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

y = 
 x + 


x	y
1	22.18
2	22.66
3	26.54
4	28.42
5	28.2
6	29.68
7	31.26
8	34.74
9	35.52
10	37.4
11	40.68
12	44.16

To calculate the **linear regression line** (also called the **line of best fit**), we will use the formula:

\[
y = mx + b
\]

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

### Step 1: Calculate the slope (\(m\))
The formula for the slope \(m\) is:

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

where:
- \(n\) is the number of data points,
- \(\sum xy\) is the sum of the products of \(x\) and \(y\),
- \(\sum x\) is the sum of the \(x\)-values,
- \(\sum y\) is the sum of the \(y\)-values,
- \(\sum x^2\) is the sum of the squares of the \(x\)-values.

### Step 2: Compute the required sums

| \(x\) | \(y\)  | \(x \cdot y\) | \(x^2\) |
|-------|--------|---------------|---------|
| 1     | 22.18  | 22.18         | 1       |
| 2     | 22.66  | 45.32         | 4       |
| 3     | 26.54  | 79.62         | 9       |
| 4     | 28.42  | 113.68        | 16      |
| 5     | 28.20  | 141.00        | 25      |
| 6     | 29.68  | 178.08        | 36      |
| 7     | 31.26  | 218.82        | 49      |
| 8     | 34.74  | 277.92        | 64      |
| 9     | 35.52  | 319.68        | 81      |
| 10    | 37.40  | 374.00        | 100     |
| 11    | 40.68  | 447.48        | 121     |
| 12    | 44.16  | 529.92        | 144     |

\[
\sum x = 78, \quad \sum y = 381.44, \quad \sum xy = 2747.70, \quad \sum x^2 = 650, \quad n = 12
\]

### Step 3: Calculate the slope (\(m\))

\[
m = \frac{12(2747.70) - (78)(381.44)}{12(650) - (78)^2}
\]

First, simplify the numerator and denominator:

\[
12(2747.70) = 32972.4, \quad (78)(381.44) = 29752.32
\]
\[
12(650) = 7800, \quad (78)^2 = 6084
\]

Now:

\[
m = \frac{32972.4 - 29752.32}{7800 - 6084} = \frac{3220.08}{1716} \approx 1.88
\]

### Step 4: Calculate the y-intercept (\(b\))

The formula for the y-intercept is:

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

Substitute the values:

\[
b = \frac{381.44 - 1.88(78)}{12}
\]

\[
1.88(78) = 146.64
\]

\[
b = \frac{381.44 - 146.64}{12} = \frac{234.80}{12} \approx 19.57
\]

### Step 5: Write the regression equation

Now that we have the slope and intercept:

\[
y = 1.88x + 19.57
\]

### Final Answer

The regression line is:

\[
y = 1.88x + 19.57
\]

---

Do you want a step-by-step graph or any further breakdown of the calculations? 

Here are five related questions to explore next:
1. How do we interpret the slope in a regression line?
2. How can we measure the goodness-of-fit for this regression line?
3. What is the predicted value of \(y\) when \(x = 15\)?
4. How do outliers affect the regression line?
5. How would adding more data points change the slope and intercept?

**Tip:** The closer the points are to the regression line, the better the line represents the relationship between \(x\) and \(y\).

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

x	y
1	22.18
2	22.66
3	26.54
4	28.42
5	28.2
6	29.68
7	31.26
8	34.74
9	35.52
10	37.4
11	40.68
12	44.16

Learn how to calculate the linear regression line based on given data points. This tutorial covers the least squares method and involves calculating the slope and intercept using a step-by-step approach. Suitable for students in Grades 10-12.

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