Class 12 Mathematics Chapter 27 Linear Regression

Key Concept: Regression lines, Perfect correlation

c) When $r = \pm 1$
[Solution Description]
The two regression lines coincide when there is perfect correlation between the variables, i.e., when the correlation coefficient $r = \pm 1$.
This is because when $r = \pm 1$, all data points lie exactly on both regression lines, making them identical.

Key Concept: Properties of Regression Coefficients

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
From the properties of regression coefficients, we know that:
$b_{yx} = r \frac{\sigma_y}{\sigma_x}$ and $b_{xy} = r \frac{\sigma_x}{\sigma_y}$.
The geometric mean of the regression coefficients is:
$\sqrt{b_{yx} \cdot b_{xy}} = \sqrt{r \frac{\sigma_y}{\sigma_x} \cdot r \frac{\sigma_x}{\sigma_y}} = \sqrt{r^2} = r$.
Thus, Assertion (A) is true.
Similarly, the product of the regression coefficients is:
$b_{yx} \cdot b_{xy} = r \frac{\sigma_y}{\sigma_x} \cdot r \frac{\sigma_x}{\sigma_y} = r^2$.
Hence, Reason (R) is also true.
However, while both Assertion and Reason are true, the Reason does not explain why the geometric mean equals $r$; it merely provides an alternative relationship between the regression coefficients and $r$. Therefore, the Reason is NOT the correct explanation of the Assertion.

Key Concept: Correlation, Line of Best Fit

b) 0.9
[Solution Description]
To find the slope $m$ of the regression line of Y on X, we use the formula:
$m = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum xy - (\sum x)^2}$
First, calculate $\sum x$, $\sum y$, $\sum xy$, $\sum x^2$, and $n$:
$\sum x = 1 + 2 + 3 + 4 + 5 = 15$
$\sum y = 2 + 3 + 5 + 4 + 6 = 20$
$\sum xy = (1)(2) + (2)(3) + (3)(5) + (4)(4) + (5)(6) = 2 + 6 + 15 + 16 + 30 = 69$
$\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55$
$n = 5$
Now, plug these into the formula:
$m = \frac{5(69) - (15)(20)}{5(55) - (15)^2} = \frac{345 - 300}{275 - 225} = \frac{45}{50} = 0.9$
The approximate slope $m$ is 0.9.

Key Concept: Regression Line Concept

d) Fitting the best curve by minimizing squared residuals
[Solution Description]
The least squares method minimizes the sum of the squares of the residuals (the differences between observed and predicted values). It is used to find the best-fitting curve or line to a given set of data points.

Key Concept: Regression Lines and Mean Point

c) $(\overline{x}, \overline{y})$
[Solution Description]
Both regression lines, the regression of Y on X and the regression of X on Y, always pass through the mean point $(\overline{x}, \overline{y})$. This is a fundamental property of regression lines where $\overline{x}$ is the mean of the x-series and $\overline{y}$ is the mean of the y-series.

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

b) 1
[Solution Description]
To find the correlation coefficient (\$r\$), we use the property that \$r\$ is the geometric mean of the two regression coefficients. Here, \$b_{yx} = 2\$ (from \$y = 2x + 3\$) and \$b_{xy} = 0.5\$ (from \$x = 0.5y - 1\$). Thus, \$r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{2 \times 0.5} = \sqrt{1} = 1\$.

Key Concept: Geometric Mean Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $r = \sqrt{b_{yx} \cdot b_{xy}}$, which is a standard property of regression coefficients. The reason states that $b_{yx} \cdot b_{xy} = r^2$, which follows directly from squaring both sides of the assertion. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.
Calculation:
- From the Geometric Mean Property: $r = \sqrt{b_{yx} \cdot b_{xy}}$
- Squaring both sides: $r^2 = b_{yx} \cdot b_{xy}$
- Thus, the reason is a direct consequence of the assertion.

Key Concept: Regression Equations, Correlation Coefficient

c) 0.8
[Solution Description]
First, calculate the mean of $x$ and $y$:
$\bar{x} = \frac{\sum x}{n} = \frac{15}{5} = 3$
$\bar{y} = \frac{\sum y}{n} = \frac{25}{5} = 5$
Next, compute the standard deviations $\sigma_x$ and $\sigma_y$:
$\sigma_x = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2} = \sqrt{\frac{55}{5} - 3^2} = \sqrt{11 - 9} = \sqrt{2}$
$\sigma_y = \sqrt{\frac{\sum y^2}{n} - \bar{y}^2} = \sqrt{\frac{135}{5} - 5^2} = \sqrt{27 - 25} = \sqrt{2}$
The slope of the regression line of $y$ on $x$ ($b_{yx}$) is given by:
$b_{yx} = r \frac{\sigma_y}{\sigma_x} = 0.8 \times \frac{\sqrt{2}}{\sqrt{2}} = 0.8$

Key Concept: Relationship Between Regression and Correlation

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion (A) states that the correlation coefficient $r$ is the geometric mean of the two regression coefficients $b_{yx}$ and $b_{xy}$, which is correct as per the formula $r = \sqrt{b_{yx} \cdot b_{xy}}$. The Reason (R) states that the arithmetic mean of the regression coefficients is always greater than the correlation coefficient, which is also correct as $\frac{b_{yx} + b_{xy}}{2} > r$. However, Reason (R) does not explain why Assertion (A) is true; it is an independent property. Thus, both are correct but Reason does not explain Assertion.

Key Concept: Regression Line Equation, Mean Deviation

c) 67.5
[Solution Description]
The regression line equation for $y$ on $x$ is:
$y - \bar{y} = b_{yx}(x - \bar{x})$
Substitute the given values:
$y - 60 = 1.5(55 - 50)$
Calculate $(55 - 50)$:
$y - 60 = 1.5 \times 5 = 7.5$
Solve for $y$:
$y = 60 + 7.5 = 67.5$
The estimated value of $y$ is 67.5.

Key Concept: Correlation Coefficient Calculation

b) 0.41
[Solution Description]
First, calculate $\sum d_x d_y = (2)(3) + (-1)(-2) + (0)(1) + (3)(-1) + (-4)(-1) = 6 + 2 + 0 - 3 + 4 = 9$.
Next, calculate $\sum d_x^2 = 2^2 + (-1)^2 + 0^2 + 3^2 + (-4)^2 = 4 + 1 + 0 + 9 + 16 = 30$.
Then, $\sum d_y^2 = 3^2 + (-2)^2 + 1^2 + (-1)^2 + (-1)^2 = 9 + 4 + 1 + 1 + 1 = 16$.
Now, plug these into the formula for $r$:
$r = \frac{\sum d_x d_y}{\sqrt{\sum d_x^2 \sum d_y^2}} = \frac{9}{\sqrt{30 \times 16}} = \frac{9}{\sqrt{480}} \approx 0.41$

Key Concept: Scatter Diagram and Correlation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a downward slope in a scatter diagram indicates a negative correlation, which is true. The reason correctly explains that a negative correlation implies an inverse relationship where one variable increases while the other decreases, leading to the observed downward trend. Thus, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Regression line equation, Mean values

b) 6.5
[Solution Description]
The regression line of $Y$ on $X$ is given by:
$y - \bar{y} = b_{yx}(x - \bar{x})$
Since the line passes through $(5, 8)$, substitute $x = 5$ and $y = 8$:
$8 - \bar{y} = 1.5(5 - 4)$
$8 - \bar{y} = 1.5 \times 1 = 1.5$
Solving for $\bar{y}$:
$\bar{y} = 8 - 1.5 = 6.5$

Key Concept: Regression Coefficient Formula

b) $\frac{\sum d_x d_y}{\sum d_x^2}$
[Solution Description]
The regression coefficient of $y$ on $x$ ($b_{yx}$) using deviations from mean is calculated as:
$b_{yx} = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}$
where $d_x = x - \bar{x}$ and $d_y = y - \bar{y}$.

Key Concept: Regression Coefficients, Correlation

c) 0.9
[Solution Description]
First, calculate the covariance and variances:
$\text{Cov}(X, Y) = \frac{\sum xy}{n} - \left(\frac{\sum x}{n}\right)\left(\frac{\sum y}{n}\right) = \frac{600}{10} - (6)(9) = 60 - 54 = 6$
$\sigma_x^2 = \frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2 = \frac{500}{10} - 6^2 = 50 - 36 = 14$
$\sigma_y^2 = \frac{\sum y^2}{n} - \left(\frac{\sum y}{n}\right)^2 = \frac{900}{10} - 9^2 = 90 - 81 = 9$
We know $b_{yx} = r \frac{\sigma_y}{\sigma_x} = 0.6$. Solving for $r$:
$r = b_{yx} \frac{\sigma_x}{\sigma_y} = 0.6 \times \frac{\sqrt{14}}{3} \approx 0.748$
Now, find $b_{xy}$:
$b_{xy} = r \frac{\sigma_x}{\sigma_y} = 0.748 \times \frac{\sqrt{14}}{3} \approx 0.933$

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 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: 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: Correlation, Regression Properties

b) $12.19^\circ$
[Solution Description]
Substitute the given values into the formula for $\tan \theta$:
$\tan \theta = \left| \frac{1 - (0.8)^2}{0.8} \cdot \frac{4 \times 3}{4^2 + 3^2} \right|.$
Calculate $1 - r^2 = 1 - 0.64 = 0.36$ and $\sigma_x \sigma_y = 12$. Also, $\sigma_x^2 + \sigma_y^2 = 16 + 9 = 25$. Substituting these:
$\tan \theta = \left| \frac{0.36}{0.8} \cdot \frac{12}{25} \right| = \left| 0.45 \times 0.48 \right| = 0.216.$
Thus, $\theta = \tan^{-1}(0.216) \approx 12.19^\circ$.

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: Line of Best Fit

b) Moderate positive correlation
[Solution Description]
Plotting the points, we observe that as \textit{x} increases, \textit{y} generally increases but not in a perfectly linear fashion. The points do not lie exactly on a straight line, indicating some variability. Hence, there is a moderate positive correlation between \textit{x} and \textit{y}.

Key Concept: Correlation Types

c) -0.95
[Solution Description]
Correlation coefficients range from -1 to 1. A coefficient close to -1 indicates a strong negative correlation, meaning as one variable increases, the other decreases. Among the options, -0.95 is closest to -1.

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 Equations in Terms of Summations, Product Moment Formula

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To determine the correctness of the assertion and reason, let's evaluate them step-by-step:
1) The assertion states that the regression coefficient $b_{yx}$ calculated using deviations from the mean ($b_{yx} = r \frac{\sigma_y}{\sigma_x}$) is always equal to the slope obtained by solving the normal equations. This is true because both methods fundamentally estimate the same linear relationship between variables, minimizing the sum of squared errors.
2) The reason provides an explanation for this equality by stating that both methods minimize the sum of squared residuals. While this is correct, it is not the primary basis for why the two forms of $b_{yx}$ are equivalent. The equivalence arises due to algebraic transformations of the same underlying statistical principle, not merely because they both minimize residuals. Thus, the reason is true but does not correctly explain the assertion.
Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Normal Equations, Regression Lines

b) $y = -2x + 14$
[Solution Description]
The regression line of $y$ on $x$ is given by $y = mx + c$, where $m = b_{yx}$ and $c$ is the intercept. Using original values:
$b_{yx} = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$
Substituting the given values:
$b_{yx} = \frac{10 \times 200 - 40 \times 60}{10 \times 180 - (40)^2} = \frac{2000 - 2400}{1800 - 1600} = \frac{-400}{200} = -2$
Next, find $c$ using $\overline{y} = m \overline{x} + c$. Calculate $\overline{x}$ and $\overline{y}$:
$\overline{x} = \frac{\sum x}{n} = \frac{40}{10} = 4, \quad \overline{y} = \frac{\sum y}{n} = \frac{60}{10} = 6$
Substitute into the equation:
$6 = (-2)(4) + c \implies c = 6 + 8 = 14$
Thus, the regression line is $y = -2x + 14$.

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 $b_{yx}$ measures the change in $Y$ for a unit change in $X$, which is true as per the definition. The reason provides the formula for $b_{yx}$, which is also correct. Moreover, the reason correctly explains how the assertion is derived using the method of least squares. Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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: Scatter Diagram, Line of Best Fit

c) For every degree increase in temperature, ice cream sales increase by 1.5 units.
[Solution Description]
The slope of the line of best fit represents the rate of change in the dependent variable (ice cream sales) with respect to the independent variable (temperature). A positive slope of 1.5 indicates that for every unit increase in temperature, ice cream sales increase by 1.5 units, showing a positive correlation.

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: Line of Best Fit, Regression

b) 1
[Solution Description]
To find the slope $m$ of the line of best fit, we can use the formula:
$m = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}$
Here, $n = 4$, $\sum x = 10 + 20 + 30 + 40 = 100$, $\sum y = 15 + 25 + 35 + 45 = 120$, $\sum xy = (10 \times 15) + (20 \times 25) + (30 \times 35) + (40 \times 45) = 150 + 500 + 1050 + 1800 = 3500$, and $\sum x^2 = 100 + 400 + 900 + 1600 = 3000$.
Substituting these values:
$m = \frac{(4 \times 3500) - (100 \times 120)}{(4 \times 3000) - (100 \times 100)} = \frac{14000 - 12000}{12000 - 10000} = \frac{2000}{2000} = 1$
So, the slope is 1.

Key Concept: Line of Best Fit

b) 13
[Solution Description]
To find the predicted value of $y$, substitute $x = 5$ into the equation: $y = 2(5) + 3 = 10 + 3 = 13$.

Key Concept: Regression Line Calculation

c) $y = 2x$
[Solution Description]
To find the regression line of $y$ on $x$, use the formula:
$y - \bar{y} = r \frac{\sigma_y}{\sigma_x} (x - \bar{x})$
Substituting the given values:
$y - 20 = 0.8 \times \frac{5}{2} (x - 10)$
Simplifying the coefficient:
$y - 20 = 2 (x - 10)$
This simplifies to:
$y = 2x - 20 + 20 \implies y = 2x$
The regression line equation is $y = 2x$.

Key Concept: Scatter Diagram, Correlation

c) Positive correlation
[Solution Description]
The scatter diagram shows that as the independent variable (study hours) increases, the dependent variable (exam scores) also increases. This indicates a positive correlation between the two variables.

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: Correlation Strength and Direction

d) Strong positive correlation
[Solution Description]
The correlation coefficient \$r = +0.85\$ indicates a strong positive correlation because it is close to \$+1\$. A strong positive correlation means that as one variable increases, the other variable tends to increase as well.

Key Concept: Regression coefficients, Normal equations

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:
$\tan \theta = \left| \frac{1 - r^2}{r} \right| \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
When $r=0$, $\tan \theta$ becomes undefined because the denominator $r=0$. This implies that $\theta=\pi/2$, meaning the lines are perpendicular. Hence, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Regression Coefficients

c) 0.63
[Solution Description]
The correlation coefficient $r$ is given by the geometric mean of the regression coefficients:
$r = \sqrt{b_{yx} \cdot b_{xy}}$
Substituting the given values:
$r = \sqrt{0.8 \times 0.5} = \sqrt{0.4} \approx 0.632$
Thus, the correlation coefficient is approximately 0.632.

Key Concept: Line of Best Fit

c) 2
[Solution Description]
The equation of the line of best fit is given by $y = mx + c$, where $m$ is the slope. Comparing with the given equation $y = 2x + 3$, the slope $m$ is 2.

Key Concept: Regression coefficient of Y on X

c) The change in $Y$ for a unit change in $X$
[Solution Description]
The regression coefficient $b_{yx}$ represents the change in the dependent variable $Y$ for a unit change in the independent variable $X$. It indicates the slope of the regression line of $Y$ on $X$.

Key Concept: Normal Equations

b) $40 = 5c + 20m$ and $200 = 20c + 120m$
[Solution Description]
The normal equations for the regression line $y = mx + c$ are:
$\sum y = nc + m \sum x$
and
$\sum xy = c \sum x + m \sum x^2$
Substituting the given values, we get:
$40 = 5c + 20m$
and
$200 = 20c + 120m$

Key Concept: Types of Regression Lines

b) $y = mx + c$
[Solution Description]
The regression line of Y on X is represented as $y = mx + c$ where m is the slope and c is the intercept. This line is used to estimate Y based on X.

Key Concept: Key Differences

d) Line graph
[Solution Description]
A line graph connects data points with straight lines, making it suitable for displaying trends over continuous intervals. This aligns with its definition in the syllabus.

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 Calculation

c) $-0.75$
[Solution Description]
The slope $m$ of the line passing through points $(x_1, y_1) = (20, 40)$ and $(x_2, y_2) = (60, 10)$ can be calculated using:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 40}{60 - 20} = \frac{-30}{40} = -0.75$

Key Concept: Sign of Regression Coefficients

b) Both $b_{yx}$ and $b_{xy}$ are negative.
[Solution Description]
From Property 4, the correlation coefficient and the two regression coefficients have the same sign. If $r$ is negative, both $b_{yx}$ and $b_{xy}$ must also be negative since they share the same sign as $r$.

Key Concept: Correlation and Regression

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A perfect negative correlation ($r = -1$) implies that as one variable increases, the other decreases linearly. The Reason states the geometric mean relationship between $r$ and the regression coefficients, which holds true for all values of $r$, including $-1$. Both Assertion and Reason are correct, and Reason explains why $r = -1$ corresponds to a perfect negative relationship mathematically.

Key Concept: Regression Coefficients and Correlation

c) 0.735
[Solution Description]
The correlation coefficient \$r\$ is the geometric mean of \$b_{yx}\$ and \$b_{xy}\$:
$r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{0.6 \times 0.9} = \sqrt{0.54} \approx 0.735.$

Key Concept: Regression coefficient using deviations

b) 0.625
[Solution Description]
The regression coefficient of $y$ on $x$ is calculated using the formula:
$b_{yx} = \frac{\sum uv - n \bar{u} \bar{v}}{\sum u^2 - n \bar{u}^2}$
Substituting the given values:
$b_{yx} = \frac{80 - 5 \times 2 \times 3}{100 - 5 \times (2)^2}$
Simplifying numerator:
$80 - 30 = 50$
Simplifying denominator:
$100 - 20 = 80$
Hence,
$b_{yx} = \frac{50}{80} = 0.625$

Key Concept: Angle Between Regression Lines

c) 0.624
[Solution Description]
The angle between the regression lines is given by:
$\tan \theta = \frac{1 - r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2} = \frac{1 - 0.25}{0.5} \cdot \frac{3 \times 4}{3^2 + 4^2} = \frac{0.75}{0.5} \cdot \frac{12}{25} = 1.5 \times 0.48 = 0.72$
Therefore, $\theta = \arctan(0.72) \approx 0.624$ radians.

Key Concept: Correlation Interpretation

b) $-0.9$ to $-1$
[Solution Description]
A strong negative linear relationship implies that as $X$ increases, $Y$ decreases consistently. The correlation coefficient $r$ ranges from $-1$ to $+1$, where $-1$ indicates a perfect negative correlation. For a strong negative correlation, $r$ will be close to $- 1$, typically in the range $-0.7$ to $-1$.

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 Coefficient Relationship

b) 1.0667
[Solution Description]
The relationship between the regression coefficients and the correlation coefficient is given by:
$b_{yx} \cdot b_{xy} = r^2$
Substitute the known values:
$0.6 \cdot b_{xy} = 0.8^2$
$0.6 \cdot b_{xy} = 0.64$
Solve for $b_{xy}$:
$b_{xy} = \frac{0.64}{0.6} = 1.0667$

Key Concept: Regression Coefficients in Terms of r, Angle Between Regression Lines

c) $\frac{9}{13}$
[Solution Description]
Using the angle formula:
$\tan \theta = \frac{1 - r^2}{|r|} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
Step 1: Calculate numerator: $1 - r^2 = 1 - 0.25 = 0.75$
Step 2: Denominator part: $\sigma_x \sigma_y = 2 \times 3 = 6$
Step 3: Second denominator: $\sigma_x^2 + \sigma_y^2 = 4 + 9 = 13$
Step 4: Combine terms: $\tan \theta = \frac{0.75}{0.5} \times \frac{6}{13} = 1.5 \times \frac{6}{13} = \frac{9}{13}$

Key Concept: Regression coefficient calculation

b) 2
[Solution Description]
The regression coefficient $b_{yx}$ is calculated as:
$b_{yx} = \frac{\sum d_x d_y}{\sum d_x^2}$
Substituting the given values:
$b_{yx} = \frac{30}{15} = 2$

Key Concept: Regression coefficients, correlation

c) 1.4
[Solution Description]
First, calculate the required sums:
- $\sum x = 1 + 2 + 3 + 4 = 10$
- $\sum y = 2 + 3 + 5 + 6 = 16$
- $\sum xy = (1 \times 2) + (2 \times 3) + (3 \times 5) + (4 \times 6) = 2 + 6 + 15 + 24 = 47$
- $\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$
Now, use the formula for $b_{yx}$:
$b_{yx} = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} = \frac{4 \times 47 - 10 \times 16}{4 \times 30 - 100} = \frac{188 - 160}{120 - 100} = \frac{28}{20} = 1.4$

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: Correlation Strength

d) Strong negative correlation
[Solution Description]
A strong negative correlation occurs when one variable decreases as the other increases, and the relationship is very clear and pronounced. This results in a steep negative slope on the scatter diagram.

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: 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: Regression coefficient and correlation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To evaluate the assertion and reason:
1. The assertion states that if both regression coefficients ($b_{yx}$ and $b_{xy}$) are positive, then the correlation coefficient ($r$) must also be positive. This is true because:
- From Property 1, $r = \sqrt{b_{yx} \cdot b_{xy}}$.
- Since both $b_{yx}$ and $b_{xy}$ are positive, their product is positive.
- Thus, $r$ will indeed be positive when taking the square root of a positive number.
2. The reason given is that the correlation coefficient $r$ is the geometric mean of the two regression coefficients. This is also true as per Property 1 mentioned in the syllabus ($r = \sqrt{b_{yx} \cdot b_{xy}}$).
3. Moreover, the reason correctly explains why the assertion is true because it connects the positivity of $r$ to the positivity of $b_{yx}$ and $b_{xy}$ through the geometric mean relationship.

Key Concept: Angle Between Regression Lines

b) 35$^\circ$
[Solution Description]
The angle between regression lines is given by:
$\tan \theta = \left| \frac{1 - r^2}{r} \right| \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}$
Substituting $r = 0.5$, $\sigma_x = 2$, $\sigma_y = 3$:
$\tan \theta = \left| \frac{1 - 0.25}{0.5} \right| \cdot \frac{6}{4 + 9} = \frac{0.75}{0.5} \cdot \frac{6}{13} = 1.5 \cdot 0.4615 \approx 0.6923$
Thus, $\theta = \arctan(0.6923) \approx 34.7^\circ$.

Key Concept: Regression Coefficients, Correlation Coefficient

c) 0.95
[Solution Description]
The regression coefficients are $b_{yx} = 1.5$ and $b_{xy} = 0.6$. The correlation coefficient $r$ is given by:
$r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{1.5 \times 0.6} = \sqrt{0.9} \approx 0.9487.$

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 Coefficient

c) 1.5
[Solution Description]
The regression coefficient $b_{yx}$ is calculated using the formula:
$b_{yx} = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$
Substituting the given values:
$b_{yx} = \frac{(5)(90) - (15)(25)}{(5)(55) - (15)^2} = \frac{450 - 375}{275 - 225} = \frac{75}{50} = 1.5$

Key Concept: Regression Coefficient

c) 1.5
[Solution Description]
The regression coefficient $b_{yx}$ is calculated as:
$$ \\ b_{yx} = r \frac{\sigma_y}{\sigma_x} = 0.6 \times \frac{5}{2} = 1.5$
$

Key Concept: Regression Line, Correlation Coefficient

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the least squares regression line of $y$ on $x$ passes through the point $(\overline{x}, \overline{y})$, which is true as it is a fundamental property of the regression line. The reason claims that the correlation coefficient $r = 0$ if and only if both regression coefficients $b_{yx}$ and $b_{xy}$ are zero. This is also true because when $r = 0$, the covariance between $x$ and $y$ is zero, leading to both regression coefficients being zero. However, the reason does not explain why the regression line passes through $(\overline{x}, \overline{y})$; it merely states another property related to correlation and regression coefficients. Therefore, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Regression Line Properties

c) 1.0
[Solution Description]
The correlation coefficient is the geometric mean of the two regression coefficients. First, calculate the product: $b_{yx} \cdot b_{xy} = 1.5 \times 0.6667 = 1.00005$. Then, take the square root: $\sqrt{1.00005} \approx 1$. Thus, the correlation coefficient is approximately 1.

Key Concept: Normal equations

c) \$ \sum Y_i = n a + b_{yx} \sum X_i \$ and \$ \sum X_i Y_i = a \sum X_i + b_{yx} \sum X_i^2 \$
[Solution Description]
The normal equations for deriving the regression coefficient \$b_{yx}\$ are obtained by minimizing the sum of squared errors. These equations are:
$\sum Y_i = n a + b_{yx} \sum X_i$
$\sum X_i Y_i = a \sum X_i + b_{yx} \sum X_i^2$
Here, \$a\$ is the intercept term, \$n\$ is the number of observations, \$\sum X_i Y_i\$ is the sum of the product of \$X\$ and \$Y\$, \$\sum X_i\$ is the sum of \$X\$ values, and \$\sum X_i^2\$ is the sum of squared \$X\$ values.

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: 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: Scatter Diagram - Correlation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that an upward trend in a scatter diagram indicates a positive correlation, which is true. The reason explains the definition of a positive correlation correctly. Since the reason accurately explains the assertion, both are true and the reason justifies the assertion.

Key Concept: Scatter Plot, Correlation

a) Strong and positive correlation
[Solution Description]
In a scatter plot, if the points cluster around an upward-sloping line, it indicates that as one variable increases, the other also increases. This describes a positive correlation. Since the points are closely clustered, the correlation is strong.

Key Concept: Angle Between Regression Lines, Correlation Coefficient, Standard Deviation

a) $27^\circ$
[Solution Description]
To find the angle $\theta$ between the regression lines, we use the formula:
$\tan \theta = \left| \frac{1 - r^2}{r} \cdot \frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2} \right|$
Substituting the given values:
$\tan \theta = \left| \frac{1 - (0.6)^2}{0.6} \cdot \frac{4 \times 3}{4^2 + 3^2} \right| \\ = \left| \frac{1 - 0.36}{0.6} \cdot \frac{12}{16 + 9} \right| \\ = \left| \frac{0.64}{0.6} \cdot \frac{12}{25} \right| \\ = \left| 1.0667 \times 0.48 \right| \\ = 0.512$
Therefore, $\theta = \arctan(0.512) \approx 27^\circ$.

Key Concept: Correlation Coefficient

b) 0.9487
[Solution Description]
The correlation coefficient $r$ is the geometric mean of the regression coefficients:
$r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{0.6 \times 1.5} = \sqrt{0.9} = 0.9487$
Thus, $r = 0.9487$.

Key Concept: Regression Coefficients, Correlation Coefficient

d) 1.7
[Solution Description]
First calculate necessary sums:
n = 5
$\sum x$ = 1+2+3+4+5 = 15
$\sum y$ = 2+4+5+7+9 = 27
$\sum xy$ = (1×2)+(2×4)+(3×5)+(4×7)+(5×9) = 2+8+15+28+45 = 98
$\sum x^2$ = 1+4+9+16+25 = 55
Using formula for $b_{yx}$:
$b_{yx} = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}$
Numerator = 5×98 - 15×27 = 490 - 405 = 85
Denominator = 5×55 - 225 = 275 - 225 = 50
Therefore, $b_{yx} = \frac{85}{50} = 1.7$

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 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: Product Moment Formula

b) 0.632
[Solution Description]
The product moment correlation coefficient is calculated using:
$r = \frac{\Sigma d_x d_y}{\sqrt{\Sigma d_x^2 \cdot \Sigma d_y^2}}$
Substituting the given values:
$r = \frac{120}{\sqrt{200 \times 180}} = \frac{120}{\sqrt{36000}} = \frac{120}{189.74} \approx 0.632$
The correlation coefficient is approximately 0.632.

Key Concept: Regression Line

d) Perfect correlation
[Solution Description] When the regression lines coincide, it indicates a perfect correlation between the two variables. This means there is a direct linear relationship without any deviation.

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: Regression Lines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that both regression lines intersect at $(\bar{x}, \bar{y})$, which is true because this is a fundamental property of regression lines. The reason given is that the means satisfy both regression equations, which is also correct and directly explains why they intersect at this point. Substituting $x = \bar{x}$ in the regression equation of $y$ on $x$ ($y = mx + c$) gives $y = m\bar{x} + c$. But from the normal equations of regression, we know that $m\bar{x} + c = \bar{y}$. Similarly, substituting $y = \bar{y}$ in the regression equation of $x$ on $y$ ($x = my + c$) gives $x = m\bar{y} + c = \bar{x}$. Therefore, both lines pass through $(\bar{x}, \bar{y})$ and hence intersect there. Thus, both assertion and reason are true, and reason correctly explains the assertion.

Key Concept: Correlation and Line of Best Fit - Freehand Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the freehand method is subjective as it depends on the observer's judgment, which is true because this method involves visual approximation rather than a mathematical formula.
The Reason claims that different observers may draw different lines for the same data due to variability in perception, which is also true and directly explains why the freehand method is subjective.
Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Line of Best Fit

c) By visually adjusting a straight line closest to all points
[Solution Description]
The line of best fit can be approximated by visually adjusting a ruler’s edge to minimize the distance from all points or using mathematical methods like least squares regression.

Key Concept: Angle Between Two Lines of Regression

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When the correlation coefficient $r = 0$, the formula for the angle $\theta$ between the two lines of regression simplifies to $\tan \theta = \infty$, which implies $\theta = \frac{\pi}{2}$. This means the two regression lines are perpendicular to each other. The assertion states this fact correctly, and the reason provides the correct explanation for it, as per the given property in the syllabus.

Key Concept: Regression equations, correlation coefficient

c) 0.95
[Solution Description]
The correlation coefficient $r$ is calculated as the geometric mean of the two regression coefficients $b_{yx}$ and $b_{xy}$. From the given regression equations, we have:
Step 1: Identify $b_{yx}$ from the regression equation of $y$ on $x$. The equation is $y - 30 = 1.5(x - 20)$. Here, $b_{yx} = 1.5$.
Step 2: Identify $b_{xy}$ from the regression equation of $x$ on $y$. The equation is $x - 20 = 0.6(y - 30)$. Here, $b_{xy} = 0.6$.
Step 3: Compute the correlation coefficient using the formula $r = \sqrt{b_{yx} \cdot b_{xy}}$.
Substituting the values, $r = \sqrt{1.5 \times 0.6} = \sqrt{0.9} = 0.9487$.
Thus, the correlation coefficient is approximately 0.95.

Key Concept: Regression properties, Correlation coefficient

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, calculate the product of the regression coefficients:
$B_{yx} \times B_{xy} = 1.6 \times 0.4 = 0.64$.
Now, find the square root of this product to get the correlation coefficient:
$r = \sqrt{0.64} = 0.8$.
The reason states a correct property that the correlation coefficient is indeed the geometric mean of the regression coefficients. However, the assertion claims "must be 0.8" without considering other constraints. One important property is that if one regression coefficient is greater than 1, the other must be less than 1, which is satisfied here (1.6 > 1 and 0.4 r$, which we can verify: $\frac{1.6 + 0.4}{2} = 1 > 0.8$. All conditions are satisfied, so both assertion and reason are true, and the reason correctly explains the assertion.

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: Regression Coefficients, Correlation Coefficient

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
We are given:
$b_{yx} = 1.5$
$b_{xy} = 0.4$
According to Property 1, the correlation coefficient $r$ is the geometric mean of $b_{yx}$ and $b_{xy}$:
$r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{1.5 \times 0.4} = \sqrt{0.6}$
Thus, Assertion is true.
Reason states the correct property that $r$ is the geometric mean of the two regression coefficients, which directly explains Assertion. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is 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 $b_{xy}$. According to the property mentioned in the syllabus, this is true because:
$\sqrt{b_{yx} \cdot b_{xy}} = \sqrt{r \frac{\sigma_y}{\sigma_x} \cdot r \frac{\sigma_x}{\sigma_y}} = r.$
The reason provides the formulas for the regression coefficients, which are correct as per the syllabus. Moreover, the reason correctly explains why the assertion is true, as the geometric mean of $b_{yx}$ and $b_{xy}$ indeed simplifies to $r$.
Therefore, both the Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

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: 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 coefficients, Normal equations

b) 0.9
[Solution Description]
First, calculate the necessary sums:
$\sum x = 1 + 2 + 3 + 4 + 5 = 15$
$\sum y = 2 + 3 + 5 + 4 + 6 = 20$
$\sum xy = (1 \times 2) + (2 \times 3) + (3 \times 5) + (4 \times 4) + (5 \times 6) = 2 + 6 + 15 + 16 + 30 = 69$
$\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55$
Now, use the formula for $b_{yx}$:
$b_{yx} = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} = \frac{5 \times 69 - 15 \times 20}{5 \times 55 - 15^2} = \frac{345 - 300}{275 - 225} = \frac{45}{50} = 0.9$

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: Regression Equations Using Means

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the regression line of $Y$ on $X$ passes through the point $(\bar{x}, \bar{y})$. This is true because in the regression equation $y - \bar{y} = b_{yx} (x - \bar{x})$, if we substitute $x = \bar{x}$, then $x - \bar{x} = 0$, which simplifies to $y = \bar{y}$. Therefore, the line passes through $(\bar{x}, \bar{y})$.
The reason correctly explains the assertion by showing how substituting $x = \bar{x}$ into the regression equation leads to $y = \bar{y}$. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Deviations from Assumed Mean, Regression Equations Using Deviations from Assumed Mean

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify, let's consider the regression coefficient formula using actual means:
$b_{yx} = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}$
Substituting $u = x - A$ and $v = y - B$, where $A$ and $B$ are arbitrary constants (assumed means), we get:
$b_{yx} = \frac{\sum (u + A - \bar{u} - A)(v + B - \bar{v} - B)}{\sum (u + A - \bar{u} - A)^2} = \frac{\sum (u - \bar{u})(v - \bar{v})}{\sum (u - \bar{u})^2}$
This can also be expressed as:
$b_{yx} = \frac{\sum uv - n\bar{u}\bar{v}}{\sum u^2 - n\bar{u}^2} = \frac{\sum uv - \frac{(\sum u)(\sum v)}{n}}{\sum u^2 - \frac{(\sum u)^2}{n}}$
Hence, the assertion is true. The reason correctly explains that the formula derived using deviations from assumed mean is equivalent to the one using actual means, making both statements true and the reason a valid explanation of the assertion.

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: 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 least squares regression are derived by minimizing the sum of squared errors. They are:
$\sum y = nc + m \sum x$
$\sum xy = c \sum x + m \sum x^2$
These are the correct normal equations.

Key Concept: Deviations from Assumed Mean

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that taking deviations from assumed mean simplifies calculations when the actual mean is fractional. This is true because working with fractional means can complicate arithmetic operations. The reason given is that deviations from assumed avoid complex arithmetic means by using simpler formulas. This is also true, as using an assumed mean reduces computational complexity. Moreover, the reason correctly explains why the assertion is true, as it provides the justification for simplification through avoidance of complex arithmetic.

Key Concept: Regression Coefficients, Geometric Mean

b) 0.40
[Solution Description]
The coefficient of correlation ($r$) is the geometric mean of the two regression coefficients $b_{yx}$ and $b_{xy}$. The formula is:
$r = \sqrt{b_{yx} \cdot b_{xy}}$
Substituting the given values:
$r = \sqrt{0.64 \times 0.25} = \sqrt{0.16} = 0.4$
Therefore, the coefficient of correlation is 0.4.

Key Concept: Regression Equation Simplification

b) $x = 0.45y + 3.9$
[Solution Description]
First, calculate the regression coefficient $b_{xy}$:
$b_{xy} = r \frac{\sigma_x}{\sigma_y} = 0.9 \times \frac{3}{6} = 0.45$
Now, write the regression equation of $x$ on $y$:
$x - \bar{x} = b_{xy} (y - \bar{y})$
Substituting the values:
$x - 12 = 0.45 (y - 18)$
Simplify the equation:
$x = 0.45y - 8.1 + 12$
$x = 0.45y + 3.9$

Key Concept: Line of Best Fit

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 minimizes the sum of the squares of the vertical distances, which is correct as this is the fundamental principle of the least squares method. The reason is also true because the method of least squares indeed provides the best linear unbiased estimator (BLUE) for the coefficients under standard assumptions. Moreover, the reason correctly explains why the line of best fit is derived using the least squares method. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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: Regression Equations

b) 0.25
[Solution Description]
The regression coefficient $b_{yx} = r \frac{\sigma_y}{\sigma_x}$. Given $b_{yx} = 0.5$ and $\frac{\sigma_y}{\sigma_x} = 2$, substituting these values into the equation gives:
$0.5 = r \times 2 \implies r = \frac{0.5}{2} = 0.25$
Thus, the coefficient of correlation $r$ is $0.25$.

Key Concept: Regression Lines Pass Through Mean

c) 1.5
[Solution Description]
The regression line of $y$ on $x$ always passes through the mean $(\overline{x}, \overline{y})$. However, here we are given that it also passes through $(4, 5)$. The regression line equation is:
$y - \overline{y} = b_{yx}(x - \overline{x})$
Substitute the given points:
$5 - 5.3 = b_{yx}(4 - 4.2)$
$-0.3 = b_{yx}(-0.2)$
$b_{yx} = \frac{-0.3}{-0.2} = 1.5$
Therefore, the value of $b_{yx}$ is 1.5.

Key Concept: Properties of Regression and Correlation

d) The correlation coefficient and regression coefficients have the same sign.
[Solution Description]
According to properties:
- The correlation coefficient and regression coefficients must have the same sign.
- If one regression coefficient is greater than 1, the other must be less than 1.
- The arithmetic mean of regression coefficients is always greater than the correlation coefficient.
The correct statement from the options aligns with these properties.

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: 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: Equation of Regression Line

b) $y = -0.5x + 11$
[Solution Description]
Using the point-slope form of a line, the equation is given by:
$y - y_1 = m(x - x_1).$
Substituting the given values $(x_1, y_1) = (6, 8)$ and slope $m = -0.5$, we get:
$y - 8 = -0.5(x - 6).$
Simplifying this equation:
$y = -0.5x + 3 + 8 \Rightarrow y = -0.5x + 11.$

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: 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: 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: Line of Best Fit by Observation

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 minimizes the sum of the squares of the vertical distances from each data point to the line, which is correct as it defines the least squares method. The reason explains that the least squares method ensures the best possible linear approximation, which directly supports the assertion. Therefore, both are true, and the reason correctly explains the assertion.

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: Correlation, Scatter Diagram

d) Weak negative correlation
[Solution Description]
A scatter diagram with points widely scattered around a downward-sloping line indicates a weak negative correlation. The downward slope shows the negative relationship, while the wide scatter indicates weakness in the correlation.

Key Concept: Regression lines and Correlation

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that as the angle between two regression lines increases, the correlation between the variables 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 but does not correctly explain why the angle affects the correlation. Instead, the angle is related to the correlation but not in a direct inverse proportionality relationship.

Key Concept: Regression Equations Using Means

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states the correct form of the regression equation of $y$ on $x$, which includes the means $\bar{y}$ and $\bar{x}$. The Reason provides the correct formula for the regression coefficient $b_{yx}$, which is derived from the correlation coefficient and standard deviations. Since the Reason correctly explains how the coefficient in the Assertion's equation is calculated, both Assertion and Reason are true, and Reason is the correct explanation of the Assertion.

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: Line of best fit, Slope calculation

b) Decrease by 3 units
[Solution Description]
The slope $m$ represents the change in y per unit change in x. Here, $m = -0.75$, meaning for every 1-unit increase in x, y decreases by 0.75 units. Therefore, for an increase of 4 units in x:
$\Delta y = m \times \Delta x = -0.75 \times 4 = -3$
The y-variable is expected to decrease by 3 units.

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 Line

a) $m = \frac{5}{9}, c = \frac{10}{3}$
[Solution Description]
Using the normal equations:
$$ \\ \Sigma y = nc + m \Sigma x \\ \\ 50 = 10c + 30m \quad \text{(1)}$
$
$$ \\ \Sigma xy = c \Sigma x + m \Sigma x^2 \\ \\ 200 = 30c + 180m \quad \text{(2)}$
$
Solve equation (1) for $c$: $c = 5 - 3m$.
Substitute $c$ in equation (2):
$$ \\ 200 = 30(5 - 3m) + 180m \\ \\ 200 = 150 - 90m + 180m \\ \\ 200 = 150 + 90m \\ \\ 50 = 90m \\ \\ m = \frac{5}{9}$
$
Now substitute $m$ back into equation for $c$: $c = 5 - 3 \times \frac{5}{9} = 5 - \frac{5}{3} = \frac{10}{3}$.

Key Concept: Point of Intersection of Regression Lines

b) $(\bar{x}, \bar{y})$
[Solution Description]
Both regression lines, $X$ on $Y$ and $Y$ on $X$, pass through the point $(\bar{x}, \bar{y})$, where $\bar{x}$ is the mean of the $x$-series and $\bar{y}$ is the mean of the $y$-series.

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 Lines, Correlation

d) 25
[Solution Description]
Since both regression lines pass through the point $(\bar{x}, \bar{y})$, we can solve the two equations simultaneously to find $\bar{y}$.
Given:
1. $y = 2x + 3$ (regression of $y$ on $x$)
2. $x = 0.4y + 1$ (regression of $x$ on $y$)
Substitute equation 2 into equation 1:
$y = 2(0.4y + 1) + 3$
Simplify:
$y = 0.8y + 2 + 3$
$y - 0.8y = 5$
$0.2y = 5$
$y = \frac{5}{0.2} = 25$
Thus, $\bar{y} = 25$.

Key Concept: Regression Line Property

c) $(\bar{x}, \bar{y})$
[Solution Description]
One of the key properties of regression lines is that they both pass through the point $(\bar{x}, \bar{y})$, where $\bar{x}$ is the mean of the x-values and $\bar{y}$ is the mean of the y-values.

Key Concept: Relationship with correlation coefficient

d) It equals 0
[Solution Description]
The regression coefficient \$b_{yx}\$ is related to the correlation coefficient \$r\$ by the formula:
$b_{yx} = r \frac{\sigma_y}{\sigma_x}$
where \$\sigma_x\$ and \$\sigma_y\$ are the standard deviations of \$X\$ and \$Y\$, respectively. If \$r = 0\$, then \$b_{yx} = 0\$. This indicates that there is no linear relationship between \$X\$ and \$Y\$.

Key Concept: Regression coefficients and correlation coefficient

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us analyze both statements carefully.
1. Assertion states that if $b_{yx}$ is negative and $b_{xy}$ is positive, then $r$ must be zero.
We know from the property that $r^2 = b_{yx} \cdot b_{xy}$. If one coefficient is negative and the other is positive, their product will indeed be negative ($-ve \times +ve = -ve$). However, $r^2$ cannot be negative since it's a squared quantity. The only way this equation holds is if $r^2 = 0$, meaning $r = 0$. So Assertion is true.
2. Reason states $b_{yx} \cdot b_{xy} = r^2$, which is a fundamental property of regression coefficients and correlation coefficient. This is absolutely true.
3. Now, does Reason correctly explain Assertion? Yes, because it's precisely this property that leads us to conclude $r$ must be zero when the product of regression coefficients is negative. Therefore, Reason is the correct explanation of Assertion.
Thus both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

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: Correlation

d) Strong and positive
[Solution Description]
In a scatter plot, if the points show an upward trend from left to right, it means that as one variable increases, the other also increases. This indicates a positive correlation.

Key Concept: Correlation interpretation, Line of Best Fit

d) This is expected due to the subjective nature of the freehand method
[Solution Description]
The freehand method is subjective as it relies on individual judgment to estimate the line that appears closest to all points. Since different observers may weigh certain data points differently or perceive the overall trend slightly differently, they may draw lines with varying slopes.

Key Concept: Regression Lines

b) $0.32$
[Solution Description]
The correlation coefficient $r$ can be calculated using the regression coefficients $b_{yx}$ and $b_{xy}$. First, note that $b_{yx} = r\left(\frac{\sigma_y}{\sigma_x}\right)$. Rearranging:
$r = b_{yx} \left(\frac{\sigma_x}{\sigma_y}\right)$
Given $b_{yx} = 0.8$, $\sigma_x = 2$, and $\sigma_y = 5$:
$r = 0.8 \left(\frac{2}{5}\right) = 0.8 \times 0.4 = 0.32$

Key Concept: Scatter Diagram Basics

b) Positive correlation
[Solution Description]
A scatter diagram shows positive correlation when both variables increase together. The data points trend upwards from left to right.

Key Concept: Line of Best Fit, Correlation Coefficient

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion claims that the line of best fit minimizes the sum of the squared horizontal distances, but this is false. The least squares method actually minimizes the sum of the squared vertical distances (errors) between observed and predicted values. The Reason correctly states that the least squares method minimizes vertical deviations, which is true. Therefore, the Assertion is false, but the Reason is true.

Key Concept: Regression

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the least squares method minimizes the sum of squared vertical deviations from the regression line, which is correct. The reason given is that this ensures the line fits the data points as closely as possible, which is also true and correctly explains why the least squares method is used. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

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: 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: Normal Equations

b) $\sum y = nc + m \sum x$ and $\sum xy = c \sum x + m \sum x^2$
[Solution Description]
The normal equations for the least squares line $y = mx + c$ are derived by minimizing the sum of squared residuals. They are:
$\sum y = nc + m \sum x$
$\sum xy = c \sum x + m \sum x^2$

Key Concept: Correlation and Regression Lines

c) 0.63
[Solution Description]
The correlation coefficient $r$ is the geometric mean of $b_{yx}$ and $b_{xy}$:
$r = \sqrt{b_{yx} \cdot b_{xy}} = \sqrt{0.8 \times 0.5} = \sqrt{0.4} \approx 0.632$
Thus, $r \approx 0.63$.

Key Concept: Basic Concept

c) $90^\circ$
[Solution Description]
When the correlation coefficient $r = 0$, the two regression lines are perpendicular to each other. Hence, the angle $\theta$ between them is $90^\circ$.

Key Concept: Perfect Correlation

c) The regression lines coincide.
[Solution Description]
When the correlation coefficient $r = \pm 1$, it indicates perfect correlation between the variables. In this case, the two regression lines ($y$ on $x$ and $x$ on $y$) coincide into a single line.

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: 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.