Answer :
We are given the data set:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 100 \\
4 & 83 \\
6 & 50 \\
8 & 35 \\
10 & 23 \\
\hline
\end{array}
\][/tex]
The goal is to find the equation of the least squares regression line, which has the form
[tex]\[
\hat{y} = bx + a,
\][/tex]
where the slope [tex]\( b \)[/tex] and intercept [tex]\( a \)[/tex] are computed by
[tex]\[
b = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}
\quad \text{and} \quad
a = \bar{y} - b\bar{x}.
\][/tex]
Step 1. Compute the Means
First, calculate the mean of [tex]\( x \)[/tex] values:
[tex]\[
\bar{x} = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6.
\][/tex]
Next, calculate the mean of [tex]\( y \)[/tex] values:
[tex]\[
\bar{y} = \frac{100 + 83 + 50 + 35 + 23}{5} = \frac{291}{5} = 58.2.
\][/tex]
Step 2. Compute the Numerator
The numerator for the slope [tex]\( b \)[/tex] is
[tex]\[
\sum (x - \bar{x})(y - \bar{y}).
\][/tex]
Calculate each term:
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
(2-6)(100-58.2) = (-4)(41.8) = -167.2.
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
(4-6)(83-58.2) = (-2)(24.8) = -49.6.
\][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
(6-6)(50-58.2) = (0)(-8.2) = 0.
\][/tex]
- For [tex]\( x = 8 \)[/tex]:
[tex]\[
(8-6)(35-58.2) = (2)(-23.2) = -46.4.
\][/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[
(10-6)(23-58.2) = (4)(-35.2) = -140.8.
\][/tex]
Summing these up:
[tex]\[
\sum (x-\bar{x})(y-\bar{y}) = -167.2 - 49.6 + 0 - 46.4 - 140.8 = -404.0.
\][/tex]
Step 3. Compute the Denominator
The denominator is
[tex]\[
\sum (x - \bar{x})^2.
\][/tex]
Again, calculate each term:
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
(2-6)^2 = (-4)^2 = 16.
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
(4-6)^2 = (-2)^2 = 4.
\][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
(6-6)^2 = 0^2 = 0.
\][/tex]
- For [tex]\( x = 8 \)[/tex]:
[tex]\[
(8-6)^2 = 2^2 = 4.
\][/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[
(10-6)^2 = 4^2 = 16.
\][/tex]
Summing these values:
[tex]\[
\sum (x-\bar{x})^2 = 16 + 4 + 0 + 4 + 16 = 40.
\][/tex]
Step 4. Calculate the Slope
Now, compute the slope:
[tex]\[
b = \frac{-404.0}{40} = -10.1.
\][/tex]
Step 5. Calculate the Intercept
The intercept is given by:
[tex]\[
a = \bar{y} - b\bar{x} = 58.2 - (-10.1 \times 6) = 58.2 + 60.6 = 118.8.
\][/tex]
Step 6. Write the Regression Equation
Substitute [tex]\( b = -10.1 \)[/tex] and [tex]\( a = 118.8 \)[/tex] into the equation of the line:
[tex]\[
\hat{y} = -10.1 x + 118.8.
\][/tex]
Thus, the equation of the least squares regression line that most closely matches the data set is
[tex]\[
\hat{y} = -10.1 x + 118.8.
\][/tex]
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 100 \\
4 & 83 \\
6 & 50 \\
8 & 35 \\
10 & 23 \\
\hline
\end{array}
\][/tex]
The goal is to find the equation of the least squares regression line, which has the form
[tex]\[
\hat{y} = bx + a,
\][/tex]
where the slope [tex]\( b \)[/tex] and intercept [tex]\( a \)[/tex] are computed by
[tex]\[
b = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}
\quad \text{and} \quad
a = \bar{y} - b\bar{x}.
\][/tex]
Step 1. Compute the Means
First, calculate the mean of [tex]\( x \)[/tex] values:
[tex]\[
\bar{x} = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6.
\][/tex]
Next, calculate the mean of [tex]\( y \)[/tex] values:
[tex]\[
\bar{y} = \frac{100 + 83 + 50 + 35 + 23}{5} = \frac{291}{5} = 58.2.
\][/tex]
Step 2. Compute the Numerator
The numerator for the slope [tex]\( b \)[/tex] is
[tex]\[
\sum (x - \bar{x})(y - \bar{y}).
\][/tex]
Calculate each term:
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
(2-6)(100-58.2) = (-4)(41.8) = -167.2.
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
(4-6)(83-58.2) = (-2)(24.8) = -49.6.
\][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
(6-6)(50-58.2) = (0)(-8.2) = 0.
\][/tex]
- For [tex]\( x = 8 \)[/tex]:
[tex]\[
(8-6)(35-58.2) = (2)(-23.2) = -46.4.
\][/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[
(10-6)(23-58.2) = (4)(-35.2) = -140.8.
\][/tex]
Summing these up:
[tex]\[
\sum (x-\bar{x})(y-\bar{y}) = -167.2 - 49.6 + 0 - 46.4 - 140.8 = -404.0.
\][/tex]
Step 3. Compute the Denominator
The denominator is
[tex]\[
\sum (x - \bar{x})^2.
\][/tex]
Again, calculate each term:
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
(2-6)^2 = (-4)^2 = 16.
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
(4-6)^2 = (-2)^2 = 4.
\][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
(6-6)^2 = 0^2 = 0.
\][/tex]
- For [tex]\( x = 8 \)[/tex]:
[tex]\[
(8-6)^2 = 2^2 = 4.
\][/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[
(10-6)^2 = 4^2 = 16.
\][/tex]
Summing these values:
[tex]\[
\sum (x-\bar{x})^2 = 16 + 4 + 0 + 4 + 16 = 40.
\][/tex]
Step 4. Calculate the Slope
Now, compute the slope:
[tex]\[
b = \frac{-404.0}{40} = -10.1.
\][/tex]
Step 5. Calculate the Intercept
The intercept is given by:
[tex]\[
a = \bar{y} - b\bar{x} = 58.2 - (-10.1 \times 6) = 58.2 + 60.6 = 118.8.
\][/tex]
Step 6. Write the Regression Equation
Substitute [tex]\( b = -10.1 \)[/tex] and [tex]\( a = 118.8 \)[/tex] into the equation of the line:
[tex]\[
\hat{y} = -10.1 x + 118.8.
\][/tex]
Thus, the equation of the least squares regression line that most closely matches the data set is
[tex]\[
\hat{y} = -10.1 x + 118.8.
\][/tex]
