Use your graphing calculator to find the equation of the trend line for the scatterplot below.

***x***

***y***

1

17

2

20

3

22

4

26

5

28

6

31

To find the equation of the trend line for the given scatterplot data, we can use the method of linear regression to find the line of best fit. The general equation of the trend line (or line of best fit) is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. 

Here's the given dataTo find the equation of the trend line for the given scatterplot data, we can use the method of linear regression to find the line of best fit. The general equation of the trend line (or line of best fit) is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. 

Here's the given 

| x | y  |
|---|----|
| 1 | 17 |
| 2 | 20 |
| 3 | 22 |
| 4 | 26 |
| 5 | 28 |
| 6 | 31 |

We'll use the formulas for the slope (\( m \)) and y-intercept (\( b \)) of the line of best fit:

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

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

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

First, we calculate the necessary sums:

\[
\sum x = 1 + 2 + 3 + 4 + 5 + 6 = 21
\]

\[
\sum y = 17 + 20 + 22 + 26 + 28 + 31 = 144
\]

\[
\sum xy = (1 \cdot 17) + (2 \cdot 20) + (3 \cdot 22) + (4 \cdot 26) + (5 \cdot 28) + (6 \cdot 31) = 17 + 40 + 66 + 104 + 140 + 186 = 553
\]

\[
\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91
\]

Substituting these values into the formulas for \( m \) and \( b \):

\[
m = \frac{6 \cdot 553 - 21 \cdot 144}{6 \cdot 91 - 21^2} = \frac{3318 - 3024}{546 - 441} = \frac{294}{105} = 2.8
\]

\[
b = \frac{144 - 2.8 \cdot 21}{6} = \frac{144 - 58.8}{6} = \frac{85.2}{6} = 14.2
\]

Therefore, the equation of the trend line is:

\[ y = 2.8x + 14.2 \]

Would you like any further details or have any questions about this calculation?

Here are five additional questions you can ask:
1. How do I interpret the slope and y-intercept of the trend line?
2. Can you show me how to plot this trend line on a scatterplot?
3. What is the correlation coefficient for this data set?
4. How can I use this trend line to make predictions?
5. What are some common applications of trend lines in data analysis?

**Tip:** When performing linear regression, it's essential to check the residuals to ensure that your model is appropriate for the data. Residuals are the differences between the observed values and the values predicted by your model.

Learn how to find the equation of the trend line for a given scatterplot data using linear regression. This problem covers calculating the slope and y-intercept for the line of best fit, essential for understanding data trends in statistics and mathematics.

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