Question

Create a linear model for the data in the table.
x
5
7
9
11
13
15

y
6
16
20
29
38
45

Answer 1

To create a linear model for the given data, we need to determine the equation of the line in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

### Step 1: Organize the data
Let's organize the data into (x, y) pairs:

\[
\begin{align*}
(5, 6) \\
(7, 16) \\
(9, 20) \\
(11, 29) \\
(13, 38) \\
(15, 45) \\
\end{align*}
\]

### Step 2: Calculate the slope (m)
The formula for the slope \( m \) is given by:

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

where \( n \) is the number of points.

Let's calculate the necessary sums:

\[
\begin{align*}
\sum x &= 5 + 7 + 9 + 11 + 13 + 15 = 60 \\
\sum y &= 6 + 16 + 20 + 29 + 38 + 45 = 154 \\
\sum xy &= (5*6) + (7*16) + (9*20) + (11*29) + (13*38) + (15*45) \\
&= 30 + 112 + 180 + 319 + 494 + 675 = 1810 \\
\sum x^2 &= 5^2 + 7^2 + 9^2 + 11^2 + 13^2 + 15^2 \\
&= 25 + 49 + 81 + 121 + 169 + 225 = 670 \\
\end{align*}
\]

Now we can plug these sums into the slope formula. The number of data points \( n = 6 \).

Calculating \( m \):

\[
m = \frac{6(1810) - (60)(154)}{6(670) - (60)^2}
\]

Calculating the numerator:

\[
6(1810) = 10860 \\
(60)(154) = 9240 \\
\text{Numerator} = 10860 - 9240 = 1620
\]

Calculating the denominator:

\[
6(670) = 4020 \\
(60)^2 = 3600 \\
\text{Denominator} = 4020 - 3600 = 420
\]

Thus,

\[
m = \frac{1620}{420} = \frac{27}{7} \approx 3.857
\]

### Step 3: Calculate the y-intercept (b)
The intercept \( b \) is calculated using the formula:

\[
b = \frac{\sum y - m \sum x}{n}
\]

Substituting the known values:

\[
b = \frac{154 - \left(\frac{27}{7} \times 60\right)}{6}
\]

Calculating \( m \sum x \):

\[
\frac{27}{7} \times 60 = \frac{1620}{7}
\]

Now substituting into the equation for \( b \):

\[
b = \frac{154 - \frac{1620}{7}}{6} = \frac{\frac{1078 - 1620}{7}}{6} = \frac{\frac{-542}{7}}{6} = \frac{-542}{42}
\]

\( b \approx -12.9 \)

### Step 4: Form the linear model
Combining the values of \( m \) and \( b \):

\[
y \approx \frac{27}{7}x - \frac{271}{21}
\]

Thus, the linear model that fits the given data is:

\[
y = \frac{27}{7}x - \frac{271}{21}
\]

You can use this model to predict other \( y \) values based on \( x \) inputs within a reasonable range of the given data.