Question

Create a linear model for the data in the table.
         
x
3
6
9
10
11
13
17

y
7
10
15
16
18
21
26
Question content area bottom
Part 1
Write a linear model for the data in the table.
yequals
  
enter your response herexplus​(
  
enter your response here​)
​(Type an integer or decimal rounded to three decimal places as​ needed.)

Answer 1

To create a linear model from the given data, we can determine the slope (m) and y-intercept (b) of the line in the form \( y = mx + b \).

Given the data:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
3 & 7 \\
6 & 10 \\
9 & 15 \\
10 & 16 \\
11 & 18 \\
13 & 21 \\
17 & 26 \\
\hline
\end{array}
\]

### Step 1: Calculate the slope (m)

The slope formula is:

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

Where:
- \( N \) is the number of data points
- \( \sum xy \) is the sum of the product of each \( x \) and \( y \)
- \( \sum x \) is the sum of \( x \) values
- \( \sum y \) is the sum of \( y \) values
- \( \sum x^2 \) is the sum of the squares of \( x \) values

### Step 2: Sum the values

1. \( \sum x = 3 + 6 + 9 + 10 + 11 + 13 + 17 = 69 \)
2. \( \sum y = 7 + 10 + 15 + 16 + 18 + 21 + 26 = 113 \)
3. \( N = 7 \)
4. \( \sum xy = (3*7) + (6*10) + (9*15) + (10*16) + (11*18) + (13*21) + (17*26) = 21 + 60 + 135 + 160 + 198 + 273 + 442 = 1290 \)
5. \( \sum x^2 = 3^2 + 6^2 + 9^2 + 10^2 + 11^2 + 13^2 + 17^2 = 9 + 36 + 81 + 100 + 121 + 169 + 289 = 805 \)

### Step 3: Plug values into the slope formula

\[
m = \frac{7(1290) - (69)(113)}{7(805) - (69)^2}
\]

Calculating the numerator:

\[
= \frac{9030 - 7797}{5635 - 4761}
= \frac{1233}{874} ≈ 1.41
\]

Calculating the denominator:

\[
= 5635 - 4761 = 874
\]

So,

\[
m ≈ 1.41
\]

### Step 4: Calculate y-intercept (b)

Once we have the slope \( m \), we can use the formula \( b = \frac{\sum y - m \sum x}{N} \)

\[
b = \frac{113 - 1.41 \times 69}{7}
\]
Calculating \( m \sum x \):

\[
1.41 \times 69 ≈ 97.29
\]

Now,

\[
b = \frac{113 - 97.29}{7} ≈ \frac{15.71}{7} ≈ 2.24
\]

### Step 5: Write the final linear model

Thus, the linear model in the form \( y = mx + b \) is:

\[
y = 1.41x + 2.24
\]

### Final response:

So the complete linear model is:

\( y = 1.41x + 2.24 \)