Use linear regression to find the equation for the line that best fits this data. Round both numbers to two decimal places. Write your final answer in a form of an equation y=mx+by=mx+b

**x**

1

2

3

4

5

6

**y**

104

130

147

177

206

213

To find the equation of the line that best fits this data, we can use the formula for linear regression:

\[
y = mx + b
\]

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

### Step 1: Calculate the slope (m)
The formula for the slope \( m \) in linear regression 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,
- \( \sum x \) is the sum of all \( x \)-values,
- \( \sum y \) is the sum of all \( y \)-values,
- \( \sum xy \) is the sum of the products of \( x \) and \( y \) for each pair,
- \( \sum x^2 \) is the sum of the squares of each \( x \)-value.

#### Calculate the values:
Given data points \((x, y)\):
- \( x \) values: \( 1, 2, 3, 4, 5, 6 \)
- \( y \) values: \( 104, 130, 147, 177, 206, 213 \)

1. **Sum of \( x \) values, \( \sum x \)**:
   \[
   \sum x = 1 + 2 + 3 + 4 + 5 + 6 = 21
   \]

2. **Sum of \( y \) values, \( \sum y \)**:
   \[
   \sum y = 104 + 130 + 147 + 177 + 206 + 213 = 977
   \]

3. **Sum of \( x^2 \) values, \( \sum x^2 \)**:
   \[
   \sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91
   \]

4. **Sum of \( xy \) values, \( \sum xy \)**:
   \[
   \sum xy = (1 \cdot 104) + (2 \cdot 130) + (3 \cdot 147) + (4 \cdot 177) + (5 \cdot 206) + (6 \cdot 213) = 4310
   \]

5. **Calculate \( m \)**:
   \[
   m = \frac{6 \cdot 4310 - 21 \cdot 977}{6 \cdot 91 - 21^2}
   \]
   \[
   m = \frac{25860 - 20517}{546 - 441} = \frac{4343}{105} \approx 41.36
   \]

### Step 2: Calculate the y-intercept (b)
The formula for the y-intercept \( b \) is:
\[
b = \frac{\sum y - m \sum x}{n}
\]

Using our values:
\[
b = \frac{977 - 41.36 \cdot 21}{6}
\]
\[
b = \frac{977 - 868.56}{6} \approx 18.41
\]

### Final Answer
The equation of the best-fit line is:
\[
y = 41.36x + 18.41
\]

Would you like more details or have any questions? Here are some related questions:

1. How is the slope in linear regression interpreted in practical terms?
2. How can we use this line equation to make predictions for unknown \( x \) values?
3. What are other ways to calculate the line of best fit, like using software tools?
4. How does linear regression handle data that doesn’t fit well on a line?
5. What is the importance of the correlation coefficient in regression analysis?

**Tip:** Linear regression minimizes the sum of squared errors, ensuring the line is as close as possible to all data points collectively.

Use linear regression to find the equation for the line that best fits this data. Round both numbers to two decimal places. Write your final answer in a form of an equation y=mx+b.

Discover how to use linear regression to derive the equation of the line that best fits the given data points. This includes calculations for slope and y-intercept based on x-values [1, 2, 3, 4, 5, 6] and y-values [104, 130, 147, 177, 206, 213].

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