Question

Create a linear model for the data in the table.
         
x
3
7
10
11
12
14
18

y
5
10
15
17
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 \( y = mx + b \) from the provided data points, we first need to calculate the slope \( m \) and the y-intercept \( b \).

1. **Data Points**: You provided the following pairs:
- (3, 5)
- (7, 10)
- (10, 15)
- (11, 17)
- (12, 18)
- (14, 21)
- (18, 26)

2. **Calculating 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 each x and y pair, \( \sum x \) is the sum of the x values, \( \sum y \) is the sum of the y values, and \( \sum x^2 \) is the sum of the squares of the x values.

Let's calculate these values:
- \( n = 7 \)
- \( \sum x = 3 + 7 + 10 + 11 + 12 + 14 + 18 = 85 \)
- \( \sum y = 5 + 10 + 15 + 17 + 18 + 21 + 26 = 112 \)
- \( \sum xy = (3*5) + (7*10) + (10*15) + (11*17) + (12*18) + (14*21) + (18*26) \)
\[
= 15 + 70 + 150 + 187 + 216 + 294 + 468 = 1400
\]
- \( \sum x^2 = 3^2 + 7^2 + 10^2 + 11^2 + 12^2 + 14^2 + 18^2 \)
\[
= 9 + 49 + 100 + 121 + 144 + 196 + 324 = 943
\]

Now plugging these values into the slope formula:
\[
m = \frac{7(1400) - (85)(112)}{7(943) - (85)^2}
\]
\[
= \frac{9800 - 9520}{6601 - 7225} = \frac{280}{-624} = -0.449
\]

3. **Calculating the y-Intercept \( b \)**:
The formula to find \( b \) is:
\[
b = \frac{\sum y - m \sum x}{n}
\]
Plugging the values into this formula:
\[
b = \frac{112 - (-0.449)(85)}{7}
\]
\[
= \frac{112 + 38.165}{7} = \frac{150.165}{7} = 21.38
\]

4. **Final Model**:
The linear model can be summarized as:
\[
y = -0.449x + 21.38
\]

### Final Answer:
Thus, the linear model in the required format is:
\[
y = -0.449x + 21.38
\]

Feel free to make adjustments or round the numbers to your required precision!