y

To find the linear function that best fits the given data points using linear regression, we need to determine the slope $m$ and the y-intercept $b$ in the equation:

$y = mx + b$

The data points given are:

$x = [1, 2, 3, 4, 5, 6]$ $y = [703, 826, 1094, 1338, 1602, 2189]$

We can use the formulas for slope $m$ and intercept $b$:

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

Where:

• $n$ is the number of data points (in this case, 6),
• $\sum x$ is the sum of all $x$ values,
• $\sum y$ is the sum of all $y$ values,
• $\sum xy$ is the sum of the product of corresponding $x$ and $y$ values,
• $\sum x^2$ is the sum of the squares of the $x$ values.

Let's calculate these values and determine the equation of the best-fit line. I'll work through the computations next.The equation for the linear function that best fits the given data is:

$y = 285.77x + 291.80$

Let me know if you'd like a more detailed explanation, or if you have any further questions!

Here are 5 related questions you might consider:

1. How is the slope of a line interpreted in a linear regression context?
2. What role does the y-intercept play in the linear regression equation?
3. Can you calculate the predicted $y$ values for $x = 7$ using the equation?
4. How would the equation change if there were an outlier in the data set?
5. How is the goodness-of-fit of this linear regression model typically measured?

Tip: In linear regression, the slope indicates how much $y$ changes for each unit increase in $x$.