Question

Write an equation for the line of best fit based on the trend line drawn.


y= mx + b




Based on the trend line what is the expected value of y when x=5

The graph shows:
(1,30)
(2,35)
(5,55)
(7,60)
(10,70)

Answer 1

To find the equation of the line of best fit, we will first need to determine the slope (m) and the y-intercept (b) from the given points.

Given points:
- (1, 30)
- (2, 35)
- (5, 55)
- (7, 60)
- (10, 70)

To find the slope (m), we can use the formula:

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

First, let's calculate the necessary summations.

### Step 1: Calculate the sums
Let’s calculate the sums:

\[
\sum x = 1 + 2 + 5 + 7 + 10 = 25
\]

\[
\sum y = 30 + 35 + 55 + 60 + 70 = 250
\]

\[
\sum xy = (1 \cdot 30) + (2 \cdot 35) + (5 \cdot 55) + (7 \cdot 60) + (10 \cdot 70) = 30 + 70 + 275 + 420 + 700 = 1495
\]

\[
\sum x^2 = 1^2 + 2^2 + 5^2 + 7^2 + 10^2 = 1 + 4 + 25 + 49 + 100 = 179
\]

The number of points (n) is 5.

### Step 2: Apply the formula for the slope
Now we can plug these values into the slope formula:

\[
m = \frac{5(1495) - (25)(250)}{5(179) - (25)^2}
\]

Calculating the numerator:

\[
5(1495) = 7475
\]
\[
(25)(250) = 6250
\]
\[
\text{Numerator} = 7475 - 6250 = 1225
\]

Calculating the denominator:

\[
5(179) = 895
\]
\[
(25)^2 = 625
\]
\[
\text{Denominator} = 895 - 625 = 270
\]

So we find:

\[
m = \frac{1225}{270} \approx 4.53
\]

### Step 3: Calculate the y-intercept (b)
Now we need to find the y-intercept (b) using the formula:

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

Substituting the known values:

\[
b = \frac{250 - 4.53 \cdot 25}{5}
\]

Calculating:

\[
4.53 \cdot 25 = 113.25
\]
\[
b = \frac{250 - 113.25}{5} = \frac{136.75}{5} \approx 27.35
\]

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

\[
y = 4.53x + 27.35
\]

### Expected value of y when x=5
To find the expected value of \(y\) when \(x = 5\):

\[
y = 4.53(5) + 27.35
\]

Calculating:

\[
y = 22.65 + 27.35 = 50
\]

### Conclusion
1. **Equation of the line of best fit:**
\(y = 4.53x + 27.35\)

2. **Expected value of y when x=5:**
\(y \approx 50\)