Use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.
x
11
13
12
15
6
5
7
9
8
14
10
 
y
10.91
10.40
10.84
8.35
5.56
3.35
7.40
9.91
8.84
9.56
10.61
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Part 1
ModifyingAbove y with caretequals
  
enter your response hereplus
  
enter your response herex​ (Round to two decimal places as​ needed.)

To find the equation of the regression line (often called the line of best fit) given the data, we'll follow these steps:

1. **Calculate the means** of \( x \) and \( y \):
   - \(\bar{x}\) (mean of \( x \))
   - \(\bar{y}\) (mean of \( y \))

2. **Compute the slope** (\( b_1 \)) of the regression line using the formula:
   \[
   b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
   \]
   
3. **Determine the y-intercept** (\( b_0 \)) of the regression line using the formula:
   \[
   b_0 = \bar{y} - b_1\bar{x}
   \]

4. **Write the equation** of the regression line:
   \[
   \hat{y} = b_0 + b_1x
   \]

Let's proceed with these calculations:

### Step 1: Calculate the means of \( x \) and \( y \)
Given \( x \) values: \( 11, 13, 12, 15, 6, 5, 7, 9, 8, 14, 10 \)

\[
\bar{x} = \frac{11 + 13 + 12 + 15 + 6 + 5 + 7 + 9 + 8 + 14 + 10}{11}
\]

Given \( y \) values: \( 10.91, 10.40, 10.84, 8.35, 5.56, 3.35, 7.40, 9.91, 8.84, 9.56, 10.61 \)

\[
\bar{y} = \frac{10.91 + 10.40 + 10.84 + 8.35 + 5.56 + 3.35 + 7.40 + 9.91 + 8.84 + 9.56 + 10.61}{11}
\]

### Step 2: Compute the slope (\( b_1 \))

\[
b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\]

### Step 3: Determine the y-intercept (\( b_0 \))

\[
b_0 = \bar{y} - b_1\bar{x}
\]

### Step 4: Write the equation of the regression line

I'll now perform the calculations to determine the regression line.

Let me do the calculations.The equation of the regression line is:

\[
\hat{y} = 3.70 + 0.50x
\]

### Characteristic of the Data Ignored by the Regression Line

One characteristic that the regression line ignores is **any potential non-linear relationship** between \( x \) and \( y \). The regression line assumes a linear relationship, meaning it fits a straight line through the data points. If the actual relationship between \( x \) and \( y \) is curved or follows a different pattern (like quadratic or exponential), the linear regression line will not capture this complexity and could lead to misleading interpretations.

Would you like to see the scatterplot and the regression line on it, or do you have any questions?

---

Here are five related questions to explore further:
1. How can you test for the significance of the slope in the regression line?
2. What methods can be used to detect non-linearity in data before applying a linear regression model?
3. How do outliers affect the regression line?
4. What are the assumptions underlying linear regression analysis?
5. How can you calculate the coefficient of determination (\(R^2\)) for the regression line?

**Tip:** Always plot your data before performing regression to check for non-linear patterns, outliers, or other irregularities that a simple linear model might miss.

Use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.
x
11
13
12
15
6
5
7
9
8
14
10

y
10.91
10.40
10.84
8.35
5.56
3.35
7.40
9.91
8.84
9.56
10.61

Discover how to calculate the equation of a regression line using data points and explore how linear regression may overlook non-linear relationships. This problem is perfect for high school students in Grades 10-12 learning about statistics and data analysis.

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