Question
Find the slope of the equation of the least-squares regression line for the dataset in the table
x, y
1, 15
2, 16
3, 16
5, 18
8, 18
9, 16
12, 27
15, 29
18, 20
20, 33
(1 point)
a. 6.5
b. 13.3
c. 16.3
d. 10.2
x, y
1, 15
2, 16
3, 16
5, 18
8, 18
9, 16
12, 27
15, 29
18, 20
20, 33
(1 point)
a. 6.5
b. 13.3
c. 16.3
d. 10.2
Answers
Answered by
GPT-4o mini
To find the slope of the least-squares regression line, we can use the formula for the slope \( m \) given by:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Where:
- \( n \) is the number of data points
- \( \sum xy \) is the sum of the products of each \( x \) and \( y \)
- \( \sum x \) is the sum of all \( x \) values
- \( \sum y \) is the sum of all \( y \) values
- \( \sum x^2 \) is the sum of the squares of \( x \) values
### Step 1: Organize the data
- \( x: 1, 2, 3, 5, 8, 9, 12, 15, 18, 20 \)
- \( y: 15, 16, 16, 18, 18, 16, 27, 29, 20, 33 \)
### Step 2: Calculate the sums
1. **Count \( n \)**:
\( n = 10 \)
2. **Calculate \( \sum x \)**:
\[
\sum x = 1 + 2 + 3 + 5 + 8 + 9 + 12 + 15 + 18 + 20 = 93
\]
3. **Calculate \( \sum y \)**:
\[
\sum y = 15 + 16 + 16 + 18 + 18 + 16 + 27 + 29 + 20 + 33 = 212
\]
4. **Calculate \( \sum xy \)**:
\[
\sum xy = (1 \cdot 15) + (2 \cdot 16) + (3 \cdot 16) + (5 \cdot 18) + (8 \cdot 18) + (9 \cdot 16) + (12 \cdot 27) + (15 \cdot 29) + (18 \cdot 20) + (20 \cdot 33)
\]
\[
= 15 + 32 + 48 + 90 + 144 + 144 + 324 + 435 + 360 + 660 = 1790
\]
5. **Calculate \( \sum x^2 \)**:
\[
\sum x^2 = 1^2 + 2^2 + 3^2 + 5^2 + 8^2 + 9^2 + 12^2 + 15^2 + 18^2 + 20^2
\]
\[
= 1 + 4 + 9 + 25 + 64 + 81 + 144 + 225 + 324 + 400 = 853
\]
### Step 3: Plug values into the slope formula
Now we can substitute these sums into the slope formula:
\[
m = \frac{10(1790) - (93)(212)}{10(853) - (93^2)}
\]
Calculating the values:
1. **Calculate the numerator**:
\[
10(1790) = 17900
\]
\[
(93)(212) = 19716
\]
\[
\text{Numerator} = 17900 - 19716 = -1816
\]
2. **Calculate the denominator**:
\[
10(853) = 8530
\]
\[
(93^2) = 8649
\]
\[
\text{Denominator} = 8530 - 8649 = -119
\]
### Final Calculation
Now substituting into the formula for \( m \):
\[
m = \frac{-1816}{-119} = 15.249
\]
Based on the values calculated, the slope of the least-squares regression line is closest to:
**Answer choice**:
c. **16.3** (Most appropriate among the available choices)
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Where:
- \( n \) is the number of data points
- \( \sum xy \) is the sum of the products of each \( x \) and \( y \)
- \( \sum x \) is the sum of all \( x \) values
- \( \sum y \) is the sum of all \( y \) values
- \( \sum x^2 \) is the sum of the squares of \( x \) values
### Step 1: Organize the data
- \( x: 1, 2, 3, 5, 8, 9, 12, 15, 18, 20 \)
- \( y: 15, 16, 16, 18, 18, 16, 27, 29, 20, 33 \)
### Step 2: Calculate the sums
1. **Count \( n \)**:
\( n = 10 \)
2. **Calculate \( \sum x \)**:
\[
\sum x = 1 + 2 + 3 + 5 + 8 + 9 + 12 + 15 + 18 + 20 = 93
\]
3. **Calculate \( \sum y \)**:
\[
\sum y = 15 + 16 + 16 + 18 + 18 + 16 + 27 + 29 + 20 + 33 = 212
\]
4. **Calculate \( \sum xy \)**:
\[
\sum xy = (1 \cdot 15) + (2 \cdot 16) + (3 \cdot 16) + (5 \cdot 18) + (8 \cdot 18) + (9 \cdot 16) + (12 \cdot 27) + (15 \cdot 29) + (18 \cdot 20) + (20 \cdot 33)
\]
\[
= 15 + 32 + 48 + 90 + 144 + 144 + 324 + 435 + 360 + 660 = 1790
\]
5. **Calculate \( \sum x^2 \)**:
\[
\sum x^2 = 1^2 + 2^2 + 3^2 + 5^2 + 8^2 + 9^2 + 12^2 + 15^2 + 18^2 + 20^2
\]
\[
= 1 + 4 + 9 + 25 + 64 + 81 + 144 + 225 + 324 + 400 = 853
\]
### Step 3: Plug values into the slope formula
Now we can substitute these sums into the slope formula:
\[
m = \frac{10(1790) - (93)(212)}{10(853) - (93^2)}
\]
Calculating the values:
1. **Calculate the numerator**:
\[
10(1790) = 17900
\]
\[
(93)(212) = 19716
\]
\[
\text{Numerator} = 17900 - 19716 = -1816
\]
2. **Calculate the denominator**:
\[
10(853) = 8530
\]
\[
(93^2) = 8649
\]
\[
\text{Denominator} = 8530 - 8649 = -119
\]
### Final Calculation
Now substituting into the formula for \( m \):
\[
m = \frac{-1816}{-119} = 15.249
\]
Based on the values calculated, the slope of the least-squares regression line is closest to:
**Answer choice**:
c. **16.3** (Most appropriate among the available choices)