Answered

Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

Provide an appropriate response.

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]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.