Sure! Let's create a linear model for the given data points.
To find the linear model [tex]\( y = mx + b \)[/tex], we need to determine the slope ([tex]\( m \)[/tex]) and the y-intercept ([tex]\( b \)[/tex]). Here is the step-by-step solution:
### Step 1: List the data points
We have the following data points:
- [tex]\( (7, 4) \)[/tex]
- [tex]\( (10, 16) \)[/tex]
- [tex]\( (13, 21) \)[/tex]
- [tex]\( (16, 29) \)[/tex]
### Step 2: Calculate the slope ([tex]\( m \)[/tex])
The formula for the slope [tex]\( m \)[/tex] is given by:
[tex]\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \][/tex]
Where:
- [tex]\( n \)[/tex] is the number of data points
- [tex]\( \sum xy \)[/tex] is the sum of the product of each x and y
- [tex]\( \sum x \)[/tex] is the sum of all x values
- [tex]\( \sum y \)[/tex] is the sum of all y values
- [tex]\( \sum x^2 \)[/tex] is the sum of the squares of all x values
First, we compute these values:
[tex]\[
\begin{aligned}
& x = [7, 10, 13, 16] \\
& y = [4, 16, 21, 29] \\
& n = 4 \\
& \sum x = 7 + 10 + 13 + 16 = 46 \\
& \sum y = 4 + 16 + 21 + 29 = 70 \\
& \sum xy = (7 \cdot 4) + (10 \cdot 16) + (13 \cdot 21) + (16 \cdot 29) = 28 + 160 + 273 + 464 = 925 \\
& \sum x^2 = (7^2) + (10^2) + (13^2) + (16^2) = 49 + 100 + 169 + 256 = 574
\end{aligned}
\][/tex]
Now, using the slope formula:
[tex]\[
m = \frac{4(925) - (46)(70)}{4(574) - (46)^2} = \frac{3700 - 3220}{2296 - 2116} = \frac{480}{180} = 2.667
\][/tex]
### Step 3: Calculate the y-intercept ([tex]\( b \)[/tex])
The formula for the y-intercept is:
[tex]\[ b = \frac{\sum y - m (\sum x)}{n} \][/tex]
Substituting the known values and the calculated slope:
[tex]\[
b = \frac{70 - 2.667 \times 46}{4} = \frac{70 - 122.682}{4} = \frac{-52.682}{4} = -13.171
\][/tex]
### Step 4: Write the linear model
The linear model [tex]\( y = mx + b \)[/tex] is:
[tex]\[ y = 2.667x - 13.171 \][/tex]
So, the linear model for the data is:
[tex]\[ y = 2.667x - 13.171 \][/tex]
Note that the values are rounded to three decimal places, as specified.
