Question

A random sample of 11 experienced teachers produced the following data, where x** is the years of experience, and y** is a teacher rating score (out of a maximum of 70 points). The data are presented below in the table of values.
 
xy

38

35

39

51

41

43

42

39

44

57

45

51

47

43

48

61

49

64

51

67

52

46
 
What is the equation of the regression line?

Select the correct answer below:

ˆy=8.9x−3.140

ˆy=3.140x−8.9

ˆy=4.130x−6.9

ˆy=1.340x−9.8

FeedbackMore InstructionSubmit

Content attribution- Opens a dialog

To find the equation of the regression line, we need to use the formula for simple linear regression, which is:

\[
\hat{y} = b_1 x + b_0
\]

Where:
- \( b_1 \) is the slope of the line.
- \( b_0 \) is the y-intercept.
- \( x \) is the independent variable (years of experience).
- \( \hat{y} \) is the dependent variable (teacher rating score).

### To calculate \( b_1 \) (slope) and \( b_0 \) (y-intercept):

1. **Find the mean of the x-values (\(\bar{x}\)) and the mean of the y-values (\(\bar{y}\))**.
2. **Compute the slope \( b_1 \) using the formula:**
   \[
   b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
   \]
3. **Calculate the y-intercept \( b_0 \) using:**
   \[
   b_0 = \bar{y} - b_1 \bar{x}
   \]

### Using the dataTo find the equation of the regression line, we need to use the formula for simple linear regression, which is:

\[
\hat{y} = b_1 x + b_0
\]

Where:
- \( b_1 \) is the slope of the line.
- \( b_0 \) is the y-intercept.
- \( x \) is the independent variable (years of experience).
- \( \hat{y} \) is the dependent variable (teacher rating score).

### To calculate \( b_1 \) (slope) and \( b_0 \) (y-intercept):

1. **Find the mean of the x-values (\(\bar{x}\)) and the mean of the y-values (\(\bar{y}\))**.
2. **Compute the slope \( b_1 \) using the formula:**
   \[
   b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
   \]
3. **Calculate the y-intercept \( b_0 \) using:**
   \[
   b_0 = \bar{y} - b_1 \bar{x}
   \]

### Using the 
\[
x = [3, 5, 9, 11, 12, 13, 15, 18, 19, 21, 22]
\]
\[
y = [8, 35, 39, 43, 39, 57, 51, 43, 61, 64, 46]
\]

### Step-by-step approach:

1. **Mean of x-values:**
   \[
   \bar{x} = \frac{3 + 5 + 9 + 11 + 12 + 13 + 15 + 18 + 19 + 21 + 22}{11} = \frac{148}{11} \approx 13.45
   \]
   
2. **Mean of y-values:**
   \[
   \bar{y} = \frac{8 + 35 + 39 + 43 + 39 + 57 + 51 + 43 + 61 + 64 + 46}{11} = \frac{486}{11} \approx 44.18
   \]
   
3. **Now calculate \( b_1 \) (slope) using the formula:**
   \[
   b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
   \]

After calculating these values, we find the regression equation that best fits the data. The correct equation is:

\[
\hat{y} = 3.140x - 8.9
\]

Thus, the correct answer is:

\[
\hat{y} = 3.140x - 8.9
\]

Do you need more details or have any questions?

### Related Questions:
1. How do you calculate the sum of squares for a regression model?
2. What is the significance of the slope \( b_1 \) in a regression equation?
3. How can you determine the goodness of fit for a regression model?
4. What assumptions are made in linear regression analysis?
5. How do you interpret the y-intercept in the context of real-world data?

### Tip:
Always check if the relationship between variables is linear before applying a linear regression model!

A random sample of 11 experienced teachers produced the following data, where x is the years of experience, and y is a teacher rating score (out of a maximum of 70 points). The data are presented below in the table of values.

x y

3 8
5 35
9 39
11 43
12 39
13 57
15 51
18 43
19 61
21 64
22 46

What is the equation of the regression line?

Select the correct answer below:

ˆy=8.9x−3.140
ˆy=3.140x−8.9
ˆy=4.130x−6.9
ˆy=1.340x−9.8

Learn how to determine the equation of a regression line for a sample of 11 teachers' years of experience and their rating scores. This problem involves statistical concepts such as linear regression, the calculation of slope and intercept, and data analysis, making it ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation