Question

Create a linear model for the data in the table.
         
x
2
6
9
10
11
13
17

y
7
10
14
17
18
21
27
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Part 1
Write a linear model for the data in the table.
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Answer 1

To create a linear model from the given data, we need to find the equation of the line in the form:

\[ y = mx + b \]

where \( m \) is the slope and \( b \) is the y-intercept.

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

The formula for the slope \( m \) 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 \( x \) and \( y \), \( \sum x \) is the sum of the \( x \) values, and \( \sum y \) is the sum of the \( y \) values.

### Step 2: Calculate Necessary Sums

We first need to compute the necessary sums:

- \( x: 2, 6, 9, 10, 11, 13, 17 \)
- \( y: 7, 10, 14, 17, 18, 21, 27 \)

1. **Calculate \( \sum x \)**:

\[
\sum x = 2 + 6 + 9 + 10 + 11 + 13 + 17 = 68
\]

2. **Calculate \( \sum y \)**:

\[
\sum y = 7 + 10 + 14 + 17 + 18 + 21 + 27 = 114
\]

3. **Calculate \( \sum xy \)**:

\[
\sum xy = (2)(7) + (6)(10) + (9)(14) + (10)(17) + (11)(18) + (13)(21) + (17)(27)
\]

Calculating each term:
- \( 2 \times 7 = 14 \)
- \( 6 \times 10 = 60 \)
- \( 9 \times 14 = 126 \)
- \( 10 \times 17 = 170 \)
- \( 11 \times 18 = 198 \)
- \( 13 \times 21 = 273 \)
- \( 17 \times 27 = 459 \)

Adding these:

\[
\sum xy = 14 + 60 + 126 + 170 + 198 + 273 + 459 = 1300
\]

4. **Calculate \( \sum x^2 \)**:

\[
\sum x^2 = 2^2 + 6^2 + 9^2 + 10^2 + 11^2 + 13^2 + 17^2 = 4 + 36 + 81 + 100 + 121 + 169 + 289 = 800
\]

### Step 3: Number of Data Points (n)

The number of data points \( n = 7 \).

### Step 4: Plug into Slope Formula

Now we can calculate \( m \):

\[
m = \frac{7(1300) - (68)(114)}{7(800) - (68)^2}
\]

Calculating the terms:

- \( 7(1300) = 9100 \)
- \( (68)(114) = 7752 \)
- \( 7(800) = 5600 \)
- \( (68)^2 = 4624 \)

So we can substitute:

\[
m = \frac{9100 - 7752}{5600 - 4624} = \frac{1348}{976} \approx 1.38
\]

### Step 5: Calculate y-intercept (b)

To find \( b \), we use the formula:

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

Substituting the values we have:

\[
b = \frac{114 - (1.38)(68)}{7}
\]
Calculating \( (1.38)(68) \approx 93.84 \):

\[
b = \frac{114 - 93.84}{7} \approx \frac{20.16}{7} \approx 2.88
\]

### Final Model

So the linear model is:

\[
y \approx 1.38x + 2.88
\]

You can round these coefficients based on your requirements. If further simplification or rounding is not necessary, we can leave them as approximate values:

#### Final Answer:

\[
y = 1.38x + 2.88
\]