To find the least-squares regression line, we first need to calculate the slope $m$ and the y-intercept $b$ of the line $\hat{y} = mx + b$. The formulas for the slope and intercept are:

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

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

where:

• $n$ is the number of data points
• $\sum x$ 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 each $x$ and $y$
• $\sum x^2$ is the sum of the squares of each $x$

Step 1: Calculate the necessary sums.

Given the data set (including the point $(9, 13)$): $(3, 6), (4, 5), (5, 7), (6, 4), (7, 6), (8, 9), (8, 8), (9, 13), (10, 8), (11, 9), (11, 7), (12, 10), (13, 12), (13, 10), (14, 11)$

1. Calculate $n$, $\sum x$, $\sum y$, $\sum xy$, and $\sum x^2$:
   • $n = 15$
   • $\sum x = 3 + 4 + 5 + 6 + 7 + 8 + 8 + 9 + 10 + 11 + 11 + 12 + 13 + 13 + 14 = 134$
   • $\sum y = 6 + 5 + 7 + 4 + 6 + 9 + 8 + 13 + 8 + 9 + 7 + 10 + 12 + 10 + 11 = 135$
   • $\sum xy = (3)(6) + (4)(5) + (5)(7) + (6)(4) + (7)(6) + (8)(9) + (8)(8) + (9)(13) + (10)(8) + (11)(9) + (11)(7) + (12)(10) + (13)(12) + (13)(10) + (14)(11) = 1304$
   • $\sum x^2 = 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 8^2 + 9^2 + 10^2 + 11^2 + 11^2 + 12^2 + 13^2 + 13^2 + 14^2 = 1270$

Step 2: Calculate the slope $m$ and intercept $b$.

Using the formulas:

$m = \frac{15(1304) - 134(135)}{15(1270) - 134^2} = \frac{19560 - 18090}{19050 - 17956} = \frac{1470}{1094} \approx 1.3449$

$b = \frac{135 - 1.3449(134)}{15} = \frac{135 - 180.2166}{15} = \frac{-45.2166}{15} \approx -3.0144$

Step 3: Write the equation of the least-squares regression line.

The equation is:

$\hat{y} = 1.3449x - 3.0144$

Rounding the slope and intercept to four decimal places, the least-squares regression line is approximately:

$\hat{y} = 1.3449x - 3.0144$

Would you like further details or have any questions about this calculation?

Related Questions:

1. How does including an outlier affect the least-squares regression line?
2. What is the meaning of the slope in a least-squares regression line?
3. How can you determine if a least-squares regression line is a good fit for the data?
4. What is the interpretation of the y-intercept in a regression line?
5. How do you calculate the coefficient of determination, $R^2$, for this data set?

Tip:

When rounding intermediate calculations, keep more decimal places than the final answer to avoid rounding errors in the final result.