What is the​ least-squares regression line with the point left parenthesis 9 comma 13 right parenthesis included in the data​ set?
Data Set   
x
y
3
6
4
5
5
7
6
4
7
6
8
9
8
8
10
8
11
9
11
7
12
10
13
12
13
10
14
11
​(This is a reading assessment question.  Be certain of your answer because you only get one attempt on this​ question.)
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Question content area right
Part 1
ModifyingAbove y with caretequals
  
enter your response herexplus
  
enter your response here
​(Type integers or decimals rounded to four decimal places as​ needed.)

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.

What is the least-squares regression line with the point (9, 13) included in the data set? Data Set: x y 3 6 4 5 5 7 6 4 7 6 8 9 8 8 10 8 11 9 11 7 12 10 13 12 13 10 14 11. (This is a reading assessment question. Be certain of your answer because you only get one attempt on this question.)

Discover how to calculate the least-squares regression line using the dataset with an additional point (9, 13). This solution involves statistics concepts like linear regression, utilizing key formulas for slope and y-intercept calculation. Ideal for students in Grades 10-12.

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