Question

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

y
3
4
5
6
7
Question content area bottom
Part 1
What is the linear function of the​ data?
yequals
  
enter your response herexplus​(
  
enter your response here​)
​(Type integers or​ decimals.)

Answer 1

To find the linear function that best fits the data, we can use the method of least squares to find the slope (m) and y-intercept (b) of the line in the form \( y = mx + b \).

### Given Data Points:
- \( (1, 3) \)
- \( (3, 4) \)
- \( (5, 5) \)
- \( (7, 6) \)
- \( (9, 7) \)

### Step 1: Calculate the necessary sums

Let \( n \) be the number of data points, and calculate the following:

1. \( n = 5 \) (the number of points)
2. Sum of \( x \):
\[
\sum x = 1 + 3 + 5 + 7 + 9 = 25
\]
3. Sum of \( y \):
\[
\sum y = 3 + 4 + 5 + 6 + 7 = 25
\]
4. Sum of \( xy \):
\[
\sum xy = (1)(3) + (3)(4) + (5)(5) + (7)(6) + (9)(7) = 3 + 12 + 25 + 42 + 63 = 145
\]
5. Sum of \( x^2 \):
\[
\sum x^2 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 + 81 = 165
\]

### Step 2: Apply the formulas for slope (m) and intercept (b)

The formulas for the slope \( m \) and y-intercept \( b \) are given by:

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

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

### Step 3: Calculate the slope (m)

Plugging in the values:

\[
m = \frac{5(145) - (25)(25)}{5(165) - (25)^2}
\]
\[
m = \frac{725 - 625}{825 - 625}
\]
\[
m = \frac{100}{200} = 0.5
\]

### Step 4: Calculate the intercept (b)

Now that we have \( m \):

\[
b = \frac{25 - 0.5(25)}{5}
\]
\[
b = \frac{25 - 12.5}{5} = \frac{12.5}{5} = 2.5
\]

### Final Equation

Thus, the linear function that best fits the data is:

\[
y = 0.5x + 2.5
\]

Thus, your answer is:
\[
y = 0.5x + 2.5
\]