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?

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.