y = 1.44(4.98) - 1.2 \approx 7.17 - 1.2 = 5.97 \)? Wait, no, 1.44*4.98 is 7.17, minus 1.2 is 5.97, but the y-value is 4.62. Wait, maybe my initial calculation of the mean was slightly off due to rounding. However, the slope calculation gave us \( m \approx 1.44 \) and \( b \approx -1.2 \), which matches the option \( y = 1.44x - 1.2 \).

Answer:

\( y = 1.44x - 1.2 \) (the option with the blue dot, which is the correct one)