To find the slope ($m$) of the linear regression line for the given data, we’ll use the formula for the slope of the least-squares regression line:

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

where:

• $n$ is the number of data points,
• $\sum xy$ is the sum of the product of each $x$ and $y$ value,
• $\sum x$ is the sum of all $x$-values,
• $\sum y$ is the sum of all $y$-values,
• $\sum x^2$ is the sum of each $x$-value squared.

The given data points are: $(x, y) = (2, 21), (3, 15), (4, 10), (5, 11), (6, 15), (7, 14)$

Let's calculate the necessary sums:

1. Sum of $x$ values ($\sum x$): $2 + 3 + 4 + 5 + 6 + 7$
2. Sum of $y$ values ($\sum y$): $21 + 15 + 10 + 11 + 15 + 14$
3. Sum of $x \cdot y$ values ($\sum xy$): Calculating each product $2 \cdot 21, 3 \cdot 15, 4 \cdot 10, \ldots$
4. Sum of $x^2$ values ($\sum x^2$): Squaring each $x$ value.

Let me compute this.The slope of the linear regression line for the given data is approximately $-0.97$.

Would you like a detailed breakdown of each step involved in this calculation, or do you have any other questions?

Here are some related questions you might explore:

1. How is the y-intercept calculated for this linear regression line?
2. What does the negative slope indicate about the relationship between $x$ and $y$ in this data?
3. How would the slope change if we added a new data point?
4. How can we use this line to make predictions for other $x$-values?
5. What is the formula for calculating the correlation coefficient?

Tip: The slope tells us the rate of change of $y$ with respect to $x$; a negative slope indicates an inverse relationship.