Question

y = 5, 9, 10, 10.5, 11.
x = 1, 2, 3, 4, 5.

у = 1.35x + 5.05
У = x+5
y =5x + 11
у = 5.05x + 1.35

determine the equation of the line of best fit for the data in this table. Round to the nearest hundredth, if needed.

Answer 1

To determine the equation of the line of best fit for the given data, we can use the method of least squares to find the best-fitting linear regression line. The given data points are:

- \( y \): 5, 9, 10, 10.5, 11
- \( x \): 1, 2, 3, 4, 5

### Step 1: Calculate the necessary sums
1. Calculate \( n \) (the number of data points):
\[
n = 5
\]

2. Calculate the sums needed:
\[
\sum x = 1 + 2 + 3 + 4 + 5 = 15
\]
\[
\sum y = 5 + 9 + 10 + 10.5 + 11 = 45.5
\]
\[
\sum xy = (1 \times 5) + (2 \times 9) + (3 \times 10) + (4 \times 10.5) + (5 \times 11) = 5 + 18 + 30 + 42 + 55 = 150
\]
\[
\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55
\]

### Step 2: Determine the slope (\( m \)) and y-intercept (\( b \))
The formulas for the slope \( m \) and y-intercept \( b \) of the line of best fit are given by:

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

Substituting the calculated sums:

1. Calculate the slope \( m \):
\[
m = \frac{5(150) - (15)(45.5)}{5(55) - (15)^2}
\]

\[
= \frac{750 - 682.5}{275 - 225} = \frac{67.5}{50} = 1.35
\]

2. Calculate the y-intercept \( b \):
\[
b = \frac{45.5 - (1.35)(15)}{5} = \frac{45.5 - 20.25}{5} = \frac{25.25}{5} = 5.05
\]

### Final Equation
Putting it all together, the equation of the line of best fit is:

\[
y = 1.35x + 5.05
\]

So, the final equation of the line of best fit is:
\[
y = 1.35x + 5.05
\]

No rounding is needed, as values are already expressed to two decimal places.