Use linear regression to find the line that best fits this data. Write your final answer in the form of an equation y^=a+bx.

**Drop Height (in)**

6

12

18

24

30

36

**Bounce Height (in)**

5

10

17

23

26

31

Use drop height as the independent variable x and bounce height as the dependent variable y. **Round to two decimal places.**

To find the line of best fit for the given data using linear regression, we will calculate the slope \( b \) and the intercept \( a \) of the regression line \( \hat{y} = a + bx \).

### Step 1: Organize the Data
We have the following data points:

- Drop Height \( x \): 6, 12, 18, 24, 30, 36
- Bounce Height \( y \): 5, 10, 17, 23, 26, 31

### Step 2: Calculate the Necessary Summations
We will need to calculate the following:

- The number of data points \( n \)
- The sum of \( x \): \( \sum{x} \)
- The sum of \( y \): \( \sum{y} \)
- The sum of \( x^2 \): \( \sum{x^2} \)
- The sum of \( xy \): \( \sum{xy} \)

I'll calculate these values first.We have the following summations:

- Number of data points \( n = 6 \)
- \( \sum{x} = 126 \)
- \( \sum{y} = 112 \)
- \( \sum{x^2} = 3276 \)
- \( \sum{xy} = 2904 \)

### Step 3: Calculate the Slope \( b \)
The formula for the slope \( b \) in linear regression is:

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

Now, I'll calculate \( b \).The slope \( b \) of the line is approximately \( 0.88 \).

### Step 4: Calculate the Intercept \( a \)
The formula for the intercept \( a \) is:

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

I'll now calculate \( a \).The intercept \( a \) is approximately \( 0.27 \).

### Step 5: Write the Equation of the Line
Thus, the equation of the line of best fit is:

\[
\hat{y} = 0.27 + 0.88x
\]

This is the linear regression equation that represents the relationship between drop height \( x \) and bounce height \( y \).

Would you like more details on any step or have any questions? 

Here are some related questions you might explore:
1. How is linear regression useful in real-life scenarios?
2. What is the significance of the slope in this context?
3. How would we compute the residuals for this data set?
4. How can we assess the goodness of fit for this regression model?
5. What happens if the data does not follow a linear trend?

**Tip:** Linear regression assumes a linear relationship between the variables, so it's crucial to visualize the data beforehand to ensure this assumption is reasonable.

Use linear regression to find the line that best fits this data. Write your final answer in the form of an equation y^=a+bx. Drop Height (in): 6, 12, 18, 24, 30, 36. Bounce Height (in): 5, 10, 17, 23, 26, 31. Use drop height as the independent variable x and bounce height as the dependent variable y. Round to two decimal places.

Learn how to use linear regression to find the line of best fit for drop height and bounce height data. Given Drop Height (in): 6, 12, 18, 24, 30, 36 and Bounce Height (in): 5, 10, 17, 23, 26, 31. Find the equation of the line, y = a + bx, with detailed steps and key formulas. Suitable for students in Grades 10-12.

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