The question asks us to find the value of \(x\) for a given value of \(y = -3\), using two linear equations representing regression lines: \(x + 4y + 1 = 0\) and \(4x + 9y + 7 = 0\).

In regression analysis, we typically have two regression lines: the regression line of \(x\) on \(y\) and the regression line of \(y\) on \(x\). The regression line of \(x\) on \(y\) is used to estimate \(x\) for a given value of \(y\), and the regression line of \(y\) on \(x\) is used to estimate \(y\) for a given value of \(x\).

To answer this question, we need to identify which of the two given equations is the regression line of \(x\) on \(y\). We can determine this by considering the properties of regression coefficients. Let \(b_{xy}\) be the regression coefficient of \(x\) on \(y\) and \(b_{yx}\) be the regression coefficient of \(y\) on \(x\). A key property is that the product of the regression coefficients must lie between 0 and 1, i.e., \(0 \le b_{xy} \cdot b_{yx} \le 1\). Also, \(b_{xy}\) and \(b_{yx}\) must have the same sign.

Let's analyze the two given equations:

Equation 1: \(x + 4y + 1 = 0\)

Equation 2: \(4x + 9y + 7 = 0\)

We can express each equation in terms of \(x\) as a function of \(y\) (to find \(b_{xy}\)) and \(y\) as a function of \(x\) (to find \(b_{yx}\)).

From Equation 1:

• Solving for \(x\) in terms of \(y\): \(x = -4y - 1\). If this is the regression of \(x\) on \(y\), then \(b_{xy} = -4\).
• Solving for \(y\) in terms of \(x\): \(4y = -x - 1 \implies y = -\frac{1}{4}x - \frac{1}{4}\). If this is the regression of \(y\) on \(x\), then \(b_{yx} = -1/4\).

From Equation 2:

• Solving for \(x\) in terms of \(y\): \(4x = -9y - 7 \implies x = -\frac{9}{4}y - \frac{7}{4}\). If this is the regression of \(x\) on \(y\), then \(b_{xy} = -9/4\).
• Solving for \(y\) in terms of \(x\): \(9y = -4x - 7 \implies y = -\frac{4}{9}x - \frac{7}{9}\). If this is the regression of \(y\) on \(x\), then \(b_{yx} = -4/9\).

Now let's consider the two possible cases for assigning the regression lines and check the condition \(0 \le b_{xy} \cdot b_{yx} \le 1\).

Case 1: Assume Equation 1 is \(x\) on \(y\) and Equation 2 is \(y\) on \(x\).

• \(b_{xy} = -4\) (from \(x = -4y - 1\))
• \(b_{yx} = -4/9\) (from \(y = -\frac{4}{9}x - \frac{7}{9}\))

The product \(b_{xy} \cdot b_{yx} = (-4) \cdot (-\frac{4}{9}) = \frac{16}{9}\). Since \(\frac{16}{9} > 1\), this case is not possible according to the property of regression coefficients.

Case 2: Assume Equation 1 is \(y\) on \(x\) and Equation 2 is \(x\) on \(y\).

• \(b_{yx} = -1/4\) (from \(y = -\frac{1}{4}x - \frac{1}{4}\))
• \(b_{xy} = -9/4\) (from \(x = -\frac{9}{4}y - \frac{7}{4}\))

The product \(b_{xy} \cdot b_{yx} = (-\frac{9}{4}) \cdot (-\frac{1}{4}) = \frac{9}{16}\). Since \(0 \le \frac{9}{16} \le 1\), this case is valid.

Therefore, the regression line of \(x\) on \(y\) is Equation 2: \(4x + 9y + 7 = 0\).

The question asks for the value of \(x\) when \(y = -3\). We substitute \(y = -3\) into the equation of the regression line of \(x\) on \(y\).

Substitute \(y = -3\):

So, the value of \(x\) when \(y = -3\) is 5.

Understanding Regression Lines

Regression lines are used in statistics to model the relationship between two variables. One line predicts \(y\) from \(x\), and the other predicts \(x\) from \(y\).

• The line of regression of \(y\) on \(x\) is written as \(y = a + b_{yx}x\), where \(b_{yx}\) is the regression coefficient of \(y\) on \(x\).
• The line of regression of \(x\) on \(y\) is written as \(x = c + b_{xy}y\), where \(b_{xy}\) is the regression coefficient of \(x\) on \(y\).

The two regression lines intersect at the point \((\bar{x}, \bar{y})\), which are the means of \(x\) and \(y\) respectively.

Key Properties of Regression Coefficients

For two variables \(x\) and \(y\):

• Regression coefficients \(b_{xy}\) and \(b_{yx}\) always have the same sign (both positive or both negative).
• The product of the regression coefficients is equal to the square of the correlation coefficient, \(r^2\). That is, \(b_{xy} \cdot b_{yx} = r^2\).
• Since the correlation coefficient \(r\) always lies between -1 and +1 (\(-1 \le r \le 1\)), the square of the correlation coefficient \(r^2\) must lie between 0 and 1 (\(0 \le r^2 \le 1\)).
• Therefore, the product of the regression coefficients must lie between 0 and 1 (\(0 \le b_{xy} \cdot b_{yx} \le 1\)). This property is crucial for identifying the correct regression line.

Step-by-Step Solution Summary

1. Identify the two given linear equations: \(x + 4y + 1 = 0\) and \(4x + 9y + 7 = 0\).
2. Assume one equation is the regression line of \(x\) on \(y\) and the other is \(y\) on \(x\).
3. Calculate the regression coefficients \(b_{xy}\) and \(b_{yx}\) for this assumption.
4. Check if the product \(b_{xy} \cdot b_{yx}\) lies between 0 and 1.
5. If the condition is not met, switch the assumption and repeat steps 2-4.
6. Once the correct regression line of \(x\) on \(y\) is identified (the one for which the product condition holds when paired with the other line as \(y\) on \(x\)), substitute the given value of \(y\) into this equation.
7. Solve the equation for \(x\).

Following these steps, we correctly identified \(4x + 9y + 7 = 0\) as the regression line of \(x\) on \(y\) and calculated \(x=5\) when \(y=-3\).

Checking products:

• If \(b_{xy} = -4\) (from eq 1) and \(b_{yx} = -4/9\) (from eq 2), product = \(16/9 > 1\). Invalid.
• If \(b_{xy} = -9/4\) (from eq 2) and \(b_{yx} = -1/4\) (from eq 1), product = \(9/16\). Valid (between 0 and 1).

Thus, \(4x + 9y + 7 = 0\) is the regression line of \(x\) on \(y\).

Substitute \(y = -3\):

Regression Line Interpretation

The regression line \(x = -\frac{9}{4}y - \frac{7}{4}\) tells us that for every one-unit increase in \(y\), \(x\) is estimated to decrease by \(9/4\) units. When \(y = -3\), we can estimate \(x\) using this relationship, which we found to be 5.

Revision Table: Regression Lines

Additional Information: Correlation vs. Regression

Correlation and regression are related statistical techniques, but they serve different purposes.

• Correlation: Measures the strength and direction of the linear relationship between two variables. The correlation coefficient (\(r\)) ranges from -1 to +1. A value near +1 indicates a strong positive linear relationship, near -1 indicates a strong negative linear relationship, and near 0 indicates a weak or no linear relationship. Correlation does not imply causation.
• Regression: Models the relationship between variables so that one variable (the dependent variable) can be predicted from one or more other variables (the independent variable(s)). Linear regression finds the "best-fit" straight line that describes the relationship. Regression helps in understanding how the dependent variable changes when the independent variable changes and is used for prediction.

In this problem, the regression lines allow us to predict the value of one variable given the value of the other, based on the observed relationship.