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To find the regression equation of \( x \) on \( y \) based on the given data, we need to follow these steps:

### Step 1: Compute Means of \( X \) and \( Y \)
Let's denote the mean of \( X \) as \( \bar{X} \) and the mean of \( Y \) as \( \bar{Y} \).

### Step 2: Calculate \( \sum XY \), \( \sum Y^2 \), \( \sum X \), \( \sum Y \)
We will compute the following:
- Sum of \( X \) values
- Sum of \( Y \) values
- Sum of products \( XY \)
- Sum of squares of \( Y \)

### Step 3: Calculate the Regression Coefficients
For the regression equation of \( x \) on \( y \), the form is:
\[
x = a + by
\]
where:
- \( b = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{\sum (Y - \bar{Y})^2} \)
- \( a = \bar{X} - b \bar{Y} \)

### Step 4: Substitute \( y = 15 \) to Estimate \( x \)

Let's proceed with these calculations. I'll start by extracting values and performing the necessary calculations.Here's the solution based on the calculations:

1. **Means of \( X \) and \( Y \):**
   - \( \bar{X} = 13.0 \)
   - \( \bar{Y} = 21.14 \) (approximately)

2. **Regression Equation of \( x \) on \( y \):**
   - Calculated values for the regression coefficients:
     - \( b = 0.435 \)
     - \( a = 3.80 \)
   - So, the regression equation is:
     \[
     x = 3.80 + 0.435 \cdot y
     \]

3. **Estimation of \( x \) when \( y = 15 \):**
   \[
   x = 3.80 + 0.435 \cdot 15 = 10.33
   \]

### Final Answer
The estimated value of \( x \) when \( y = 15 \) is approximately **10.33**.

Would you like further details or have any questions?

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Here are five related questions to explore this topic further:

1. How do we calculate the regression equation of \( y \) on \( x \)?
2. What does the slope of the regression line indicate about the relationship between \( x \) and \( y \)?
3. How is the correlation coefficient related to the regression coefficients?
4. What are the assumptions underlying the least-squares method for linear regression?
5. How can we assess the goodness of fit of a regression line?

**Tip:** For a better understanding of regression, try visualizing data points and the regression line on a scatter plot.

Find the regression equation of x on y for the following data. Estimate x when y = 15.

| X  | 10 | 12 | 14 | 19 | 8  | 11 | 17 |
| Y  | 20 | 24 | 25 | 21 | 16 | 22 | 20 |

Learn how to find the regression equation of x on y for the provided data set and estimate the value of x when y is 15. This solution covers key statistical concepts like linear regression, slope, and intercept calculation, and is suitable for students in Grades 11-12.

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