To calculate the slope of the least-squares regression line for the given data, we use the following formula for the slope $m$:

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

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 product of corresponding $x$ and $y$-values,
• $\sum x^2$ is the sum of the squares of the $x$-values.

Extracting data from the table:

From the table:

• $n = 16$ (number of houses),
• $\sum x = 144.6$ (sum of distances),
• $\sum y = 3775.5$ (sum of selling prices),
• $\sum xy = 45091.2$ (sum of products $xy$),
• $\sum x^2 = 1531.4$ (sum of squares of distances).

Step-by-Step Calculation:

1. Plug values into the slope formula:

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

Substitute the known values:

$m = \frac{16(45091.2) - (144.6)(3775.5)}{16(1531.4) - (144.6)^2}$

2. Simplify the terms:

• $16 \cdot 45091.2 = 721459.2$,
• $144.6 \cdot 3775.5 = 545882.3$,
• $16 \cdot 1531.4 = 24502.4$,
• $(144.6)^2 = 20907.16$.

Now substitute these values back into the equation:

$m = \frac{721459.2 - 545882.3}{24502.4 - 20907.16}$

3. Perform the arithmetic:

• Numerator: $721459.2 - 545882.3 = 175576.9$,
• Denominator: $24502.4 - 20907.16 = 3595.24$.

Thus:

$m = \frac{175576.9}{3595.24}$

$m \approx 48.84$

Final Answer:

The slope of the least-squares regression line is approximately 48.84.

Do you want me to find the equation of the regression line or explain further steps? Let me know!

Related Questions:

1. What is the full equation of the least-squares regression line?
2. How do you interpret the slope of this regression line in context?
3. What is the correlation coefficient $r$, and what does it tell us about the data?
4. Can you calculate the predicted selling price of a house 10 miles from the beach?
5. How do you calculate the coefficient of determination $r^2$ for this data?

Tip:

The slope of a regression line represents how much the dependent variable (selling price) changes for each unit increase in the independent variable (distance from the beach).