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The following summary of bivariate data shows the final exam scores of 10 randomly selected statistics students and the number of hours they studied for the exam. Find the equation of the regression line for the data.

[tex]\[
\left(\Sigma x=42, \Sigma y=773, \Sigma x^2=208, \Sigma y^2=60875, \Sigma xy=3406, n=10\right)
\][/tex]

Select one:
A. [tex]\(\hat{y}=-5.044x + 56.113\)[/tex]
B. [tex]\(\hat{y}=-56.113x - 5.044\)[/tex]
C. [tex]\(\hat{y}=5.044x + 56.113\)[/tex]
D. [tex]\(\hat{y}=56.113x - 5.044\)[/tex]

Sagot :

GinnyAnswer
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]
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