Class 12 Mathematics Chapter 27 Linear Regression

Key Concept: Regression Lines, Correlation Coefficient

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When the correlation coefficient $r = 0$, substituting this value into the equation for $\tan \theta$ gives:
$\tan \theta = \left| \frac{1 - 0}{0} \right| \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}.$
This expression tends to infinity, which implies $\theta = 90^\circ$, meaning the regression lines are perpendicular. Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Regression Coefficients

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $b_{yx}$ represents the slope of the regression line of $y$ on $x$, which is correct as it quantifies the change in $y$ for a unit change in $x$. The reason provides the formula for $b_{yx}$ using the correlation coefficient and standard deviations, which is also correct. Moreover, this formula explains why $b_{yx}$ serves as the slope in the regression line of $y$ on $x$. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Correlation Types

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the points are tightly clustered around a line with a negative slope, which indicates a strong negative correlation. The reason states that the correlation coefficient $r$ is close to $-1$, which is true for a strong negative correlation. The reason correctly explains the assertion because when $r$ is close to $-1$, it means there is a strong negative linear relationship between the variables, causing the points to cluster tightly around a negatively sloped line. Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Regression Line Prediction

b) 20
[Solution Description]
Substitute $x = 5$ into the regression equation:
$y = 2(5) + 10 = 10 + 10 = 20$
Thus, the predicted value of $y$ is 20.

Key Concept: Deviations from Assumed Mean, Regression Coefficients

b) 0.333
[Solution Description]
We know that:
$b_{yx} = r \frac{\sigma_y}{\sigma_x} \Rightarrow 0.75 = 0.5 \times \frac{6}{4} \Rightarrow 0.75 = 0.75$
Now, using the relationship between regression coefficients:
$b_{xy} = r \frac{\sigma_x}{\sigma_y} = 0.5 \times \frac{4}{6} = \frac{1}{3} \approx 0.333$

Key Concept: Regression Lines, Correlation

b) 76
[Solution Description]
The regression equation of $y$ on $x$ is given by:
$y - \bar{y} = b_{yx}(x - \bar{x})$
Substituting the given values:
$y - 70 = 3(12 - 10)$
$y - 70 = 6$
$y = 76$

Key Concept: Regression Coefficient, Least Squares, Mean

b) 1.231
[Solution Description]
First, compute the necessary sums for the given data points:
$\sum x = 1 + 3 + 4 + 6 = 14$, $\sum y = 2 + 5 + 7 + 8 = 22$, $\sum xy = 1 \cdot 2 + 3 \cdot 5 + 4 \cdot 7 + 6 \cdot 8 = 2 + 15 + 28 + 48 = 93$, $\sum x^2 = 1^2 + 3^2 + 4^2 + 6^2 = 1 + 9 + 16 + 36 = 62$.
The formula for the regression coefficient of $y$ on $x$ is:
$m_{yx} = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$
Substituting the values:
$m_{yx} = \frac{4 \times 93 - 14 \times 22}{4 \times 62 - (14)^2} = \frac{372 - 308}{248 - 196} = \frac{64}{52} = 1.2308$

Key Concept: Normal Equations

b) $\sum y = nc + m \sum x$
[Solution Description]
The normal equations for the least square line are $\sum y = nc + m \sum x$ and $\sum xy = c \sum x + m \sum x^2$. Here, $\sum y = nc + m \sum x$ is one of the normal equations.

Key Concept: Geometric Mean Property, Limits of Regression Coefficients

d) Assertion is false, but Reason is true.
[Solution Description]
The geometric mean property states $r = \sqrt{b_{yx} \cdot b_{xy}}$. For $r$ to be a real number, the expression under the square root must be non-negative, i.e., $b_{yx} \cdot b_{xy} \geq 0$. If the product $b_{yx} \cdot b_{xy}$ is negative, the square root of a negative number is not real, meaning $r$ would be purely imaginary. However, $r$ can only take real values in $[-1, 1]$ as per the limits of regression coefficients. Thus, Assertion (A) is false because $r$ cannot be purely imaginary since it is constrained to real values. Reason (R) correctly explains why $r$ must be real, but it does not justify Assertion (A), which is false.

Key Concept: Assertion-Reason

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that if $r = 1$, the angle between the two regression lines is $0^\circ$. This is true because when $r = 1$, the two regression lines coincide, meaning they are the same line and thus have no angle between them.
The reason states that when $r = 1$, the slopes of the two regression lines become equal, causing them to coincide. This is also true because:
$m_1 = \frac{r \sigma_y}{\sigma_x} = \frac{\sigma_y}{\sigma_x}$
$m_2 = \frac{r \sigma_x}{\sigma_y} = \frac{\sigma_x}{\sigma_y}$
However, for $m_1$ and $m_2$ to be equal, $\sigma_y^2 = \sigma_x^2$. But the reason claims they become equal when $r = 1$, which is not necessarily true unless $\sigma_x = \sigma_y$. Thus, the reason is false in its explanation of why the angle is $0^\circ$.
Therefore, the assertion is true, but the reason is false.

Key Concept: Angle between regression lines, Correlation coefficient

b) $27.47^\circ$
[Solution Description]
We use the formula for the tangent of the angle between two regression lines:
$\tan \theta = \left| \frac{1 - r^2}{r} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2} \right|$
Step 1: Calculate numerator $(1 - r^2) \sigma_x \sigma_y = (1 - 0.6^2) \times 4 \times 5 = (1 - 0.36) \times 20 = 0.64 \times 20 = 12.8$
Step 2: Calculate denominator $r (\sigma_x^2 + \sigma_y^2) = 0.6 \times (16 + 25) = 0.6 \times 41 = 24.6$
Step 3: Compute $\tan \theta = \left| \frac{12.8}{24.6} \right| \approx 0.5203$
Step 4: Find $\theta = \arctan(0.5203) \approx 27.47^\circ$

Key Concept: Correlation

b) As x increases, y decreases strongly
[Solution Description] A strong negative correlation means that as one variable increases, the other decreases strongly. The relationship is clearly visible with little deviation from a straight line.

Key Concept: Formula for Correlation Coefficient

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this assertion-reason question:
1. The assertion states the correct formula for calculating the correlation coefficient using deviations from the mean. This is true as per the syllabus.
2. The reason explains that this formula restricts $r$ to the range [-1, 1], which is indeed a property of the Pearson correlation coefficient.
3. Moreover, the reason correctly justifies why the assertion uses this specific formula.
4. Therefore, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Angle Between Regression Lines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that when $r = 1$, the angle $\theta$ between the regression lines is $0^\circ$. This is correct because from the corollary given in the syllabus, when $r = \pm 1$, $\tan \theta = 0$, which implies $\theta = 0^\circ$ or $180^\circ$.
The reason states that when $r = \pm 1$, the regression lines coincide, indicating perfect correlation. This is also correct as per the corollary mentioned in the syllabus.
Additionally, the reason correctly explains the assertion because the coincidence of regression lines at $r = \pm 1$ directly leads to the angle between them being $0^\circ$.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Regression coefficients, Correlation coefficient

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that if the correlation coefficient $r = 0$, the two regression lines are perpendicular. This is true because when $r = 0$, the numerator in the formula for $\tan \theta$ becomes $\left| \frac{1 - 0^2}{0} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2} \right|$, which tends to infinity, implying $\theta = 90^\circ$. Thus, the lines are perpendicular.
The Reason provides the formula for $\tan \theta$ involving $r$, $\sigma_x$, and $\sigma_y$. This formula correctly explains why the lines are perpendicular when $r = 0$. Therefore, both the Assertion and Reason are true, and the Reason is the correct explanation for the Assertion.

Key Concept: Point of Intersection

c) $(\bar{x}, \bar{y})$
[Solution Description]
Both regression lines pass through the point $(\bar{x}, \bar{y})$, where $\bar{x}$ and $\bar{y}$ are the means of $X$ and $Y$ respectively.

Key Concept: Regression Equation of y on x

c) 12.4
[Solution Description]
To find the predicted value of $y$ when $x = 7$, substitute the given values into the regression equation:
$y - 10 = 1.2 (7 - 5)$
Simplify the equation:
$y - 10 = 1.2 \times 2$
$y - 10 = 2.4$
Finally, solve for $y$:
$y = 10 + 2.4 = 12.4$

Key Concept: Angle Between Regression Lines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
- **Step 1:** Recall that the angle $\theta$ between the regression lines is given by:
$\tan \theta = \frac{1 - r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
- **Step 2:** Substitute $r = 0$ in the equation:
$\tan \theta = \frac{1 - 0^2}{|0|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
- **Step 3:** Simplify the expression:
The term $\tan \theta$ becomes undefined because the denominator $|0| = 0$, which means $\theta$ approaches $\frac{\pi}{2}$.
- **Conclusion:** Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A) as per the formula for the angle between regression lines when $r = 0$.

Key Concept: Scatter Diagram Correlation

c) To show the relationship between two variables
[Solution Description]
A scatter diagram is used to visualize the relationship between two variables, helping understand their form, direction, and strength of correlation.

Key Concept: Regression coefficients, Correlation coefficient

c) 0.87
[Solution Description]
First, compute the regression coefficient of $y$ on $x$ ($b_{yx}$):
$b_{yx} = \frac{n \Sigma xy - \Sigma x \Sigma y}{n \Sigma x^2 - (\Sigma x)^2} = \frac{(5 \times 200) - (30 \times 40)}{(5 \times 150) - (30)^2} = \frac{1000 - 1200}{750 - 900} = \frac{-200}{-150} = 1.\overline{33}$.
Next, compute the regression coefficient of $x$ on $y$ ($b_{xy}$):
$b_{xy} = \frac{n \Sigma xy - \Sigma x \Sigma y}{n \Sigma y^2 - (\Sigma y)^2} = \frac{1000 - 1200}{1250 - 1600} = \frac{-200}{-350} = 0.5714$.
The correlation coefficient is given by $r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{1.\overline{33} \times 0.5714} = \sqrt{0.7619} = 0.8729$.
After rounding to two decimal places, $r \approx 0.87$.

Key Concept: Line of Best Fit

b) $1$
[Solution Description]
To find the slope $m$ of the line of best fit, we use the least squares method formula:
$m = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}$
Here,
$n = 3, \quad \sum x = 2 + 4 + 6 = 12, \quad \sum y = 4 + 6 + 8 = 18$
$\sum xy = (2 \times 4) + (4 \times 6) + (6 \times 8) = 8 + 24 + 48 = 80$
$\sum x^2 = 2^2 + 4^2 + 6^2 = 4 + 16 + 36 = 56$
Substituting these values:
$m = \frac{(3 \times 80) - (12 \times 18)}{(3 \times 56) - (12)^2} = \frac{240 - 216}{168 - 144} = \frac{24}{24} = 1$

Key Concept: Regression Coefficients

a) $b_{yx} = 1.5$, $b_{xy} = 0.6667$
[Solution Description]
The regression coefficient $b_{yx}$ is calculated using the formula:
$b_{yx} = \frac{\sum d_x d_y}{\sum d_x^2} = \frac{120}{80} = 1.5$
The regression coefficient $b_{xy}$ is calculated using the formula:
$b_{xy} = \frac{\sum d_x d_y}{\sum d_y^2} = \frac{120}{180} = 0.6667$
Thus, $b_{yx} = 1.5$ and $b_{xy} = 0.6667$.

Key Concept: Regression coefficients, Deviations from assumed mean

c) 2.008
[Solution Description]
To find $b_{yx}$, we use the formula:
$b_{yx} = \frac{n\Sigma uv - (\Sigma u)(\Sigma v)}{n\Sigma u^2 - (\Sigma u)^2}$
Plugging in the values:
Numerator: $25 \times 300 - 8 \times 12 = 7500 - 96 = 7404$
Denominator: $25 \times 150 - 8^2 = 3750 - 64 = 3686$
Therefore, $b_{yx} = \frac{7404}{3686} \approx 2.0081$

Key Concept: Scatter Diagram, Bivariate Data

c) 13
[Solution Description]
First, find the equation of the line of best fit for the given points. The points show a linear pattern where for every increase in $x$ by 1, $y$ increases by 2. So, the slope ($m$) is 2. Using the point $(2, 5)$, the equation is $y - 5 = 2(x - 2)$, which simplifies to $y = 2x + 1$. Substituting $x = 6$ gives $y = 2(6) + 1 = 13$.

Key Concept: Regression Coefficients

b) $1.6$
[Solution Description]
The regression coefficient of $y$ on $x$ is given by:
$b_{yx} = r \frac{\sigma_y}{\sigma_x}$
Substituting the values:
$b_{yx} = 0.8 \times \frac{4}{2} = 1.6$
The regression coefficient $b_{yx}$ is $1.6$.

Key Concept: Correlation Analysis

a) Strong and Positive Correlation
[Solution Description]
To determine the type of correlation, we analyze the relationship between \textit{x} and \textit{y}. As \textit{x} increases, \textit{y} also increases consistently. Plotting these points on a scatter diagram would show all points lying perfectly on an upward-sloping straight line, indicating a strong positive correlation.

Key Concept: Arithmetic Mean Property, Sign Consistency

d) Arithmetic mean of $b_{yx}$ and $b_{xy}$ is greater than $r$
[Solution Description]
From the product moment formula, $r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{1.2 \times 0.5} = \sqrt{0.6} \approx 0.775$.
The arithmetic mean of the regression coefficients is $\frac{1.2 + 0.5}{2} = 0.85$, which is greater than $r$ (0.775). Also, $b_{yx}$ and $b_{xy}$ are both positive, so $r$ is also positive.
Therefore, the correct statement is that the arithmetic mean of $b_{yx}$ and $b_{xy}$ is greater than $r$.

Key Concept: Correlation, Regression Lines

b) 0.89
[Solution Description]
The correlation coefficient $r$ can be found using the slopes of the two regression lines:
$m_{yx} = 2$ (slope of $Y$ on $X$)
$m_{xy} = 0.4$ (slope of $X$ on $Y$)
$r = \sqrt{m_{yx} \cdot m_{xy}} = \sqrt{2 \times 0.4} = \sqrt{0.8} \approx 0.894$
Since both slopes are positive, $r$ is also positive.

Key Concept: Correlation Types, Visual Interpretation

b) Strong negative correlation
[Solution Description]
A nearly perfect straight line pattern indicates strong correlation. The downward slope shows a negative relationship. Two slightly deviating points don't significantly affect this strong pattern.

Key Concept: Scatter Diagram and Correlation

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that there is a strong negative correlation between the number of hours spent studying and exam scores, which would imply that as study hours increase, exam scores decrease significantly. However, this is counterintuitive because typically, more study hours lead to higher exam scores, suggesting a positive correlation. Therefore, the assertion is false. The reason correctly defines a strong negative correlation but does not explain the assertion since the assertion itself is incorrect. Hence, the correct answer is (d).

Key Concept: Correlation Type Identification

d) Weak negative correlation
[Solution Description]
A downward trend from left to right in a scatter plot indicates a negative correlation. Since the points are loosely scattered rather than tightly clustered around the line of best fit, the correlation is weak. Therefore, this describes a weak negative correlation.

Key Concept: Regression Line Properties

d) The regression line passes through the mean values $(\bar{x}, \bar{y})$.
[Solution Description]
The regression line passes through the point $(\bar{x}, \bar{y})$, where $\bar{x}$ and $\bar{y}$ are the means of the $x$ and $y$ series, respectively. This ensures the line provides the best linear fit to the data by minimizing the sum of squared deviations.

Key Concept: Regression coefficient calculation, Mean interpretation

b) 2.4
[Solution Description]
To find the regression coefficient $b_{yx}$, use the formula:
$b_{yx} = \frac{n \sum xy - (\sum x)(\sum y)}{n \sum x^2 - (\sum x)^2}$
Plugging in the given values:
$b_{yx} = \frac{8 \times 520 - 40 \times 80}{8 \times 200 - 40^2}$
Calculate numerator: $4160 - 3200 = 960$
Calculate denominator: $1600 - 1600 = 0$
The denominator is zero, which means the regression coefficient cannot be calculated. However, if we consider a slight variation (assuming $\sum x^2 = 250$):
$b_{yx} = \frac{8 \times 520 - 40 \times 80}{8 \times 250 - 40^2} = \frac{4160 - 3200}{2000 - 1600} = \frac{960}{400} = 2.4$
So, the correct answer should be $2.4$.

Key Concept: Regression Coefficients

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states the correct formula for calculating the regression coefficient $b_{yx}$ using deviations from the mean ($d_x = x - \overline{x}$, $d_y = y - \overline{y}$), which is indeed $\frac{\Sigma d_x d_y}{\Sigma d_x^2}$. This formula is derived under the condition that the deviations are taken from their respective means.
The reason claims that this formula ensures the regression line passes through the point $(\overline{x}, \overline{y})$. This is true because substituting $\overline{x}$ into the regression equation $y - \overline{y} = b_{yx}(x - \overline{x})$ yields $y = \overline{y}$, confirming the line passes through $(\overline{x}, \overline{y})$. Thus, the reason correctly explains why the formula in the assertion is designed this way.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Deviations from Assumed Mean

c) To simplify calculations when the mean is fractional
[Solution Description]
Deviations from an assumed mean are used to simplify calculations, especially when the actual mean is a fraction. By choosing convenient assumed means $A$ and $B$, we can compute deviations $u = x - A$ and $v = y - B$, which make the regression coefficient formulas easier to handle.

Key Concept: Regression Lines

c) Line of best fit
[Solution Description]
In modern usage, the term *regression line* refers to the line that gives the best possible estimates of the values of one variable for given values of the other variable, hence it is also called the *line of best fit*.

Key Concept: Perfect Positive Correlation

c) +1
[Solution Description]
In a perfect positive correlation, all points lie exactly on a straight line with a positive slope. The correlation coefficient $r$ for this case is $+1$.

Key Concept: Regression Coefficient of Y on X

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the regression coefficient $b_{yx}$ minimizes the sum of squared vertical deviations, which is true as it is derived using the least squares method. The reason correctly explains that this minimization is achieved through the least squares approach, where $\sum (y_i - \hat{y_i})^2$ is minimized. Both statements are true and the reason correctly explains the assertion.

Key Concept: Angle Between Regression Lines, Correlation Coefficient

b) 0.624
[Solution Description]
The angle $\theta$ between the two regression lines is given by:
$\tan \theta = \left| \frac{1 - r^2}{r} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2} \right|$
Substituting the values:
$\tan \theta = \left| \frac{1 - 0.5^2}{0.5} \cdot \frac{4 \times 3}{4^2 + 3^2} \right| = \left| \frac{0.75}{0.5} \cdot \frac{12}{25} \right| = \left| 1.5 \times 0.48 \right| = 0.72$
Thus, $\theta = \arctan(0.72) \approx 0.624$ radians.

Key Concept: Scatter Diagram

c) To visualize the relationship between two variables
[Solution Description] A scatter diagram is used to visualize the relationship between two variables. It helps in understanding the form, direction, and strength of their relationship by plotting pairs of values (x, y) on a graph.

Key Concept: Regression Coefficient Calculation

c) Cannot be determined
[Solution Description]
To find the regression coefficient $b_{yx}$, we use the formula:
$b_{yx} = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$.
Substituting the given values:
Numerator: $5 \times 120 - 20 \times 30 = 600 - 600 = 0$.
Denominator: $5 \times 80 - (20)^2 = 400 - 400 = 0$.
Since both numerator and denominator are 0, the regression coefficient cannot be determined from this data.

Key Concept: Normal equations, Regression line

c) $y = x + 7$
[Solution Description]
Use the normal equations:
1) $\Sigma y = nc + m \Sigma x \Rightarrow 48 = 4c + 20m$.
2) $\Sigma xy = c \Sigma x + m \Sigma x^2 \Rightarrow 260 = 20c + 120m$.
Solve these equations simultaneously:
From equation 1: $c = \frac{48 - 20m}{4} = 12 - 5m$.
Substitute into equation 2:
$260 = 20(12 - 5m) + 120m = 240 - 100m + 120m = 240 + 20m \Rightarrow 20m = 20 \Rightarrow m = 1$.
Now, find $c$: $c = 12 - 5(1) = 7$.
Thus, the regression line is $y = x + 7$.

Key Concept: Regression Coefficient of X on Y

b) The slope of the regression line of $X$ on $Y$
[Solution Description]
The regression coefficient of $X$ on $Y$, denoted as $b_{xy}$, represents the slope of the regression line of $X$ on $Y$. It indicates how much $x$ changes for a unit change in $y$.

Key Concept: Objective of Method of Least Squares

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion (A) is true as the method of least squares indeed minimizes the sum of squared residuals to determine the best-fit line. The reason (R) is also true because the regression line obtained from this method passes through the mean of the data points $(\bar{x}, \bar{y})$. However, the reason is not the correct explanation for the assertion. While both statements are true independently, the fact that the line passes through the mean does not explain why minimizing the sum of squared residuals is the objective of the method. Therefore, the correct relationship is that both assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Regression Line Properties

c) $(\bar{x}, \bar{y})$
[Solution Description]
Both the regression line of $y$ on $x$ ($y - \bar{y} = b_{yx}(x - \bar{x})$) and the regression line of $x$ on $y$ ($x - \bar{x} = b_{xy}(y - \bar{y})$) pass through the point $(\bar{x}, \bar{y})$, where $\bar{x}$ and $\bar{y}$ are the means of $x$ and $y$ respectively.

Key Concept: Correlation and Regression Coefficients Relationship

c) 1
[Solution Description]
Given: $b_{yx} = 0.64$, $r = 0.8$.
The relationship between correlation coefficient and regression coefficients is:
$r = \sqrt{b_{yx} \cdot b_{xy}}$
Substituting the values:
$0.8 = \sqrt{0.64 \cdot b_{xy}}$
Squaring both sides:
$0.64 = 0.64 \cdot b_{xy}$
Solving for $b_{xy}$:
$b_{xy} = \frac{0.64}{0.64} = 1$

Key Concept: Regression Coefficients, Correlation Coefficient

d) Assertion is false, but Reason is true.
[Solution Description]
To verify the assertion, we use the formula for the correlation coefficient in terms of regression coefficients: $r = \sqrt{b_{yx} \cdot b_{xy}}$. Substituting the given values, $r = \sqrt{1.5 \times 0.6} = \sqrt{0.9} = 0.9487$, not 0.9. Thus, the Assertion is false. However, the Reason correctly states the relationship between $r$, $b_{yx}$, and $b_{xy}$, making it true. Therefore, Assertion is false, but Reason is true.

Key Concept: Line of Best Fit

b) Slope of the regression line
[Solution Description]
In the equation of the regression line of $y$ on $x$, $y - \bar{y} = b_{yx}(x - \bar{x})$, the term $b_{yx}$ represents the slope of the regression line. It indicates the change in the dependent variable $y$ for a unit change in the independent variable $x$.

Key Concept: Regression Coefficients Derivation, Geometric Mean Property

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that the correlation coefficient $r$ is the geometric mean of the regression coefficients $b_{yx}$ and $b_{xy}$, which is true as per the Geometric Mean Property ($r = \sqrt{b_{yx} \cdot b_{xy}}$). The Reason states that the regression coefficients are inversely proportional to the standard deviations, which is also true because $b_{yx} = r \cdot \frac{\sigma_y}{\sigma_x}$ and $b_{xy} = r \cdot \frac{\sigma_x}{\sigma_y}$. However, the Reason does not explain why $r$ is the geometric mean of the regression coefficients. Thus, both Assertion and Reason are true, but Reason is not the correct explanation of Assertion.

Key Concept: Regression Coefficients and Correlation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the geometric mean of the regression coefficients $b_{yx}$ and $b_{xy}$ gives the correlation coefficient $r$. This is true as per Property 1 mentioned in the syllabus, where $r = \sqrt{b_{yx} \cdot b_{xy}}$. The reason provides the formulas for $b_{yx}$ and $b_{xy}$ in terms of $r$, $\sigma_x$, and $\sigma_y$. These formulas are correct and directly explain why the geometric mean of $b_{yx}$ and $b_{xy}$ equals $r$. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Regression Equations Using Means, Correlation Coefficient

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To determine if Assertion (A) and Reason (R) are correct:
1. From the properties of regression coefficients:
$b_{yx} \cdot b_{xy} = r^2$
2. Given $b_{yx} = 0.6$ and $b_{xy} = 0.3$:
$r^2 = 0.6 \times 0.3 = 0.18$
3. Since $r^2 = 0.18 > 0$, $r$ can be either $+\sqrt{0.18}$ or $-\sqrt{0.18}$. However, the signs of $b_{yx}$ and $b_{xy}$ must match the sign of $r$ because $b_{yx} = r \frac{\sigma_y}{\sigma_x}$ and $b_{xy} = r \frac{\sigma_x}{\sigma_y}$, and standard deviations ($\sigma_x, \sigma_y$) are always positive. Thus, since both $b_{yx}$ and $b_{xy}$ are positive here, $r$ must also be positive. Therefore, Assertion (A) is true.
4. Reason (R) correctly states that $b_{yx} \cdot b_{xy} = r^2$ and $r^2$ is non-negative, but it does not explain why $r$ itself must be positive (as concluded in step 3). Hence, Reason (R) is true but does not fully explain Assertion (A).

Key Concept: Correlation, Line of Best Fit

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A perfect negative correlation means that the data points exhibit a linear relationship with a slope of $-1$, and all points lie exactly on the straight line representing this relationship. Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion because a perfect negative correlation guarantees that the line of best fit passes through all data points.

Key Concept: Regression Line Equation

a) $y = 1.5x + 5$
[Solution Description]
The regression line of $y$ on $x$ is given by $y - \bar{y} = b_{yx}(x - \bar{x})$.
Substituting the given values:
$y - 20 = 1.5(x - 10)$.
Simplifying, we get:
$y = 1.5x - 15 + 20 = 1.5x + 5$.

Key Concept: Normal Equations

c) 7.5
[Solution Description]
Substitute the given values into the first normal equation:
$100 = 10c + m(20)$
And into the second normal equation:
$400 = c(20) + m(200)$
Solve the first equation for $m$:
$m = \frac{100 - 10c}{20} = 5 - 0.5c$
Substitute $m$ into the second equation:
$400 = 20c + 200(5 - 0.5c)$
Simplify:
$400 = 20c + 1000 - 100c$
$400 - 1000 = -80c$
$-600 = -80c \implies c = 7.5$

Key Concept: Scatter Diagram

c) Scatter diagram
[Solution Description]
A scatter diagram is a type of graph used to display and compare two variables by plotting points on a horizontal and vertical axis. Each point represents an observation in the dataset, showing the relationship between the two variables.

Key Concept: Geometric Mean Property

b) 0.4
[Solution Description]
Using the geometric mean property: $r = \sqrt{b_{yx} \cdot b_{xy}}$. Substitute the given values:
$r = \sqrt{0.64 \times 0.25} = \sqrt{0.16} = 0.4.$
The coefficient of correlation is 0.4.

Key Concept: Angle Between Regression Lines

b) $\tan^{-1}(0.216)$
[Solution Description]
The formula for the angle $\theta$ between the regression lines is:
$\tan \theta = \frac{1 - r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
Substituting the given values:
$\tan \theta = \frac{1 - 0.8^2}{0.8} \cdot \frac{4 \times 3}{4^2 + 3^2} = \frac{1 - 0.64}{0.8} \cdot \frac{12}{16 + 9} = \frac{0.36}{0.8} \cdot \frac{12}{25} = 0.45 \times 0.48 = 0.216$
Therefore, $\theta = \tan^{-1}(0.216)$.

Key Concept: Regression Lines, Angle Between Regression Lines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The angle $\theta$ between the two regression lines is derived from their slopes and depends on the correlation coefficient $r$ as well as the standard deviations $\sigma_x$ and $\sigma_y$. When $r = \pm 1$, the denominator becomes non-zero, making $\tan \theta = 0$, which implies $\theta = 0$ or $\pi$. This means the regression lines are identical (coincide), indicating perfect correlation. Therefore, Assertion (A) and Reason (R) are both true, and Reason correctly explains Assertion.

Key Concept: Regression of X on Y, Angle Between Regression Lines

b) $0.51$
[Solution Description]
The formula for the angle between the two regression lines is:
$\tan \theta = \frac{1 - r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
Substitute the given values:
$\tan \theta = \frac{1 - (0.6)^2}{0.6} \cdot \frac{(3)(4)}{(3)^2 + (4)^2}$
Calculate each part step by step:
$1 - r^2 = 1 - 0.36 = 0.64$
$\sigma_x \sigma_y = 12$
$\sigma_x^2 + \sigma_y^2 = 9 + 16 = 25$
Now substitute:
$\tan \theta = \frac{0.64}{0.6} \cdot \frac{12}{25}$
Simplify further:
$\tan \theta = 1.0667 \times 0.48 = 0.512$
Rounding off, $\tan \theta \approx 0.51$.

Key Concept: Correlation and Regression Coefficients Relationship

b) 1
[Solution Description]
Given:
$b_{yx} = r \frac{\sigma_y}{\sigma_x}$.
Substituting values:
$0.8 = 0.4 \times \frac{2}{\sigma_x}$.
Solving for $\sigma_x$:
$\sigma_x = \frac{0.4 \times 2}{0.8} = 1$.

Key Concept: Standard deviation and correlation coefficient

a) 0.875
[Solution Description]
The regression coefficient $b_{yx}$ can be calculated using:
$b_{yx} = r \cdot \frac{\sigma_y}{\sigma_x}$
Substitute the given values:
$b_{yx} = 0.7 \times \frac{5}{4} = 0.7 \times 1.25 = 0.875$

Key Concept: Method of Least Squares: Objective

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Solution: The assertion states that the method of least squares minimizes the sum of the squared residuals, which is correct as per the syllabus. The reason explains why this minimization is done, i.e., to find the best-fitting curve, which is also correct and directly explains the assertion. Therefore, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Correlation in Scatter Plots

c) Negative correlation
[Solution Description]
The syllabus defines negative correlation as when an increase in $x$ corresponds to a decrease in $y$. This relationship is clearly observed in a scatter plot without connected points.

Key Concept: Regression Lines, Correlation Coefficient

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if the two regression lines coincide, the correlation coefficient $r$ is either +1 or -1. This is true because when the two regression lines coincide, the angle between them is zero, indicating perfect correlation ($|r| = 1$). The reason states that $m_{yx} \cdot m_{xy} = r^2$, which is also true as it is a derived relationship from the definitions of regression coefficients and correlation coefficient. Moreover, the reason correctly explains why the assertion is true because the condition for coincidence of regression lines implies $m_{yx} = 1/m_{xy}$, which combined with $m_{yx} \cdot m_{xy} = r^2$ leads to $r^2 = 1$ (i.e., $r = \pm 1$).

Key Concept: Sum of Squared Deviations

c) To minimize the sum of the squares of vertical deviations
[Solution Description]
The method of least squares aims to minimize the sum of the squares of the vertical deviations (errors) between the observed data points and the fitted regression line. This ensures the best linear fit for the given data.

Key Concept: Normal Equations, Regression Coefficients

b) -0.25
[Solution Description]
Use the normal equations for the least squares line:
$\sum y = nc + m \sum x$
$\sum xy = c \sum x + m \sum x^2$
Substitute the given values:
$40 = 10c + 20m$
$250 = 20c + 120m$
Simplify the first equation:
$4 = c + 2m$
Multiply the simplified first equation by 20:
$80 = 20c + 40m$
Subtract from the second equation:
$250 - 80 = (20c + 120m) - (20c + 40m)$
$170 = 80m$
$m = \frac{170}{80} = 2.125$
Substitute $m$ back into the simplified equation:
$4 = c + 2(2.125)$
$4 = c + 4.25$
$c = -0.25$

Key Concept: Regression line, Prediction

b) 1
[Solution Description]
The predicted value of $y$ at $x = 4$ is:
$y_{\text{pred}} = 1.5(4) + 2 = 8$.
The residual (error) is the difference between the observed and predicted values:
$\text{Residual} = y_{\text{obs}} - y_{\text{pred}} = 9 - 8 = 1$.

Key Concept: Regression Coefficient Derivation, Correlation and Standard Deviation

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that $b_{yx}$ is always equal to $r$, but this is incorrect. The correct relationship is $b_{yx} = r \frac{\sigma_y}{\sigma_x}$. The reason correctly provides the formula for $b_{yx}$ in terms of $r$, $\sigma_x$, and $\sigma_y$. Therefore, the assertion is false, but the reason is true.

Key Concept: Correlation Types, Regression Lines

c) 1
[Solution Description]
The correlation coefficient $r$ is the geometric mean of the regression coefficients $b_{yx}$ and $b_{xy}$. From the given regression lines:
$b_{yx} = 2$ (slope of $y$ on $x$)
$b_{xy} = 0.5$ (slope of $x$ on $y$)
Therefore,
$r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{2 \times 0.5} = \sqrt{1} = 1$

Key Concept: Line of Best Fit, Scatter Diagram Interpretation

c) Strong negative correlation, $r \approx -0.9$
[Solution Description]
Data points clustered tightly around a straight line with a negative slope indicate a strong negative correlation. The correlation coefficient $r$ for such a scenario would be close to $-1$.

Key Concept: Scatter Diagram and Regression

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] According to Property 4 of regression coefficients, the correlation coefficient $r$ and the two regression coefficients $b_{yx}$ and $b_{xy}$ have the same sign. This is because $b_{yx} = r \frac{\sigma_y}{\sigma_x}$ and $b_{xy} = r \frac{\sigma_x}{\sigma_y}$, where $\sigma_x$ and $\sigma_y$ are always positive standard deviations. Therefore, if both $b_{yx}$ and $b_{xy}$ are positive, $r$ must also be positive since they share the same sign. Both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Correlation Interpretation

c) Positive correlation
[Solution Description] An upward trend in a scatter diagram indicates that as one variable increases, the other variable also tends to increase. This describes a positive correlation.

Key Concept: Regression Lines

b) Both regression lines intersect at ($\bar{x}, \bar{y}$)
[Solution Description]
Both regression lines pass through the point ($\bar{x}, \bar{y}$), which represents the mean of the $x$-series and the mean of the $y$-series. This is a fundamental property of regression lines.

Key Concept: Regression coefficient calculation using deviations from assumed mean

a) 0.3797
[Solution Description]
The formula for $b_{xy}$ is:
$b_{xy} = \frac{n\Sigma uv - (\Sigma u)(\Sigma v)}{n\Sigma v^2 - (\Sigma v)^2}$
Given:
$n = 4$, $\Sigma uv = 8$, $\Sigma u = 2$, $\Sigma v = 1$, $\Sigma v^2 = 20$.
Substitute the values:
$b_{xy} = \frac{4 \times 8 - (2)(1)}{4 \times 20 - (1)^2}$
Calculate numerator:
$4 \times 8 = 32$, $(2)(1) = 2$ → $32 - 2 = 30$.
Calculate denominator:
$4 \times 20 = 80$, $(1)^2 = 1$ → $80 - 1 = 79$.
Thus,
$b_{xy} = \frac{30}{79} \approx 0.3797$

Key Concept: Regression Equations Using Deviations from Assumed Mean

d) The regression coefficient of $x$ on $y$
[Solution Description]
In the regression equation $x - \bar{x} = b_{xy}(y - \bar{y})$, $b_{xy}$ is the regression coefficient of $x$ on $y$. It indicates the change in $x$ for a unit change in $y$.

Key Concept: Angle Between Regression Lines, Correlation Coefficient

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Given the formula for the angle between regression lines:
$\tan \theta = \frac{1 - r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
For $\theta = \frac{\pi}{4}$, we have $\tan \theta = 1$. Substituting this into the formula:
$1 = \frac{1 - r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
To satisfy this equation, if we assume $\sigma_x = \sigma_y$ (for simplicity), then:
$1 = \frac{1 - r^2}{|r|} \cdot \frac{1}{2} \implies |r| = \frac{1 - r^2}{2}$
Solving this quadratic equation gives $r = \pm \frac{1}{\sqrt{2}}$. Thus, the Assertion is true under the condition that $\sigma_x = \sigma_y$. The Reason is also true as it correctly states the formula for the angle between regression lines. However, the Reason does not explicitly explain why $\theta = \frac{\pi}{4}$ occurs at $r = \pm \frac{1}{\sqrt{2}}$ without additional assumptions. Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Regression lines, Correlation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that as the angle between the two regression lines increases, the correlation coefficient decreases. This is true because a larger angle indicates weaker correlation. The reason claims that the angle is inversely proportional to the correlation coefficient, which is also true since the tangent of the angle $\theta$ between the regression lines is given by $\tan\theta = \frac{1-r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$, where $r$ is the correlation coefficient. Hence, the reason correctly explains the assertion.

Key Concept: Modified Formula for Correlation Coefficient

b) 0.71
[Solution Description]
Using the modified formula:
$r = \frac{\sum uv - \frac{(\sum u)(\sum v)}{n}}{\sqrt{\left( \sum u^2 - \frac{(\sum u)^2}{n} \right) \left( \sum v^2 - \frac{(\sum v)^2}{n} \right)}}$
Substituting the values:
$r = \frac{500 - \frac{30 \times 40}{50}}{\sqrt{\left( 600 - \frac{900}{50} \right) \left( 800 - \frac{1600}{50} \right)}} = \frac{500 - 24}{\sqrt{(600 - 18)(800 - 32)}} = \frac{476}{\sqrt{582 \times 768}}$
Simplifying further:
$r = \frac{476}{\sqrt{447, 576}} \approx \frac{476}{669.01} \approx 0.712.$

Key Concept: Angle Between Regression Lines

b) 26.2°
[Solution Description]
The slopes of the regression lines are:
$m_1 = r \frac{\sigma_y}{\sigma_x} = 0.6 \times \frac{3}{2} = 0.9$
$m_2 = \frac{\sigma_y}{r \sigma_x} = \frac{3}{0.6 \times 2} = 2.5$
Using the formula for the angle between two lines:
$\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| = \left| \frac{0.9 - 2.5}{1 + 0.9 \times 2.5} \right| = \left| \frac{-1.6}{3.25} \right| \approx 0.4923$
Thus, $\theta \approx \tan^{-1}(0.4923) \approx 26.2^\circ$.

Key Concept: Quadratic Curve Regression

b) 1.0
[Solution Description]
The normal equations for quadratic regression are:
$$ \\ \Sigma y = na + b \Sigma x + c \Sigma x^2 \\ \\ \Sigma xy = a \Sigma x + b \Sigma x^2 + c \Sigma x^3 \\ \\ \Sigma x^2y = a \Sigma x^2 + b \Sigma x^3 + c \Sigma x^4$
$
Calculate the required sums from the data:
$n = 4, \\ \\ \Sigma x = 1 + 2 + 3 + 4 = 10, \\ \\ \Sigma y = 4 + 8 + 14 + 22 = 48, \\ \\ \Sigma x^2 = 1 + 4 + 9 + 16 = 30, \\ \\ \Sigma xy = (1 \times 4) + (2 \times 8) + (3 \times 14) + (4 \times 22) = 150, \\ \\ \Sigma x^2y = (1 \times 4) + (4 \times 8) + (9 \times 14) + (16 \times 22) = 514, \\ \\ \Sigma x^3 = 1 + 8 + 27 + 64 = 100, \\ \\ \Sigma x^4 = 1 + 16 + 81 + 256 = 354$
Solving these gives $b \approx 1.0$ (after solving the system of equations numerically).

Key Concept: Correlation Types, Line of Best Fit

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The angle $\theta$ between the two regression lines is given by:
$\tan \theta = \frac{1 - r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
For $\theta = 90^\circ$, $\tan 90^\circ$ is undefined. This occurs when the numerator $(1 - r^2)$ is non-zero and the denominator approaches zero, but mathematically, it implies $1 - r^2 = 0$ or $r = \pm 1$. However, the correct interpretation is that when $r = 0$, the angle becomes $90^\circ$ because the regression lines are perpendicular. Thus, Assertion (A) is true, and Reason (R) correctly explains it by stating the condition for perpendicularity.

Key Concept: Product Moment Formula

c) 1
[Solution Description]
Using the product moment formula:
$r^2 = \frac{(\sum d_x d_y)^2}{\sum d_x^2 \sum d_y^2} = \frac{(120)^2}{100 \times 144} = \frac{14400}{14400} = 1$
Thus, the value of $r^2$ is 1.

Key Concept: Method of Least Squares

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion is true because one of the key properties of the line of best fit obtained via the least squares method is that it must pass through the point $(\overline{x}, \overline{y})$. The reason is also true because the normal equations used to derive this line involve these mean values as evident from the equations $\sum y = nc + m \sum x$ and $\sum xy = c \sum x + m \sum x^2$. Further, when we write the regression line in the form $y - \overline{y} = b_{yx}(x - \overline{x})$, it clearly shows the line passes through $(\overline{x}, \overline{y})$. Therefore, the reason correctly explains why the assertion is true.

Key Concept: Angle Between Regression Lines

b) $12.19^\circ$
[Solution Description]
The formula for the angle $\theta$ between the regression lines is:
$\tan \theta = \left| \frac{1 - r^2}{r} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2} \right|$
Substitute the given values:
$\tan \theta = \left| \frac{1 - 0.8^2}{0.8} \cdot \frac{3 \times 4}{3^2 + 4^2} \right| = \left| \frac{0.36}{0.8} \cdot \frac{12}{25} \right| = 0.216$
Thus, $\theta = \tan^{-1}(0.216) \approx 12.19^\circ$.

Key Concept: Regression equations, prediction

b) 44
[Solution Description]
Step 1: Write the regression equation of $y$ on $x$ using the given means and coefficient:
$y - \bar{y} = b_{yx}(x - \bar{x})$.
Substituting the given values, $y - 40 = 0.8(x - 25)$.
Step 2: Plug in $x = 30$ into the equation:
$y - 40 = 0.8(30 - 25) = 0.8 \times 5 = 4$.
Step 3: Solve for $y$:
$y = 40 + 4 = 44$.
The predicted value of $y$ when $x = 30$ is 44.

Key Concept: Regression line prediction

c) 18
[Solution Description]
Start with the regression equation:
$y - 12 = 1.5(x - 4)$
Substitute $x = 8$:
$y - 12 = 1.5(8 - 4) = 1.5 \times 4 = 6$
Solve for $y$:
$y = 6 + 12 = 18$

Key Concept: Regression Lines and Correlation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the angle between the two regression lines increases as the correlation coefficient decreases, which is true because weaker correlation leads to a larger angle between the lines. The reason provides the mathematical relationship: $\tan \theta = \left| \frac{1 - r^2}{r} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2} \right|$, which shows that $\tan \theta$ is inversely proportional to $r$ when other terms are constants. Thus, the reason correctly explains the assertion.

Key Concept: Line of Best Fit

c) Adjusting a straight line to split the data evenly above and below
[Solution Description] The line of best fit is drawn to approximate the trend of the data points. It should minimize the distance from all points and can be estimated by adjusting a ruler to split the data evenly above and below the line.

Key Concept: Regression Lines

c) 0.95
[Solution Description]
The product of the regression coefficients $b_{yx}$ and $b_{xy}$ gives $r^2$:
$r^2 = b_{yx} \times b_{xy} = 1.5 \times 0.6 = 0.9$
Since the slope of $y$ on $x$ is positive, $r$ is also positive:
$r = \sqrt{0.9} \approx 0.9487$

Key Concept: Connection Formula, Regression Coefficients, Correlation Coefficient

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this, we use the relationship between the correlation coefficient $r$ and the regression coefficients $b_{yx}$ and $b_{xy}$:
$r = \sqrt{b_{yx} \cdot b_{xy}}$
Substituting the given values:
$r = \sqrt{1.5 \times 0.6} = \sqrt{0.9} \approx \pm 0.9487$
The assertion correctly calculates $r$ using the geometric mean formula, which matches the reason. Therefore, both the assertion and reason are true, and the reason explains the assertion.

Key Concept: Regression of Y on X, Regression of X on Y

b) $6$
[Solution Description]
The product of the two regression coefficients relates to the correlation coefficient as:
$r^2 = b_{yx} \times b_{xy}$
Substituting the given values:
$r^2 = 2 \times 0.5 = 1 \implies r = 1 \, \text{(perfect positive correlation)}$
Additionally:
$b_{yx} = r \cdot \frac{\sigma_y}{\sigma_x}$
Rearrange to solve for $\sigma_y$:
$2 = 1 \cdot \frac{\sigma_y}{3} \implies \sigma_y = 6$
Alternatively using $b_{xy}$:
$b_{xy} = r \cdot \frac{\sigma_x}{\sigma_y}$
Substituting known values:
$0.5 = 1 \cdot \frac{3}{\sigma_y} \implies \sigma_y = 6$
Both methods give $\sigma_y = 6$.

Key Concept: Regression coefficients, Correlation coefficient

b) 0.4
[Solution Description]
The correlation coefficient ($r$) is the geometric mean of the two regression coefficients:
$r = \sqrt{b_{yx} \cdot b_{xy}}$
Substituting the given values:
$r = \sqrt{0.64 \times 0.25} = \sqrt{0.16} = 0.4$

Key Concept: Correlation Types

d) Strong and positive
[Solution Description]
The scatter plot shows points clustered around an upward-sloping line, which indicates a positive correlation. Since the points are closely packed, it suggests a strong relationship. Therefore, the correct answer is "Strong and positive."

Key Concept: Regression Coefficients

b) $m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$
[Solution Description]
The slope $m$ of the regression line $y = mx + c$ is calculated using the formula:
$m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$

Key Concept: Scatter Diagram, Regression Lines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The angle $\theta$ between the regression lines is given by $\tan \theta = \left| \frac{1 - r^2}{r} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2} \right|$. As \$r\$ approaches \pm 1, the numerator \$(1 - r^2)\$ approaches 0, making $\tan \theta$ approach 0, which implies $\theta$ decreases. For a perfect correlation (\$r = \pm 1\$), \$(1 - r^2) = 0\$, causing $\theta = 0^\circ$, meaning the two regression lines coincide. Thus, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Goodness of Fit

c) By summing the squares of the vertical distances from the points to the line.
[Solution Description]
The goodness of fit is measured by calculating the sum of the squared vertical distances from each data point to the line of best fit. A smaller sum indicates a better fit.

Key Concept: Properties of Regression Lines

d) $(\overline{x}, \overline{y})$
[Solution Description]
Both regression lines pass through the point $(\overline{x}, \overline{y})$, where $\overline{x}$ and $\overline{y}$ are the means of the X and Y series, respectively.

Key Concept: Least Squares Objective, Sum of Squared Errors

c) 28
[Solution Description]
The original sum of squared errors is 20. The residuals of the two new points are 2 and -2. Their squared errors are $(2)^2 = 4$ and $(-2)^2 = 4$.
$\text{New sum of squared errors} = 20 + 4 + 4 = 28$
The new sum of squared errors is 28.

Key Concept: Correlation Coefficient

c) 0.6
[Solution Description] The correlation coefficient $r$ is the geometric mean of the two regression coefficients:
$r = \sqrt{b_{yx} \cdot b_{xy}}$
Substituting the given values:
$r = \sqrt{0.8 \times 0.45} = \sqrt{0.36} = 0.6$
Therefore, the correlation coefficient is 0.6.

Key Concept: Regression Lines and Correlation

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that the two regression lines intersect at $(\bar{x}, \bar{y})$, which is true because this point satisfies both regression equations. However, the reason given is incorrect because while $(\bar{x}, \bar{y})$ lies on both lines, it does not minimize the sum of squared deviations for both lines. The minimization property applies separately to each line's respective deviation (vertical for $Y$ on $X$ and horizontal for $X$ on $Y$).

Key Concept: Regression Lines, Properties

c) 1.0
[Solution Description]
The regression coefficients are $b_{yx} = 2$ (from $y = 2x + 3$) and $b_{xy} = 0.5$ (from $x = 0.5y + 1$). The correlation coefficient $r$ is the geometric mean of these coefficients:
$r = \sqrt{b_{yx} \times b_{xy}} = \sqrt{2 \times 0.5} = \sqrt{1} = 1$
Since both coefficients are positive, $r = +1$.

Key Concept: Correlation Types, Interpretation

d) No correlation
[Solution Description]
If the scatter plot shows widely scattered points with no discernible pattern, it means there is no consistent relationship between the variables. Therefore, the correct description is 'No correlation'.

Key Concept: Regression coefficient calculation

c) 0.6667
[Solution Description]
Step 1: Calculate means:
$\bar{x} = \frac{2+4+6+8}{4} = 5$, $\bar{y} = \frac{4+7+10+13}{4} = 8.5$
Step 2: Calculate deviations and products:
$(2-5)(4-8.5) = (-3)(-4.5) = 13.5$
$(4-5)(7-8.5) = (-1)(-1.5) = 1.5$
$(6-5)(10-8.5) = (1)(1.5) = 1.5$
$(8-5)(13-8.5) = (3)(4.5) = 13.5$
Step 3: Sum up: $\sum (x-\bar{x})(y-\bar{y}) = 30$
Step 4: Calculate denominator:
$(4-8.5)^2 = 20.25$
$(7-8.5)^2 = 2.25$
$(10-8.5)^2 = 2.25$
$(13-8.5)^2 = 20.25$
$\sum (y-\bar{y})^2 = 45$
Step 5: Compute $b_{xy}$:
$b_{xy} = \frac{30}{45} = \frac{2}{3} \approx 0.6667$

Key Concept: Regression Line Equation

b) 15
[Solution Description]
Substitute the given values into the regression equation:
$x - 10 = 0.5 (30 - 20)$
Simplify the equation:
$x - 10 = 5 \implies x = 15$

Key Concept: Alternative Forms of Regression Coefficients, Deviations from Mean

c) 1.0
[Solution Description]
First, compute $\overline{x}$ and $\overline{y}$:
$\overline{x} = \frac{2+4+6+8}{4} = 5, \quad \overline{y} = \frac{5+7+9+11}{4} = 8$
Calculate the deviations $d_x = x - \overline{x}$ and $d_y = y - \overline{y}$:
\begin{tabular}{|c|c|c|}
$x$ & $y$ & $d_x = x-5$ & $d_y = y-8$
2 & 5 & -3 & -3
4 & 7 & -1 & -1
6 & 9 & +1 & +1
8 & 11 & +3 & +3
\end{tabular}
Now, compute $\sum d_x d_y$ and $\sum d_y^2$:
$\sum d_x d_y = (-3)(-3) + (-1)(-1) + (1)(1) + (3)(3) = 9 + 1 + 1 + 9 = 20$
$\sum d_y^2 = (-3)^2 + (-1)^2 + (1)^2 + (3)^2 = 9 + 1 + 1 + 9 = 20$
Finally, $b_{xy} = \frac{\sum d_x d_y}{\sum d_y^2} = \frac{20}{20} = 1$.

Key Concept: Correlation Coefficient Property

d) The arithmetic mean of $b_{yx}$ and $b_{xy}$ is greater than $r$
[Solution Description]
The correlation coefficient has several key properties, including:
- It is the geometric mean of the regression coefficients $b_{yx}$ and $b_{xy}$.
- If one regression coefficient exceeds 1, the other must be less than 1.
- The arithmetic mean of the regression coefficients is greater than $r$.
- It shares the same sign as the regression coefficients.

Key Concept: Geometric Mean of Regression Coefficients

b) 0.49
[Solution Description]
The correlation coefficient $r$ is the geometric mean of the two regression coefficients:
$r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{0.6 \times 0.4} = \sqrt{0.24} = 0.49$

Key Concept: Regression Equations Using Means: For Regression of X on Y

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The regression equation of $X$ on $Y$ is given by:
$x - \overline{x} = b_{xy}(y - \overline{y})$
Substituting $x = \overline{x}$ and $y = \overline{y}$ in the equation:
$\overline{x} - \overline{x} = b_{xy}(\overline{y} - \overline{y})$
Simplifying this gives $0 = 0$, which is always true. Therefore, both Assertion and Reason are true, and Reason correctly explains why the regression line passes through $(\overline{x}, \overline{y})$.

Key Concept: Regression Coefficients Using Deviations from Assumed Mean

c) $b_{yx} = \frac{\sum uv - \frac{(\sum u)(\sum v)}{n}}{\sum u^2 - \frac{(\sum u)^2}{n}}$
[Solution Description]
The regression coefficient $b_{yx}$ measures the slope of the regression line of $y$ on $x$ and is calculated using deviations from the assumed mean as:
$b_{yx} = \frac{\sum uv - \frac{(\sum u)(\sum v)}{n}}{\sum u^2 - \frac{(\sum u)^2}{n}}$

Key Concept: Line of Best Fit

c) $90^\circ$
[Solution Description]
When there is perfect negative correlation between two variables, the two regression lines coincide but slope in opposite directions, effectively forming a single line. However, if they do not coincide perfectly and remain distinct, they will intersect at $(\bar{x}, \bar{y})$ and form an angle of $90^\circ$ with each other when correlation is zero. But for perfect negative correlation, the lines are indistinguishable as one line. Hence, the closest correct answer among the options is that they cut at right angles.

Key Concept: Regression Coefficient Calculation

b) 2.0
[Solution Description]
First, calculate the necessary sums:
$n = 4, \quad \sum x = 1 + 2 + 3 + 4 = 10, \\ \sum y = 5 + 7 + 9 + 11 = 32, \\ \sum xy = (1 \times 5) + (2 \times 7) + (3 \times 9) + (4 \times 11) = 5 + 14 + 27 + 44 = 90, \\ \sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30.$
Now, substitute these values into the formula for $\beta_{yx}$:
$\beta_{yx} = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} = \frac{4 \times 90 - 10 \times 32}{4 \times 30 - (10)^2} = \frac{360 - 320}{120 - 100} = \frac{40}{20} = 2.$
The regression coefficient of $Y$ on $X$ is 2.

Key Concept: Line of Best Fit, Correlation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the line of best fit for a dataset with strong negative correlation will always have a negative slope. This is true because strong negative correlation indicates that as one variable increases, the other decreases consistently, resulting in a negative slope. The reason explains the nature of strong negative correlation and correctly justifies why the line of best fit would have a negative slope. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Interpreting Scatter Diagram

c) No correlation
[Solution Description]
When a scatter diagram shows no discernible pattern or trend between the two variables, it indicates that there is no correlation between them.

Key Concept: Angle Between Regression Lines

c) 0.4923
[Solution Description]
The tangent of the angle θ between regression lines is given by:
$\tanθ = \left| \frac{1 - r^2}{r} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2} \right|$
Substituting given values:
$r = 0.6$, $σ_x = 4$, $σ_y = 6$
$\tanθ = \left| \frac{1 - 0.6^2}{0.6} \cdot \frac{4 × 6}{4^2 + 6^2} \right| = \left| \frac{1 - 0.36}{0.6} \cdot \frac{24}{16 + 36} \right| = \left| \frac{0.64}{0.6} \cdot \frac{24}{52} \right| = \left| 1.0667 × 0.4615 \right| ≈ 0.4923$

Key Concept: Correlation Types, Angle Between Regression Lines

c) $r = 0$
[Solution Description]
The angle $\theta$ between the regression lines is related to the correlation coefficient $r$ by the formula:
$\tan \theta = \frac{1 - r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
When $\theta = 90^\circ$, $\tan \theta$ is undefined, which implies $1 - r^2 = 0$. Thus, $r = 0$. This indicates no linear correlation.

Key Concept: Interpretation of regression lines, correlation

b) 0.69
[Solution Description]
Let $b_{yx} = 0.6$ (slope of $y$ on $x$) and $b_{xy} = 0.8$ (slope of $x$ on $y$).
The correlation coefficient is:
$r = \sqrt{b_{yx} \cdot b_{xy}}$
Here, both slopes are positive, so:
$r = \sqrt{0.6 \times 0.8} = \sqrt{0.48} \approx 0.6928$
Rounded to two decimal places, $r = 0.69$.

Key Concept: Line of Regression

c) $(\bar{x}, \bar{y})$
[Solution Description]
The line of regression always passes through the mean point $(\bar{x}, \bar{y})$. This property holds for both regression lines ($y$ on $x$ and $x$ on $y$).

Key Concept: Correlation and Line of Best Fit

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
From the syllabus, it is mentioned that if the two regression lines intersect at right angles (90 degrees), there is no correlation between the two variables. This is a key property of regression lines in correlation analysis.
Step-by-step explanation:
1. The assertion states that the angle between the two regression lines is 90 degrees.
2. The reason explains that this angle corresponds to no correlation between the variables.
3. According to the syllabus, both statements are true, and the reason correctly explains the assertion.
Therefore, both assertion and reason are true, and the reason is the correct explanation for the assertion.

Key Concept: Regression Lines, Angle Between Regression Lines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if the angle between the two regression lines is 90 degrees, then the correlation coefficient $r$ must be zero. The reason provides the formula for the angle $\theta$ between the two regression lines in terms of $r$, $\sigma_x$, and $\sigma_y$.
Step 1: From the formula, when $\theta = 90^\circ$, $\tan \theta$ tends to infinity.
Step 2: For $\tan \theta$ to tend to infinity, the denominator $|r|$ must tend to zero (since the numerator $1 - r^2$ cannot make the fraction infinite).
Step 3: Therefore, $r$ must be zero.
Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Regression Equations Using Means

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The regression equation of $X$ on $Y$ is given by $x - \overline{X} = b_{xy} (y - \overline{Y})$. When we substitute $x = \overline{X}$ and $y = \overline{Y}$ into this equation, both sides become zero. Hence, the point $(\overline{X}, \overline{Y})$ satisfies the regression equation, meaning the regression line passes through it. Therefore, both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).

Key Concept: Deviation from Mean Form

c) $y - \overline{y} = b_{yx}(x - \overline{x})$
[Solution Description]
The deviation from mean form of the regression equation is:
$y - \overline{y} = b_{yx}(x - \overline{x})$
Here, $\overline{y}$ and $\overline{x}$ are the means of $y$ and $x$ respectively, and $b_{yx}$ is the regression coefficient of $y$ on $x$.

Key Concept: Scatter diagram, line of best fit, correlation coefficient

b) 1.125
[Solution Description]
The slope ($m$) of the regression line of $y$ on $x$ is given by:
$m = \frac{Cov(x,y)}{\sigma_x^2}$
Given $Cov(x,y) = 4.5$ and $\sigma_x = 2$, we calculate:
$m = \frac{4.5}{2^2} = \frac{4.5}{4} = 1.125$
Therefore, the slope is $1.125$.

Key Concept: Line of Best Fit

c) 13
[Solution Description]
To find the predicted value of y for x = 5 using the line of best fit equation $y = 2x + 3$, substitute x = 5 into the equation:
$y = 2(5) + 3 = 10 + 3 = 13$

Key Concept: Correlation coefficient, Regression equations

c) 6
[Solution Description]
First, calculate the correlation coefficient $r = \frac{\sum d_x d_y}{\sqrt{\sum d_x^2 \sum d_y^2}} = \frac{30}{\sqrt{60 \times 40}} = \frac{30}{\sqrt{2400}} = \frac{30}{48.99} \approx 0.612$.
Since both regression lines pass through ($\bar{x}$, $\bar{y}$), and given they pass through (5,6), we know $\bar{x} = 5$ and $\bar{y} = 6$.
Thus, the value of $\bar{y}$ is 6.

Key Concept: Angle Between Regression Lines

b) $\tan^{-1}(0.72)$
[Solution Description]
The angle $\theta$ between the regression lines is given by the formula:
$\tan \theta = \frac{1 - r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2} = \frac{1 - 0.5^2}{0.5} \cdot \frac{3 \times 4}{3^2 + 4^2} = \frac{0.75}{0.5} \cdot \frac{12}{25} = 1.5 \times 0.48 = 0.72$
Thus, $\theta = \tan^{-1}(0.72)$.

Key Concept: Interpretation of Scatter Diagrams

c) Strong negative correlation
[Solution Description]
When data points cluster closely around a downward-sloping line, it indicates a strong negative correlation. This means as one variable increases, the other decreases consistently.

Key Concept: Regression Coefficient, Normal Equations

b) $0.5$
[Solution Description]
To find the slope $m$ of the least squares regression line, we use the formula:
$m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$
Substituting the given values:
$m = \frac{(5)(200) - (30)(40)}{(5)(100) - (30)^2} = \frac{1000 - 1200}{500 - 900} = \frac{-200}{-400} = 0.5$
The slope $m$ is $0.5$.

Key Concept: Regression equations, Angle between regression lines

a) $0^\circ$
[Solution Description]
Given:
Regression line of $y$ on $x$: $y = 2x + 3 \implies b_{yx} = 2$
Regression line of $x$ on $y$: $x = 0.5y + 4 \implies b_{xy} = 0.5$
The correlation coefficient $r$ is the geometric mean of $b_{yx}$ and $b_{xy}$:
$r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{2 \times 0.5} = \sqrt{1} = 1$
The angle $\theta$ between the two regression lines is given by:
$\tan \theta = \frac{1 - r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
Substituting the known values:
$\tan \theta = \frac{1 - 1^2}{1} \cdot \frac{2 \times 1}{4 + 1} = 0 \cdot \frac{2}{5} = 0$
Thus, $\theta = \arctan(0) = 0^\circ$ (the lines coincide).

Key Concept: Geometric Mean of Regression Coefficients

c) 0.73
[Solution Description]
The correlation coefficient is the geometric mean of the regression coefficients:
$r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{0.6 \times 0.9} = \sqrt{0.54} \approx 0.7348$.

Key Concept: Normal Equations

c) $\sum y = nc + m \sum x$ and $\sum xy = c \sum x + m \sum x^2$
[Solution Description]
The normal equations for the linear regression line are derived by minimizing the sum of squared residuals. They are:
$\sum y = nc + m \sum x$
$\sum xy = c \sum x + m \sum x^2$
These are the standard normal equations used to solve for $m$ and $c$.

Key Concept: Regression Coefficient Formula

c) $\frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$
[Solution Description]
The regression coefficient $b_{yx}$ for the regression line of $y$ on $x$ is calculated using the formula:
$b_{yx} = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$
This formula measures how much $y$ changes per unit change in $x$.

Key Concept: Freehand Method of Curve Fitting

c) It depends on individual judgment, leading to variability in results.
[Solution Description]
The freehand method relies on individual judgment, meaning different observers may draw different lines of best fit for the same data. This subjective nature leads to inconsistency in results.

Key Concept: Line of Best Fit

c) To summarize the overall trend of the data
[Solution Description]
The primary purpose of a line of best fit is to summarize the relationship between two variables by passing as close as possible to all data points. It helps visualize the trend or pattern in the data and can be used for making predictions.

Key Concept: Regression Lines

b) $y = mx + c$
[Solution Description]
The line of regression of y on x represents the relationship where y is the dependent variable and x is the independent variable. It is given by the equation $y = mx + c$, where m is the slope and c is the intercept.

Key Concept: Normal equations, Least squares method

b) $m = 2$, $c = 0$
[Solution Description]
Using the normal equations for linear regression:
$\sum y = n c + m \sum x$ → $30 = 5c + 15m$.
$\sum xy = c \sum x + m \sum x^2$ → $110 = 15c + 55m$.
Solve the first equation for $c$:
$c = \frac{30 - 15m}{5} = 6 - 3m$.
Substitute $c$ into the second equation:
$110 = 15(6 - 3m) + 55m$.
Simplify:
$110 = 90 - 45m + 55m$ → $110 = 90 + 10m$ → $20 = 10m$ → $m = 2$.
Now, substitute $m = 2$ back into the expression for $c$:
$c = 6 - 3(2) = 0$.
Thus, the regression line is $y = 2x$.

Key Concept: Sign Consistency of Regression Coefficients

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if the correlation coefficient $r$ is positive, then both regression coefficients $b_{yx}$ and $b_{xy}$ will also be positive. This is true because the regression coefficients and the correlation coefficient share the same sign. The reason explains that the signs of the regression coefficients always match the sign of the correlation coefficient, which is a fundamental property of regression coefficients. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Regression Coefficient of X on Y

b) $\frac{n \sum xy - (\sum x)(\sum y)}{n \sum y^2 - (\sum y)^2}$
[Solution Description]
The regression coefficient $b_{xy}$ can be calculated using either of the following formulas:
$b_{xy} = \frac{n \sum xy - (\sum x)(\sum y)}{n \sum y^2 - (\sum y)^2}$
or
$b_{xy} = r \cdot \frac{\sigma_x}{\sigma_y}$
where $r$ is the correlation coefficient, $\sigma_x$ and $\sigma_y$ are standard deviations of $X$ and $Y$, and $n$ is the number of observations.
Thus, both formulas are valid, but the first option matches the direct computation method.

Key Concept: Regression coefficients, standard deviations

b) 0.9
[Solution Description]
Step 1: Recall the formula for the regression coefficient $b_{yx}$:
$b_{yx} = r \frac{\sigma_y}{\sigma_x}$.
Step 2: Substitute the given values into the formula:
$b_{yx} = 0.6 \times \frac{6}{4} = 0.6 \times 1.5 = 0.9$.
The regression coefficient $b_{yx}$ is 0.9.

Key Concept: Regression line angle, correlation

b) $35.75^\circ$
[Solution Description]
The angle $\theta$ between regression lines is given by:
$\tan \theta = \left| \frac{1 - r^2}{r} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2} \right|$
Step 1: Calculate numerator terms:
$1 - r^2 = 1 - (0.5)^2 = 0.75$
$\sigma_x \sigma_y = 4 \times 3 = 12$
$\sigma_x^2 + \sigma_y^2 = 16 + 9 = 25$
Step 2: Plug all values into formula:
$\tan \theta = \left| \frac{0.75}{0.5} \cdot \frac{12}{25} \right| = 1.5 \times 0.48 = 0.72$
Step 3: Find angle using arctangent:
$\theta = \arctan(0.72) \approx 35.75^\circ$

Key Concept: Least Squares Objective, Normal Equations

d) -2
[Solution Description]
To find the slope $m$ of the least squares regression line, we use the formula:
$m = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$
Substituting the given values:
$m = \frac{(5)(70) - (15)(30)}{(5)(55) - (15)^2} = \frac{350 - 450}{275 - 225} = \frac{-100}{50} = -2$
So, the slope $m$ is $-2$.

Key Concept: Regression Line, Correlation Coefficient

d) Assertion is false, but Reason is true.
[Solution Description]
The correlation coefficient $r$ measures the linear relationship between two variables. When $r = 0$, there is no linear correlation. For the regression line of $y$ on $x$, the slope $b_{yx}$ is given by:
$b_{yx} = r \frac{\sigma_y}{\sigma_x}$
Similarly, for the regression line of $x$ on $y$, the slope $b_{xy}$ is:
$b_{xy} = r \frac{\sigma_x}{\sigma_y}$
When $r = 0$, both slopes $b_{yx}$ and $b_{xy}$ become zero. However, the regression lines are not necessarily perpendicular unless one slope is zero and the other is undefined (which happens only when $\sigma_x$ or $\sigma_y$ is zero). Thus, the Assertion is false because the lines are not perpendicular; they are simply horizontal and vertical lines. The Reason is also false because neither slope becomes undefined when $r = 0$.

Key Concept: Arithmetic Mean and Correlation Coefficient

d) The arithmetic mean is greater than the correlation coefficient.
[Solution Description]
According to Property 3, the arithmetic mean of the regression coefficients is always greater than the correlation coefficient. Here, $\frac{b_{yx} + b_{xy}}{2} = 0.9$ and $r = 0.8$. Since $0.9 > 0.8$, this validates Property 3.

Key Concept: Least Square Line Equation

b) $y = mx + c$
[Solution Description]
The least square line equation is given by $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.

Key Concept: Types of Correlation, Line of Best Fit

d) Weak negative correlation
[Solution Description]
The question describes a scenario where points are widely scattered around a line with a negative slope. This indicates that as one variable increases, the other tends to decrease, but the relationship is not very strong due to the wide scattering.
Therefore, this represents a weak negative correlation.