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.