Question

Find the slope of the equation of the least-squares regression line for the dataset in the table.
x y
1 15
3 17
6 18
7 18
9 20
15 24
16 23
22 26
23 27
28 30
33 32
(1 point)
a. 0.52
b. 1.91
c. 15.14
d. -28.91

Answer 1

To find the slope of the least-squares regression line for the given dataset, we can use the formula for the slope \( b \) of the regression line:

\[
b = \frac{n(\sum{xy}) - (\sum{x})(\sum{y})}{n(\sum{x^2}) - (\sum{x})^2}
\]

Where:
- \( n \) = number of data points
- \( \sum{xy} \) = sum of the product of each pair of \( x \) and \( y \)
- \( \sum{x} \) = sum of the \( x \) values
- \( \sum{y} \) = sum of the \( y \) values
- \( \sum{x^2} \) = sum of the squares of the \( x \) values

### Step 1: Organize the Data
We first extract the data from the table.

\[
\begin{align*}
(x, y) \\
(1, 15) \\
(3, 17) \\
(6, 18) \\
(7, 18) \\
(9, 20) \\
(15, 24) \\
(16, 23) \\
(22, 26) \\
(23, 27) \\
(28, 30) \\
(33, 32)
\end{align*}
\]

### Step 2: Calculate the necessary summations

1. **Calculate \( n \) (the number of points)**:
- \( n = 11 \)

2. **Calculate \( \sum{x} \)**:
- \( \sum{x} = 1 + 3 + 6 + 7 + 9 + 15 + 16 + 22 + 23 + 28 + 33 = 163 \)

3. **Calculate \( \sum{y} \)**:
- \( \sum{y} = 15 + 17 + 18 + 18 + 20 + 24 + 23 + 26 + 27 + 30 + 32 = 239 \)

4. **Calculate \( \sum{xy} \)**:
- \( \sum{xy} = 1 \cdot 15 + 3 \cdot 17 + 6 \cdot 18 + 7 \cdot 18 + 9 \cdot 20 + 15 \cdot 24 + 16 \cdot 23 + 22 \cdot 26 + 23 \cdot 27 + 28 \cdot 30 + 33 \cdot 32 \)
- \( = 15 + 51 + 108 + 126 + 180 + 360 + 368 + 572 + 621 + 840 + 1056 = 3227 \)

5. **Calculate \( \sum{x^2} \)**:
- \( \sum{x^2} = 1^2 + 3^2 + 6^2 + 7^2 + 9^2 + 15^2 + 16^2 + 22^2 + 23^2 + 28^2 + 33^2 \)
- \( = 1 + 9 + 36 + 49 + 81 + 225 + 256 + 484 + 529 + 784 + 1089 = 2259 \)

### Step 3: Substitute into the slope formula

Now we can substitute our values into the slope formula:

\[
b = \frac{11(3227) - (163)(239)}{11(2259) - (163)^2}
\]

Calculate the terms:

**Numerator:**
\[
11(3227) = 35497
\]
\[
(163)(239) = 38957
\]
\[
\text{Numerator} = 35497 - 38957 = -3440
\]

**Denominator:**
\[
11(2259) = 24849
\]
\[
(163)^2 = 26569
\]
\[
\text{Denominator} = 24849 - 26569 = -1720
\]

Thus, the slope \( b \) becomes:

\[
b = \frac{-3440}{-1720} \approx 2
\]

### Conclusion

Based on the above calculations, the slope of the least-squares regression line is approximately \( 2 \). However, that means none of the provided options seem to match closely. Would you like me to recalculate or check any assumptions?