Based on the data shown below, calculate the regression line (each value to at least two decimal places)

y =  x + 



xy

4

27.79

5

27.38

6

25.37

7

23.06

8

21.45

9

21.14

10

22.73

11

19.02

12

20.11

13

16.5

To calculate the regression line, we need to find the equation of the form:

\[
y = mx + b
\]

where \( m \) is the slope of the line and \( b \) is the y-intercept. 

The slope \( m \) is calculated using the formula:

\[
m = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}
\]

The y-intercept \( b \) is calculated using the formula:

\[
b = \frac{\sum{y} - m\sum{x}}{n}
\]

Where:
- \( n \) is the number of data points,
- \( x \) and \( y \) are the individual values of the independent and dependent variables, respectively.

Let's compute these values step by step.

### Given dataTo calculate the regression line, we need to find the equation of the form:

\[
y = mx + b
\]

where \( m \) is the slope of the line and \( b \) is the y-intercept. 

The slope \( m \) is calculated using the formula:

\[
m = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}
\]

The y-intercept \( b \) is calculated using the formula:

\[
b = \frac{\sum{y} - m\sum{x}}{n}
\]

Where:
- \( n \) is the number of data points,
- \( x \) and \( y \) are the individual values of the independent and dependent variables, respectively.

Let's compute these values step by step.

### Given 

\[
(x, y) = \{(4, 27.79), (5, 27.38), (6, 25.37), (7, 23.06), (8, 21.45), (9, 21.14), (10, 22.73), (11, 19.02), (12, 20.11), (13, 16.5)\}
\]

I will calculate the sums and apply the formulas to find the regression line.The regression line equation is:

\[
y = -1.12x + 32.01
\]

This means that for each unit increase in \( x \), \( y \) decreases by approximately 1.12 units, and the y-intercept is 32.01.

Would you like more details on the calculations or further explanation?

Here are 5 related questions to deepen your understanding:
1. How does the slope of a regression line affect predictions?
2. What is the significance of the y-intercept in this context?
3. Can you calculate the predicted value of \( y \) when \( x = 15 \)?
4. How would the regression change if we removed an outlier?
5. How do we assess the goodness-of-fit for this regression line?

**Tip:** Always plot your data and regression line to visually confirm the relationship!

Based on the data shown below, calculate the regression line (each value to at least two decimal places)

xy
4 27.79
5 27.38
6 25.37
7 23.06
8 21.45
9 21.14
10 22.73
11 19.02
12 20.11
13 16.5

Learn how to calculate the regression line using the least squares method. This step-by-step guide covers the data points (4, 27.79), (5, 27.38), ..., (13, 16.5) and provides a solution for y = -1.12x + 32.01, explaining the slope and y-intercept in detail.

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