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To determine the equation of the regression line, we need to find the slope ([tex]\( b \)[/tex]) and y-intercept ([tex]\( a \)[/tex]) for the line equation [tex]\(\hat{y} = b x + a\)[/tex].
Here are the steps to calculate the required values:
1. Calculate the means of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\( \bar{x} \)[/tex] = mean of [tex]\( x \)[/tex]
[tex]\( \bar{y} \)[/tex] = mean of [tex]\( y \)[/tex]
2. Compute the slope [tex]\( b \)[/tex]:
[tex]\( b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)[/tex]
3. Compute the y-intercept [tex]\( a \)[/tex]:
[tex]\( a = \bar{y} - b\bar{x} \)[/tex]
Let's break this down step by step:
1. Calculate the means of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ \bar{x} = \frac{-5 + (-3) + 4 + 1 + (-1) + (-2) + 0 + 2 + 3 + (-4)}{10} = \frac{-5 - 3 + 4 + 1 - 1 - 2 + 0 + 2 + 3 - 4}{10} = \frac{-5}{10} = -0.5 \][/tex]
[tex]\[ \bar{y} = \frac{-10 + (-8) + 9 + 1 + (-2) + (-6) + (-1) + 3 + 6 + (-8)}{10} = \frac{-10 - 8 + 9 + 1 - 2 - 6 - 1 + 3 + 6 - 8}{10} = \frac{-16}{10} = -1.6 \][/tex]
2. Compute the slope [tex]\( b \)[/tex]:
We need to calculate the sums for the numerator and the denominator of the slope formula.
[tex]\[ \sum(x_i - \bar{x})(y_i - \bar{y}) = (-5 + 0.5)(-10 + 1.6) + (-3 + 0.5)(-8 + 1.6) + \ldots + (-4 + 0.5)(-8 + 1.6) \][/tex]
[tex]\[ = (-4.5 \times -8.4) + (-2.5 \times -6.4) + (4.5 \times 10.6) + (1.5 \times 2.6) + (-0.5 \times -0.4) + (-1.5 \times -4.4) + (0 \times -2.6) + (2.5 \times 4.6) + (3.5 \times 7.6) + (-3.5 \times -6.4) \][/tex]
[tex]\[ = 37.8 + 16 + 47.7 + 3.9 + 0.2 + 6.6 + 0 + 11.5 + 26.6 + 22.4 = 172.7 \][/tex]
[tex]\[ \sum(x_i - \bar{x})^2 = (-5 + 0.5)^2 + (-3 + 0.5)^2 + (4 + 0.5)^2 + (1 + 0.5)^2 + (-1 + 0.5)^2 + (-2 + 0.5)^2 + (0 + 0.5)^2 + (2 + 0.5)^2 + (3 + 0.5)^2 + (-4 + 0.5)^2 \][/tex]
[tex]\[ = 20.25 + 6.25 + 20.25 + 2.25 + 0.25 + 2.25 + 0.25 + 6.25 + 12.25 + 12.25 = 82.5 \][/tex]
Now, the slope [tex]\( b \)[/tex]:
[tex]\[ b = \frac{172.7}{82.5} \approx 2.097 \][/tex]
3. Compute the y-intercept [tex]\( a \)[/tex]:
[tex]\[ a = \bar{y} - b\bar{x} = -1.6 - (2.097 \times -0.5) = -1.6 + 1.0485 \approx -0.5515 \][/tex]
Approximating to 3 decimal places, the intercept [tex]\( a \approx -0.552 \)[/tex].
So, the equation of the regression line is:
[tex]\[ \hat{y} = 2.097x - 0.552 \][/tex]
Therefore, the correct answer is:
C. [tex]\(\hat{y} = 2.097 x - 0.552\)[/tex].
Here are the steps to calculate the required values:
1. Calculate the means of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\( \bar{x} \)[/tex] = mean of [tex]\( x \)[/tex]
[tex]\( \bar{y} \)[/tex] = mean of [tex]\( y \)[/tex]
2. Compute the slope [tex]\( b \)[/tex]:
[tex]\( b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)[/tex]
3. Compute the y-intercept [tex]\( a \)[/tex]:
[tex]\( a = \bar{y} - b\bar{x} \)[/tex]
Let's break this down step by step:
1. Calculate the means of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ \bar{x} = \frac{-5 + (-3) + 4 + 1 + (-1) + (-2) + 0 + 2 + 3 + (-4)}{10} = \frac{-5 - 3 + 4 + 1 - 1 - 2 + 0 + 2 + 3 - 4}{10} = \frac{-5}{10} = -0.5 \][/tex]
[tex]\[ \bar{y} = \frac{-10 + (-8) + 9 + 1 + (-2) + (-6) + (-1) + 3 + 6 + (-8)}{10} = \frac{-10 - 8 + 9 + 1 - 2 - 6 - 1 + 3 + 6 - 8}{10} = \frac{-16}{10} = -1.6 \][/tex]
2. Compute the slope [tex]\( b \)[/tex]:
We need to calculate the sums for the numerator and the denominator of the slope formula.
[tex]\[ \sum(x_i - \bar{x})(y_i - \bar{y}) = (-5 + 0.5)(-10 + 1.6) + (-3 + 0.5)(-8 + 1.6) + \ldots + (-4 + 0.5)(-8 + 1.6) \][/tex]
[tex]\[ = (-4.5 \times -8.4) + (-2.5 \times -6.4) + (4.5 \times 10.6) + (1.5 \times 2.6) + (-0.5 \times -0.4) + (-1.5 \times -4.4) + (0 \times -2.6) + (2.5 \times 4.6) + (3.5 \times 7.6) + (-3.5 \times -6.4) \][/tex]
[tex]\[ = 37.8 + 16 + 47.7 + 3.9 + 0.2 + 6.6 + 0 + 11.5 + 26.6 + 22.4 = 172.7 \][/tex]
[tex]\[ \sum(x_i - \bar{x})^2 = (-5 + 0.5)^2 + (-3 + 0.5)^2 + (4 + 0.5)^2 + (1 + 0.5)^2 + (-1 + 0.5)^2 + (-2 + 0.5)^2 + (0 + 0.5)^2 + (2 + 0.5)^2 + (3 + 0.5)^2 + (-4 + 0.5)^2 \][/tex]
[tex]\[ = 20.25 + 6.25 + 20.25 + 2.25 + 0.25 + 2.25 + 0.25 + 6.25 + 12.25 + 12.25 = 82.5 \][/tex]
Now, the slope [tex]\( b \)[/tex]:
[tex]\[ b = \frac{172.7}{82.5} \approx 2.097 \][/tex]
3. Compute the y-intercept [tex]\( a \)[/tex]:
[tex]\[ a = \bar{y} - b\bar{x} = -1.6 - (2.097 \times -0.5) = -1.6 + 1.0485 \approx -0.5515 \][/tex]
Approximating to 3 decimal places, the intercept [tex]\( a \approx -0.552 \)[/tex].
So, the equation of the regression line is:
[tex]\[ \hat{y} = 2.097x - 0.552 \][/tex]
Therefore, the correct answer is:
C. [tex]\(\hat{y} = 2.097 x - 0.552\)[/tex].
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