To find the least-squares regression line, we need to fit a line of the form:

$y = mx + b$

where $m$ is the slope of the line, and $b$ is the y-intercept. The least-squares method minimizes the sum of the squared residuals between the observed values and the values predicted by the line.

Given Data Points:

$(5, 69.0), (20, 67.5), (25, 66.8), (40, 65.7), (60, 64.2), (84, 63.2), (105, 61.0)$

We will follow these steps:

1. Calculate the means of the x-values ($\bar{x}$) and y-values ($\bar{y}$).

2. Calculate the slope $m$ using the formula:

   $m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}$

3. Calculate the y-intercept $b$ using the formula:

   $b = \bar{y} - m \bar{x}$

Let's calculate these values step by step.

Step 1: Calculate the means

$\bar{x} = \frac{5 + 20 + 25 + 40 + 60 + 84 + 105}{7}$ $\bar{y} = \frac{69.0 + 67.5 + 66.8 + 65.7 + 64.2 + 63.2 + 61.0}{7}$

Step 2: Slope $m$

We calculate each term in the numerator and denominator:

$m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}$

Step 3: Y-intercept $b$

After calculating the slope, we can substitute it into the equation to find the intercept.

Let me compute this for you.The least-squares regression line has the following equation:

$y = -0.0749x + 68.968$

Interpretation:

• Slope $m$: -0.0749 indicates that for each increase of 1 unit in $x$, $y$ decreases by approximately 0.0749 units.
• Intercept $b$: 68.968 represents the predicted $y$-value when $x = 0$.

Would you like to explore the details or have any questions?

Here are five follow-up questions:

1. How can we use the regression line to make predictions?
2. What does the slope tell us about the relationship between $x$ and $y$?
3. How would you test the accuracy of this regression model?
4. What is the meaning of residuals in the context of regression?
5. Can this regression line be applied to values of $x$ far outside the given range?

Tip: Always check for outliers, as they can significantly impact the regression line.