Question

Find the linear function that is the best fit for the data.
x
1
3
5
7
9

y
5
8
11
14
17
Question content area bottom
Part 1
What is the linear function of the​ data?

Answer 1

To find the linear function that best fits the given data points, we first recognize that we are looking for a linear equation of the form:

\[ y = mx + b \]

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

### Given Data:
- \( x: \) 1, 3, 5, 7, 9
- \( y: \) 5, 8, 11, 14, 17

### Step 1: Calculate the slope (m)
The slope \( m \) of the line can be calculated using the formula:

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

where \( N \) is the number of data points.

**Calculating necessary sums:**

- \( N = 5 \) (number of data points)
- \( \sum x = 1 + 3 + 5 + 7 + 9 = 25 \)
- \( \sum y = 5 + 8 + 11 + 14 + 17 = 55 \)
- \( \sum xy = (1)(5) + (3)(8) + (5)(11) + (7)(14) + (9)(17) = 5 + 24 + 55 + 98 + 153 = 335 \)
- \( \sum x^2 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 + 81 = 165 \)

### Step 2: Plug the values into the slope formula

Now, substituting into the formula for \( m \):

\[
m = \frac{5(335) - (25)(55)}{5(165) - (25)^2}
\]

Calculating the components:

- \( 5(335) = 1675 \)
- \( (25)(55) = 1375 \)
- \( 5(165) = 825 \)
- \( (25)^2 = 625 \)

Now substituting:

\[
m = \frac{1675 - 1375}{825 - 625} = \frac{300}{200} = 1.5
\]

### Step 3: Calculate the y-intercept (b)

The y-intercept \( b \) can be found using the formula:

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

Substituting the known values:

\[
b = \frac{55 - 1.5(25)}{5}
\]

Calculating:

- \( 1.5(25) = 37.5 \)

So,

\[
b = \frac{55 - 37.5}{5} = \frac{17.5}{5} = 3.5
\]

### Conclusion

The best-fit linear function for the given data points is:

\[
y = 1.5x + 3.5
\]