To find the equation of the regression line for the given data, we need to use the following formulas for the slope [tex]\(m\)[/tex] and the intercept [tex]\(b\)[/tex] of the regression line [tex]\(\hat{y} = mx + b\)[/tex].
1. Calculate the means of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
- Mean of [tex]\(x\)[/tex] ([tex]\(\bar{x}\)[/tex]): [tex]\(\bar{x} = \frac{\Sigma x}{n} = \frac{42}{10} = 4.2\)[/tex]
- Mean of [tex]\(y\)[/tex] ([tex]\(\bar{y}\)[/tex]): [tex]\(\bar{y} = \frac{\Sigma y}{n} = \frac{773}{10} = 77.3\)[/tex]
2. Calculate the slope ([tex]\(m\)[/tex]) of the regression line:
- Slope [tex]\(m\)[/tex] is given by:
[tex]\[
m = \frac{n \Sigma xy - (\Sigma x)(\Sigma y)}{n \Sigma x^2 - (\Sigma x)^2}
\][/tex]
- Substitute the given values:
[tex]\[
m = \frac{10 \cdot 3406 - 42 \cdot 773}{10 \cdot 208 - 42^2} = \frac{34060 - 32466}{2080 - 1764} = \frac{1594}{316} \approx 5.044
\][/tex]
3. Calculate the intercept ([tex]\(b\)[/tex]) of the regression line:
- Intercept [tex]\(b\)[/tex] is given by:
[tex]\[
b = \bar{y} - m \cdot \bar{x}
\][/tex]
- Substitute the calculated slope and means:
[tex]\[
b = 77.3 - 5.044 \cdot 4.2 \approx 77.3 - 21.185 \approx 56.115
\][/tex]
Therefore, the equation of the regression line is:
[tex]\[
\hat{y} = 5.044x + 56.113
\][/tex]
Looking at the given options, the correct answer is:
C. [tex]\(\hat{y} = 5.044x + 56.113\)[/tex]
