Question
Create a linear model for the data in the table.
x
4
7
10
13
16
19
x
4
7
10
13
16
19
Answers
GPT-4o mini
To create a linear model from the given data, we would typically express the relationship between the variable \( x \) and some dependent variable \( y \). However, you only provided values for \( x \) without corresponding \( y \) values.
For the purpose of creating a linear model, let’s assume some hypothetical \( y \) values associated with the given \( x \) values. Here’s an example with arbitrary \( y \) values:
| \( x \) | \( y \) |
|---------|---------|
| 4 | 2 |
| 7 | 3 |
| 10 | 5 |
| 13 | 7 |
| 16 | 10 |
| 19 | 12 |
### Step 1: Identify Data Points
From the above hypothetical table, our data points are:
- (4, 2)
- (7, 3)
- (10, 5)
- (13, 7)
- (16, 10)
- (19, 12)
### Step 2: Find the Equation of the Line
To find the linear model \( y = mx + b \), we will use the method of least squares to compute the slope \( m \) and the y-intercept \( b \).
1. **Calculate the necessary sums**:
\[
n = 6 \text{ (number of points)}
\]
\(
\sum x = 4 + 7 + 10 + 13 + 16 + 19 = 69
\)
\(
\sum y = 2 + 3 + 5 + 7 + 10 + 12 = 39
\)
\(
\sum xy = (4*2) + (7*3) + (10*5) + (13*7) + (16*10) + (19*12) = 8 + 21 + 50 + 91 + 160 + 228 = 558
\)
\(
\sum x^2 = 4^2 + 7^2 + 10^2 + 13^2 + 16^2 + 19^2 = 16 + 49 + 100 + 169 + 256 + 361 = 951
\)
2. **Calculate the slope \( m \) and intercept \( b \)**:
The slope \( m \) is given by the formula:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Substituting the values:
\[
m = \frac{6(558) - (69)(39)}{6(951) - (69)^2}
\]
\[
m = \frac{3348 - 2691}{5706 - 4761} = \frac{657}{945} \approx 0.695
\]
The intercept \( b \) is given by:
\[
b = \frac{\sum y - m \sum x}{n}
\]
Substituting the values:
\[
b = \frac{39 - 0.695(69)}{6} = \frac{39 - 48.045}{6} \approx \frac{-9.045}{6} \approx -1.5075
\]
### Step 3: Write the Final Linear Model
Using the values calculated for \( m \) and \( b \):
\[
y \approx 0.695x - 1.5075
\]
### Conclusion
So, the linear model for the given data (with hypothetical \( y \) values) is:
\[
y \approx 0.695x - 1.5075
\]
If you have actual \( y \) values to use, replace the hypothetical values and repeat the calculations for the slope and intercept to get your linear model.
For the purpose of creating a linear model, let’s assume some hypothetical \( y \) values associated with the given \( x \) values. Here’s an example with arbitrary \( y \) values:
| \( x \) | \( y \) |
|---------|---------|
| 4 | 2 |
| 7 | 3 |
| 10 | 5 |
| 13 | 7 |
| 16 | 10 |
| 19 | 12 |
### Step 1: Identify Data Points
From the above hypothetical table, our data points are:
- (4, 2)
- (7, 3)
- (10, 5)
- (13, 7)
- (16, 10)
- (19, 12)
### Step 2: Find the Equation of the Line
To find the linear model \( y = mx + b \), we will use the method of least squares to compute the slope \( m \) and the y-intercept \( b \).
1. **Calculate the necessary sums**:
\[
n = 6 \text{ (number of points)}
\]
\(
\sum x = 4 + 7 + 10 + 13 + 16 + 19 = 69
\)
\(
\sum y = 2 + 3 + 5 + 7 + 10 + 12 = 39
\)
\(
\sum xy = (4*2) + (7*3) + (10*5) + (13*7) + (16*10) + (19*12) = 8 + 21 + 50 + 91 + 160 + 228 = 558
\)
\(
\sum x^2 = 4^2 + 7^2 + 10^2 + 13^2 + 16^2 + 19^2 = 16 + 49 + 100 + 169 + 256 + 361 = 951
\)
2. **Calculate the slope \( m \) and intercept \( b \)**:
The slope \( m \) is given by the formula:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Substituting the values:
\[
m = \frac{6(558) - (69)(39)}{6(951) - (69)^2}
\]
\[
m = \frac{3348 - 2691}{5706 - 4761} = \frac{657}{945} \approx 0.695
\]
The intercept \( b \) is given by:
\[
b = \frac{\sum y - m \sum x}{n}
\]
Substituting the values:
\[
b = \frac{39 - 0.695(69)}{6} = \frac{39 - 48.045}{6} \approx \frac{-9.045}{6} \approx -1.5075
\]
### Step 3: Write the Final Linear Model
Using the values calculated for \( m \) and \( b \):
\[
y \approx 0.695x - 1.5075
\]
### Conclusion
So, the linear model for the given data (with hypothetical \( y \) values) is:
\[
y \approx 0.695x - 1.5075
\]
If you have actual \( y \) values to use, replace the hypothetical values and repeat the calculations for the slope and intercept to get your linear model.