Maharashtra Board Class 12 Maths Part 2 Chapter 3 Linear Regression Miscellaneous Solutions

(I) Choose the correct alternative.

Question 1. Regression analysis is the theory of
(a) Estimation
(b) Prediction
(c) Both a and b
(d) Calculation
Answer: (c) Both a and b
In simple words: Regression analysis helps in estimating and predicting values of one variable based on another, encompassing both estimation and prediction.

🎯 Exam Tip: Remember that regression is fundamentally about both estimation and prediction in statistical analysis.

Question 2. We can estimate the value of one variable with the help of other known variable only if they are
(a) Correlated
(b) Positively correlated
(c) Negatively correlated
(d) Uncorrelated
Answer: (a) Correlated
In simple words: For regression analysis to be effective, the variables must have some degree of correlation; otherwise, one cannot reliably predict the other.

🎯 Exam Tip: A fundamental requirement for regression analysis is that the variables involved must be correlated. If they are uncorrelated, no linear relationship can be established for estimation or prediction.

Question 3. There are _______ types of regression equation
(a) 4
(b) 2
(c) 3
(d) 1
Answer: (b) 2
In simple words: There are two main types of linear regression equations: regression of Y on X and regression of X on Y.

🎯 Exam Tip: Recall the two primary regression lines: Y on X and X on Y. This is a common theoretical concept.

Question 4. In the regression equation of Y on X
(a) X is independent and Y is dependent
(b) Y is independent and X is dependent
(c) Both X and Y are independent
(d) Both X and Y are dependent.
Answer: (a) X is independent and Y is dependent
In simple words: When we regress Y on X, X is considered the predictor (independent variable) and Y is the outcome (dependent variable).

🎯 Exam Tip: Understanding which variable is independent and which is dependent is crucial for correctly interpreting a regression equation (Y on X means X is independent, Y is dependent).

Question 5. In the regression equation of X on Y
(a) X is independent and Y is dependent
(b) Y is independent and X is dependent
(c) Both X and Y are independent
(d) Both X and Y are dependent
Answer: (b) Y is independent and X is dependent
In simple words: When we regress X on Y, Y is the predictor (independent variable) and X is the outcome (dependent variable).

🎯 Exam Tip: For the regression of X on Y, Y acts as the explanatory variable, and X is the variable being explained or predicted.

Question 6. \(b_{xy}\) is __________
(a) Regression coefficient of Y on X
(b) Regression coefficient of X on Y
(c) Correlation coefficient between X and Y
(d) Covariance between X and Y
Answer: (b) Regression coefficient of X on Y
In simple words: \(b_{xy}\) represents the change in X for a unit change in Y, hence it is the regression coefficient of X on Y.

🎯 Exam Tip: Distinguish between \(b_{xy}\) and \(b_{yx}\) carefully; the subscripts indicate which variable is dependent on which.

Question 7. \(b_{yx}\) is __________
(a) Regression coefficient of Y on X
(b) Regression coefficient of X on Y
(c) Correlation coefficient between X and Y
(d) Covariance between X and Y
Answer: (a) Regression coefficient of Y on X
In simple words: \(b_{yx}\) measures the expected change in Y for a unit change in X, making it the regression coefficient of Y on X.

🎯 Exam Tip: Knowing the notation \(b_{yx}\) for the regression coefficient of Y on X is fundamental for solving regression problems.

Question 8. 'r' is __________
(a) Regression coefficient of Y on X
(b) Regression coefficient of X on Y
(c) Correlation coefficient between X and Y
(d) Covariance between X and Y
Answer: (c) Correlation coefficient between X and Y
In simple words: 'r' is the symbol for the correlation coefficient, which quantifies the strength and direction of a linear relationship between two variables.

🎯 Exam Tip: The symbol 'r' is universally used to denote the Pearson product-moment correlation coefficient, indicating the linear association between two variables.

Question 9. \(b_{xy} \cdot b_{yx}\) = __________
(a) v
(b) yx
(c) \(r^2\)
(d) \((yy)^2\)
Answer: (c) \(r^2\)
In simple words: The product of the two regression coefficients is equal to the square of the correlation coefficient.

🎯 Exam Tip: This is a key property linking correlation and regression coefficients, often used to find 'r' if both regression coefficients are known.

Question 10. If \(b_{yx} > 1\) then \(b_{xy}\) is __________
(a) > 1
(b) < 1
(c) > 0
(d) < 0
Answer: (b) < 1
In simple words: Since the product \(b_{yx} \cdot b_{xy} = r^2\) and \(r^2 \le 1\), if one regression coefficient is greater than 1, the other must be less than 1 (assuming positive coefficients).

🎯 Exam Tip: Remember the property \(0 \le r^2 \le 1\), which implies that if one regression coefficient is greater than 1, the other must be less than 1 (for non-negative coefficients, which is implied by \(b_{yx} > 1\)).

Question 11. \(|b_{xy} + b_{yx}| \geq\) __________
(a) r
(b) 2|r|
(c) r
(d) 2r
Answer: (b) 2|r|
In simple words: This inequality is a property derived from the relationship between regression coefficients and the correlation coefficient, stating that the absolute sum is greater than or equal to twice the absolute correlation coefficient.

🎯 Exam Tip: This inequality, \(|b_{xy} + b_{yx}| \geq 2|r|\), is an important theoretical property in regression analysis and should be memorized.

Question 12. \(b_{xy}\) and \(b_{yx}\) are __________
(a) Independent of change of origin and scale
(b) Independent of change of origin but not of the scale
(c) Independent of change of scale but not of origin
(d) Affected by change of origin and scale
Answer: (b) Independent of change of origin but not of the scale
In simple words: Regression coefficients do not change if you add or subtract a constant from the variables (change of origin), but they do change if you multiply or divide by a constant (change of scale).

🎯 Exam Tip: This is a key property of regression coefficients: they are independent of origin but dependent on scale changes, affecting how they are calculated and interpreted.

Question 13. If \(u = \frac{x-a}{c}\) and \(v = \frac{y-b}{d}\) then \(b_{yx}\) = __________
(a) \(\frac{d}{c} b_{vu}\)
(b) \(\frac{c}{d} b_{vu}\)
(c) \(\frac{d}{b} b_{vu}\)
(d) \(\frac{c}{a} b_{vu}\)
Answer: (a) \(\frac{d}{c} b_{vu}\)
In simple words: When variables are transformed by changing both origin and scale, the new regression coefficient \(b_{vu}\) can be related back to the original \(b_{yx}\) using the scale factors c and d.

🎯 Exam Tip: Understanding the impact of change of origin and scale on regression coefficients is crucial for simplifying calculations. The formula \(b_{yx} = \frac{d}{c} b_{vu}\) is vital.

Question 14. If \(u = \frac{x-a}{c}\) and \(v = \frac{y-b}{d}\) then \(b_{xy}\) = __________
(a) \(\frac{d}{c} b_{uv}\)
(b) \(\frac{c}{d} b_{uv}\)
(c) \(\frac{b}{a} b_{uv}\)
(d) \(\frac{a}{d} b_{uv}\)
Answer: (b) \(\frac{c}{d} b_{uv}\)
In simple words: Similar to \(b_{yx}\), the regression coefficient \(b_{xy}\) can also be expressed in terms of the transformed variables' coefficient \(b_{uv}\) using their respective scale factors.

🎯 Exam Tip: Pay close attention to the order of scale factors (c, d) in the numerator and denominator when converting between \(b_{xy}\) and \(b_{uv}\) to avoid errors.

Question 15. Corr(x, x) = __________
(a) 0
(b) 1
(c) -1
(d) can't be found
Answer: (b) 1
In simple words: The correlation of a variable with itself is always perfect and positive, meaning it is 1.

🎯 Exam Tip: The correlation coefficient of any variable with itself is always +1, indicating perfect positive linear relationship.

Question 16. Corr (x, y) = __________
(a) corr(x, x)
(b) corr(y, y)
(c) corr(y, x)
(d) cov(y, x)
Answer: (c) corr(y, x)
In simple words: The correlation between X and Y is the same as the correlation between Y and X, as correlation is a symmetric measure.

🎯 Exam Tip: Correlation is a symmetric measure; the correlation of X with Y is identical to the correlation of Y with X.

Question 17. Corr\((\frac{x-a}{c}, \frac{y-b}{d})\) = -corr(x, y) if,
(a) c and d are opposite in sign
(b) c and d are same in sign
(c) a and b are opposite in sign
(d) a and b are same in sign
Answer: (a) c and d are opposite in sign
In simple words: The sign of the correlation coefficient changes if the scale factors c and d have opposite signs, indicating a reversal in the direction of the relationship.

🎯 Exam Tip: The sign of the correlation coefficient is affected by the signs of the scale factors (c, d) in the transformation. If their signs are opposite, the correlation sign flips.

Question 18. Regression equation of X and Y is
(a) \(y - \bar{y} = b_{yx} (x - \bar{x})\)
(b) \(x - \bar{x} = b_{xy} (y - \bar{y})\)
(c) \(y - \bar{y} = b_{xy} (x - \bar{x})\)
(d) \(x - \bar{x} = b_{yx} (y - \bar{y})\)
Answer: (b) \(x - \bar{x} = b_{xy} (y - \bar{y})\)
In simple words: This is the standard form of the regression equation of X on Y, where X is the dependent variable predicted by Y.

🎯 Exam Tip: Ensure you correctly associate the regression coefficient \(b_{xy}\) with the equation of X on Y, where X is the dependent variable.

Question 19. Regression equation of Y and X is
(a) \(y - \bar{y} = b_{yx} (x - \bar{x})\)
(b) \(x - \bar{x} = b_{xy} (y - \bar{y})\)
(c) \(y - \bar{y} = b_{xy} (x - \bar{x})\)
(d) \(x - \bar{x} = b_{yx} (y - \bar{y})\)
Solution:
Answer: (a) \(y - \bar{y} = b_{yx} (x - \bar{x})\)
In simple words: This is the standard form of the regression equation of Y on X, where Y is the dependent variable predicted by X.

🎯 Exam Tip: It is crucial to remember the correct formula for the regression line of Y on X, which involves \(b_{yx}\) and predicts Y based on X.

Question 20. \(b_{yx} =\) __________
(a) \(r \frac{\sigma_x}{\sigma_y}\)
(b) \(r \frac{\sigma_y}{\sigma_x}\)
(c) \(\frac{1}{r} \frac{\sigma_y}{\sigma_x}\)
(d) \(\frac{1}{r} \frac{\sigma_x}{\sigma_y}\)
Answer: (b) \(r \frac{\sigma_y}{\sigma_x}\)
In simple words: The regression coefficient of Y on X is calculated by multiplying the correlation coefficient 'r' by the ratio of the standard deviation of Y to the standard deviation of X.

🎯 Exam Tip: This formula connects the regression coefficient, correlation coefficient, and standard deviations, making it useful when all these statistics are given.

Question 21. \(b_{xy} =\) __________
(a) \(r \frac{\sigma_x}{\sigma_y}\)
(b) \(r \frac{\sigma_y}{\sigma_x}\)
(c) \(\frac{1}{r} \frac{\sigma_y}{\sigma_x}\)
(d) \(\frac{1}{r} \frac{\sigma_x}{\sigma_y}\)
Answer: (a) \(r \frac{\sigma_x}{\sigma_y}\)
In simple words: The regression coefficient of X on Y is found by multiplying the correlation coefficient 'r' by the ratio of the standard deviation of X to the standard deviation of Y.

🎯 Exam Tip: Be careful not to confuse the formulas for \(b_{yx}\) and \(b_{xy}\). The numerator's standard deviation always corresponds to the dependent variable.

Question 22. Cov (x, y) = __________
(a) \(\sum(x - \bar{x})(y - \bar{y})\)
(b) \(\frac{\sum(x - \bar{x})(y - \bar{y})}{n}\)
(c) \(\frac{\sum xy}{n} - \bar{x}\bar{y}\)
(d) b and c both
Answer: (d) b and c both
In simple words: Covariance measures how much two variables change together, and it can be calculated using either the sum of products of deviations from means divided by n, or using the formula involving the mean of products and the product of means.

🎯 Exam Tip: Both formulas for covariance are valid and can be used depending on the available data. It's important to recognize both forms.

Question 23. If \(b_{xy} < 0\) and \(b_{yx} < 0\) then 'r' is __________
(a) > 0
(b) < 0
(c) > 1
(d) not found
Answer: (b) < 0
In simple words: The correlation coefficient 'r' always has the same sign as the regression coefficients; if both \(b_{xy}\) and \(b_{yx}\) are negative, 'r' must also be negative.

🎯 Exam Tip: A key property is that the correlation coefficient 'r' always shares the same sign as both regression coefficients (\(b_{xy}\) and \(b_{yx}\)).

Question 24. If equation of regression lines are \(3x + 2y - 26 = 0\) and \(6x + y - 31 = 0\) then means of x and y are __________
(a) (7, 4)
(b) (4, 7)
(c) (2, 9)
(d) (-4, 7)
Answer: (b) (4, 7)
In simple words: The mean values of x and y (\(\bar{x}\) and \(\bar{y}\)) correspond to the point of intersection of the two regression lines. By solving the system of equations, we find these mean values.

🎯 Exam Tip: The intersection point of the two regression lines always yields the mean values of X and Y (\(\bar{x}\) and \(\bar{y}\)). Solve the simultaneous equations to find these values.

(II) Fill in the blanks:

Question 1. If \(b_{xy} < 0\) and \(b_{yx} < 0\) then 'r' is __________
Answer: negative
In simple words: The sign of the correlation coefficient is determined by the sign of the regression coefficients; if they are both negative, 'r' is also negative.

🎯 Exam Tip: Remember the rule: the sign of 'r' is always the same as the sign of \(b_{xy}\) and \(b_{yx}\).

Question 2. Regression equation of Y on X is __________
Answer: \((y - \bar{y}) = b_{yx} (x - \bar{x})\)
In simple words: This is the formula used to predict the value of Y based on the value of X, using the mean values and the regression coefficient of Y on X.

🎯 Exam Tip: Be sure to recall the standard form of the regression equation of Y on X, which passes through the means of X and Y.

Question 3. Regression equation of X on Y is __________
Answer: \((x - \bar{x}) = b_{xy} (y - \bar{y})\)
In simple words: This equation allows us to predict the value of X based on the value of Y, using their respective means and the regression coefficient of X on Y.

🎯 Exam Tip: Correctly identifying the formula for the regression equation of X on Y is vital for accurate predictions and understanding the relationship between variables.

Question 4. There are __________ types of regression equations.
Answer: 2
In simple words: There are two distinct types of linear regression equations: one for predicting Y from X, and another for predicting X from Y.

🎯 Exam Tip: Always remember that for two variables, there are two regression lines: Y on X and X on Y.

Question 5. Corr \((x_1 - x)\) = __________
Answer: -1
In simple words: The correlation between a variable and its negative (or its deviation from a fixed point if the sign flips) would be perfectly negative, meaning -1.

🎯 Exam Tip: The correlation of X with \(-X\) (or similar transformations that effectively reverse the relationship direction) is -1, signifying perfect inverse correlation.

Question 6. If \(u = \frac{x-a}{c}\) and \(v = \frac{y-b}{d}\) then \(b_{xy}\) = __________
Answer: \(\frac{c}{d} b_{uv}\)
In simple words: When transforming variables X and Y into U and V using scale factors c and d, the regression coefficient \(b_{xy}\) relates to \(b_{uv}\) through these scale factors.

🎯 Exam Tip: Master the relationship between regression coefficients and scale factors for transformed variables, remembering the ratio of scale factors \(c/d\) for \(b_{xy}\).

Question 7. If \(u = \frac{x-a}{c}\) and \(v = \frac{y-b}{d}\) then \(b_{yx}\) = __________
Answer: \(\frac{d}{c} b_{vu}\)
In simple words: Similarly, the regression coefficient \(b_{yx}\) is related to \(b_{vu}\) by the inverse ratio of the scale factors c and d.

🎯 Exam Tip: Ensure you use the correct ratio of scale factors, \(d/c\), when deriving \(b_{yx}\) from \(b_{vu}\) after variable transformation.

Question 8. \(|b_{xy} + b_{yx}| \geq\) __________
Answer: 2|r|
In simple words: This is a known inequality in regression analysis that defines a lower bound for the absolute sum of the regression coefficients in terms of the correlation coefficient.

🎯 Exam Tip: This inequality is a standard theoretical result. It is important to know that the absolute sum of regression coefficients is always greater than or equal to twice the absolute correlation coefficient.

Question 9. If \(b_{yx} > 1\) then \(b_{xy}\) is __________
Answer: < 1
In simple words: Given that the product of regression coefficients, \(b_{yx} \cdot b_{xy}\), equals \(r^2\) (which is always between 0 and 1), if one coefficient is greater than 1, the other must necessarily be less than 1.

🎯 Exam Tip: This property is a direct consequence of \(b_{yx} \cdot b_{xy} = r^2\) and \(|r| \leq 1\). It helps check the consistency of given regression coefficients.

Question 10. \(b_{xy} \cdot b_{yx}\) = __________
Answer: \(r^2\)
In simple words: The geometric mean of the two regression coefficients is the absolute value of the correlation coefficient, meaning their product equals the square of the correlation coefficient.

🎯 Exam Tip: This fundamental identity is critical for understanding the relationship between correlation and regression and is frequently tested.

(III) State whether each of the following is True or False.

Question 1. Corr (x, x) = 1.
Answer: True
In simple words: A variable is perfectly and positively correlated with itself.

🎯 Exam Tip: The correlation of any variable with itself is always 1, indicating a perfect positive relationship.

Question 2. Regression equation of X on Y is \(y - \bar{y} = b_{xy} (x - \bar{x})\).
Answer: False
In simple words: The regression equation of X on Y should be \(x - \bar{x} = b_{xy} (y - \bar{y})\). The given equation is for Y on X, but incorrectly uses \(b_{xy}\).

🎯 Exam Tip: Carefully distinguish between the regression equations of Y on X and X on Y. The dependent variable and the corresponding regression coefficient must match.

Question 3. Regression equation of Y on X is \(y - \bar{y} = b_{yx} (x - \bar{x})\).
Answer: True
In simple words: This is the correct standard form for the regression equation predicting Y from X.

🎯 Exam Tip: This formula is a fundamental concept; memorize it for correctly setting up the regression line of Y on X.

Question 4. Corr (x, y) = Corr (y, x).
Answer: True
In simple words: The correlation coefficient is a symmetric measure, meaning the relationship between X and Y is the same as between Y and X.

🎯 Exam Tip: Correlation is a symmetric relationship; switching the order of variables does not change its value.

Question 5. \(b_{xy}\) and \(b_{yx}\) are independent of change of origin and scale.
Answer: False
In simple words: Regression coefficients are independent of change of origin but are affected by changes in scale.

🎯 Exam Tip: Remember that regression coefficients are scale-dependent, unlike the correlation coefficient which is scale-independent. They are both origin-independent.

Question 6. 'r' is the regression coefficient of Y on X.
Answer: False
In simple words: 'r' is the correlation coefficient, not a regression coefficient. \(b_{yx}\) is the regression coefficient of Y on X.

🎯 Exam Tip: Do not confuse the correlation coefficient 'r' with the regression coefficients \(b_{yx}\) or \(b_{xy}\). They measure different aspects of the relationship.

Question 7. \(b_{yx}\) is the correlation coefficient between X and Y.
Answer: False
In simple words: \(b_{yx}\) is the regression coefficient of Y on X, while 'r' is the correlation coefficient.

🎯 Exam Tip: Clearly differentiate between the regression coefficient \(b_{yx}\) and the correlation coefficient 'r'; they are distinct statistical measures.

Question 8. If \(u = x - a\) and \(v = y - b\) then \(b_{xy} = b_{uv}\).
Answer: True
In simple words: Since \(u\) and \(v\) are formed by only changing the origin (subtracting constants a and b, respectively) without changing the scale, the regression coefficient remains the same.

🎯 Exam Tip: Regression coefficients are invariant under change of origin (adding or subtracting a constant), meaning \(b_{xy}\) equals \(b_{uv}\) if only origin is shifted.

Question 9. If \(u = x - a\) and \(v = y - b\) then \(r_{xy} = r_{uv}\).
Answer: True
In simple words: The correlation coefficient is independent of change of origin and change of scale, so shifting the origin (subtracting constants) does not affect it.

🎯 Exam Tip: The correlation coefficient is robust to linear transformations, meaning it is unaffected by changes in origin or scale.

Question 10. In the regression equation of Y on X, \(b_{yx}\) represents the slope of the line.
Answer: True
In simple words: The regression coefficient \(b_{yx}\) quantifies the rate of change in Y for every unit change in X, which is precisely what the slope of a line represents.

🎯 Exam Tip: The regression coefficient \(b_{yx}\) is indeed the slope of the regression line of Y on X, indicating the steepness and direction of the relationship.

(IV) Solve the following problems.

Question 1. The data obtained on X, the length of time in weeks that a promotional project has been in progress at a small business, and Y the percentage increase in weekly sales over the period just prior to the beginning of the campaign.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह तालिका एक प्रचार परियोजना की प्रगति (X, सप्ताह में) और साप्ताहिक बिक्री में प्रतिशत वृद्धि (Y) के बीच संबंध दर्शाती है। इसमें प्रत्येक अवलोकन के लिए X और Y के मान, साथ ही गणना में उपयोग के लिए परिवर्तित चर u (x-3) और v (y-15) तथा उनके गुणनफल (u.v) और u के वर्ग (\(u^2\)) को दिखाया गया है। तालिका के अंत में सभी स्तंभों का योग भी दिया गया है।

Find the equation of regression line to predict percentage increase in sales if the company has been in progress for 1.5 weeks.
Solution:
Let \(u = x - 3, v = y - 15\)
\(\bar{x} = \frac{\sum x}{n} = \frac{30}{12} = 2.5\)
\(\bar{y} = \frac{\sum y}{n} = \frac{176}{12} = 14.67\)
\(b_{yx} = b_{vu} = \frac{n \sum (uv) - (\sum u)(\sum v)}{n \sum u^2 - (\sum u)^2}\)
\( = \frac{12(41) - (-6)(-4)}{12(18) - (-6)^2}\)
\( = \frac{492 - 24}{216 - 36}\)
\( = \frac{468}{180}\)
\( = 2.6\)
\(\therefore\) Regression equation of Y on X is
\((y - \bar{y}) = b_{yx} (x - \bar{x})\)
\((y - 14.67) = 2.6(x - 2.5)\)
\(y - 14.67 = 2.6x - 6.5\)
\(y = 2.6x + 8.17\)
When \(x = 1.5\)
\(y = (2.6)(1.5) + 8.17\)
\( = 3.9 + 8.17\)
\( = 12.07\)
In simple words: We calculated the regression equation of Y on X by transforming variables to simplify computations, then used this equation to predict sales increase for a project in progress for 1.5 weeks.

🎯 Exam Tip: When given raw data, consider transforming variables (changing origin/scale) to simplify calculations for the regression coefficients before forming the regression equation. Remember to substitute the predicted X value back into the final Y on X equation.

Question 2. The regression equation of y on x is given by \(3x + 2y - 26 = 0\). Find \(b_{yx}\).
Solution:
Given, regression equation of Y on X is
\(3x + 2y - 26 = 0\)
\(\therefore 2y = -3x + 26\)
\(y = -\frac{3}{2}x + 13\)
\(\therefore b_{yx} = -\frac{3}{2}\)
In simple words: To find \(b_{yx}\) from the given linear equation, we rearrange it into the slope-intercept form (\(y = mx + c\)), where the slope 'm' is \(b_{yx}\).

🎯 Exam Tip: To find \(b_{yx}\) from a linear equation of the form \(Ax + By + C = 0\), always isolate Y on one side (i.e., express Y in terms of X) and the coefficient of X will be \(b_{yx}\).

Question 3. If for a bivariate data \(\bar{x} = 10, \bar{y} = 12, v(x) = 9, \sigma_y = 4\) and \(r = 0.6\). Estimate y when \(x = 5\).
Solution:
Given, \(V(x) = 9\)
\(\therefore \sigma_x = \sqrt{9} = 3\)
\(b_{yx} = r \frac{\sigma_y}{\sigma_x}\)
\( = 0.6 \times \frac{4}{3}\)
\( = 0.8\)
\(\therefore\) Regression equation of Y on X is
\((y - \bar{y}) = b_{yx} (x - \bar{x})\)
\((y - 12) = 0.8(5 - 10)\)
\(y - 12 = 0.8(-5)\)
\(y - 12 = -4\)
\(y = 8\)
In simple words: We first calculated the standard deviation of X and then \(b_{yx}\) using the given 'r' and standard deviations. With \(b_{yx}\) and the means, we formed the regression equation of Y on X and predicted Y for \(x=5\).

🎯 Exam Tip: When asked to estimate a value, first calculate the required regression coefficient using the provided statistical measures (like r, standard deviations), then construct the relevant regression equation using the means.

Question 4. The equation of the line of regression of y on x is \(y = \frac{2}{9}x\) and x on y is \(x = \frac{1}{2}y + \frac{7}{6}\).
Find (i) r (ii) \(\sigma_x^2\) if \(\sigma_y^2 = 4\).
Solution:
(i) Regression equation of Y on X is \(y = \frac{2}{9}x\)
\(\therefore b_{yx} = \frac{2}{9}\)
Regression equation of X on Y is \(x = \frac{1}{2}y + \frac{7}{6}\)
\(\therefore b_{xy} = \frac{1}{2}\)
\(r^2 = b_{yx} \cdot b_{xy}\)
\( = \frac{2}{9} \times \frac{1}{2}\)
\( = \frac{1}{9}\)
\(r = \pm\sqrt{\frac{1}{9}}\)
\(r = \pm\frac{1}{3}\)
Since \(b_{yx}\) and \(b_{xy}\) are positive, \(\therefore r = \frac{1}{3}\)
(ii) \(\sigma_y^2 = 4 \implies \sigma_y = 2\)
\(b_{yx} = r \frac{\sigma_y}{\sigma_x}\)
\(\frac{2}{9} = \frac{1}{3} \times \frac{2}{\sigma_x}\)
\(\frac{2}{9} = \frac{2}{3\sigma_x}\)
\(3\sigma_x = 9\)
\(\sigma_x = 3\)
\(\therefore \sigma_x^2 = 3^2 = 9\)
In simple words: We first extracted \(b_{yx}\) and \(b_{xy}\) from the given equations, calculated \(r^2\) and then 'r'. Using the formula for \(b_{yx}\) and the given \(\sigma_y^2\), we then found \(\sigma_x\), and thus \(\sigma_x^2\).

🎯 Exam Tip: Ensure that the sign of 'r' matches the signs of \(b_{yx}\) and \(b_{xy}\). When calculating \(\sigma_x\), verify the correct use of the \(b_{yx}\) formula with 'r' and standard deviations.

Question 5. Identify the regression equations of x on y and y on x from the following equations.
\(2x + 3y = 6\) and \(5x + 7y - 12 = 0\)
Solution:
Assume \(2x + 3y = 6\) is Y on X equation.
\(3y = -2x + 6\)
\(y = -\frac{2}{3}x + 2\)
\(\therefore b_{yx} = -\frac{2}{3}\)
Assume \(5x + 7y - 12 = 0\) is X on Y equation.
\(5x = -7y + 12\)
\(x = -\frac{7}{5}y + \frac{12}{5}\)
\(\therefore b_{xy} = -\frac{7}{5}\)
Now check the condition \(r^2 = b_{yx} \cdot b_{xy}\) and \(0 \le r^2 \le 1\)
\(b_{yx} \cdot b_{xy} = (-\frac{2}{3}) \times (-\frac{7}{5})\)
\( = \frac{14}{15}\)
Since \(\frac{14}{15} \in [0, 1]\)
\(\therefore\) Our assumption is correct
\(\therefore\) Regression equation of Y on X is \(2x + 3y = 6\)
\(\therefore\) Regression equation of X on Y is \(5x + 7y - 12 = 0\)
In simple words: We assumed which equation represented Y on X and X on Y, calculated their respective regression coefficients, and then verified our assumption using the condition that the product of the coefficients (\(r^2\)) must lie between 0 and 1.

🎯 Exam Tip: When given two regression lines and asked to identify them, calculate \(b_{yx}\) and \(b_{xy}\) for both possible assignments. The correct assignment will result in \(0 \le b_{yx} \cdot b_{xy} \le 1\).

Question 6. (i) If for a bivariate data \(b_{yx} = -1.2\) and \(b_{xy} = -0.3\) then find r.
(ii) From the two regression equations \(y = 4x - 5\) and \(3x = 2y + 5\), find \(\bar{x}\) and \(\bar{y}\).
Solution:
(i) \(r^2 = b_{yx} \cdot b_{xy}\)
\(r^2 = (-1.2) \times (-0.3)\)
\(r^2 = 0.36\)
\(r = \pm\sqrt{0.36}\)
\(r = \pm0.6\)
Since, \(b_{yx}\) and \(b_{xy}\) are negative, \(r = -0.6\)
(ii) Also, \((\bar{x}, \bar{y})\) is the point of intersection of the regression lines
Given equations:
\(y = 4x - 5 \implies 4x - y = 5\) (Equation 1)
\(3x = 2y + 5 \implies 3x - 2y = 5\) (Equation 2)
Multiply Equation 1 by 2: \(8x - 2y = 10\) (Equation 3)
Subtract Equation 2 from Equation 3:
\((8x - 2y) - (3x - 2y) = 10 - 5\)
\(5x = 5\)
\(x = 1\)
Substituting \(x = 1\) in Equation 1:
\(y = 4(1) - 5\)
\(y = 4 - 5\)
\(y = -1\)
\(\therefore \bar{x} = 1, \bar{y} = -1\)
In simple words: We first found 'r' using the product of the given regression coefficients, ensuring its sign matches theirs. Then, we solved the system of the two regression equations to find their intersection point, which gives the mean values of X and Y.

🎯 Exam Tip: Remember to assign the correct sign to 'r' based on the signs of the regression coefficients. For finding means, treat the regression equations as simultaneous linear equations and solve for x and y.

Question 7. The equation of the two lines of regression are \(3x + 2y - 26 = 0\) and \(6x + y - 31 = 0\). Find
(i) Means of X and Y
(ii) Correlation coefficient between X on Y
(iii) Estimate of Y for X = 2
(iv) var (X) if var (Y) = 36
Solution:
(i) Since \((\bar{x}, \bar{y})\) is the point of intersection of regression lines
\(3x + 2y = 26\) .......(i)
\(6x + y = 31\) .......(ii)
Multiply equation (ii) by 2: \(12x + 2y = 62\) ........(iii)
Subtract equation (i) from equation (iii):
\((12x + 2y) - (3x + 2y) = 62 - 26\)
\(9x = 36\)
\(x = 4\)
Substituting \(x = 4\) in equation (i):
\(3(4) + 2y = 26\)
\(12 + 2y = 26\)
\(2y = 14\)
\(y = 7\)
\(\therefore \bar{x} = 4, \bar{y} = 7\)
(ii) From \(3x + 2y - 26 = 0\)
\(2y = -3x + 26\)
\(y = -\frac{3}{2}x + 13\)
\(b_{yx} = -\frac{3}{2}\)
From \(6x + y - 31 = 0\)
\(y = -6x + 31\)
In this case, if this is Y on X, \(b_{yx}' = -6\). If it is X on Y, \(x = -\frac{1}{6}y + \frac{31}{6}\), so \(b_{xy}' = -\frac{1}{6}\).
Let's assume the first equation is Y on X and the second is X on Y:
\(b_{yx} = -\frac{3}{2}\) from \(3x+2y-26=0\)
\(x = -\frac{1}{6}y + \frac{31}{6}\) from \(6x+y-31=0\), so \(b_{xy} = -\frac{1}{6}\)
Check \(r^2 = b_{yx} \cdot b_{xy} = (-\frac{3}{2}) \times (-\frac{1}{6}) = \frac{3}{12} = \frac{1}{4}\)
Since \(r^2 = \frac{1}{4} \in [0, 1]\), this assumption is correct.
\(\therefore r = \pm\sqrt{\frac{1}{4}} = \pm\frac{1}{2}\)
Since \(b_{yx}\) and \(b_{xy}\) are negative, \(r = -0.5\)
(iii) Regression equation of Y on X is
\((y - \bar{y}) = b_{yx} (x - \bar{x})\)
\((y - 7) = -\frac{3}{2} (x - 4)\)
For \(x = 2\):
\((y - 7) = -\frac{3}{2} (2 - 4)\)
\((y - 7) = -\frac{3}{2} (-2)\)
\((y - 7) = 3\)
\(y = 10\)
(iv) Var (Y) = 36
\(\therefore \sigma_y = \sqrt{36} = 6\)
We have \(b_{yx} = r \frac{\sigma_y}{\sigma_x}\)
\(-\frac{3}{2} = (-0.5) \frac{6}{\sigma_x}\)
\(-\frac{3}{2} = -\frac{1}{2} \frac{6}{\sigma_x}\)
\(-\frac{3}{2} = -\frac{3}{\sigma_x}\)
\(3\sigma_x = 6\)
\(\sigma_x = 2\)
\(\therefore \text{Var(X)} = \sigma_x^2 = 2^2 = 4\)
In simple words: We found the means by solving the simultaneous regression equations. Then, we identified the correct regression coefficients and calculated the correlation coefficient 'r'. Finally, we used the Y on X equation to estimate Y for a given X, and derived \(\text{Var(X)}\) using the formula for \(b_{yx}\) and the given \(\text{Var(Y)}\).

🎯 Exam Tip: This question tests multiple concepts: finding means from intersection, identifying regression lines, calculating 'r', using the regression equation for estimation, and finding variance using \(b_{yx}\) formula. Practice each step thoroughly.

Question 8. Find the line of regression of X on Y for the following data:
n = 8, ∑(xᵢ - x)- = 36, Σ(yᵢ - y)- = 44, Σ(xᵢ - x)(yᵢ - y) = 24
Solution:
\( b_{xy} = \frac{\sum(x - \bar{x}) (y - \bar{y})}{\sum(y - \bar{y})^2} \)
\( = \frac{24}{44} \)
\( = \frac{6}{11} \)
Regression equation of X on Y is
\( (x - \bar{x}) = b_{xy} (y - \bar{y}) \)
\( (x - \bar{x}) = \frac{6}{11} (y - \bar{y}) \)
In simple words: This problem requires calculating the regression coefficient of X on Y, \( b_{xy} \), using the given sums of squares and products of deviations from means. Once \( b_{xy} \) is found, it is used in the standard regression equation formula to express X in terms of Y.

🎯 Exam Tip: Remember to correctly identify which regression coefficient is needed (X on Y or Y on X) and use the appropriate formula. Pay close attention to the numerator and denominator sums of squares and products.

Question 9. Find the equation of line regression of Y on X for the following data:
n = 8, ∑(xᵢ - x)(yᵢ - ỹ) = 120, x = 20, y = 36, σx = 2, σy = 3.
Solution:
\( b_{yx} = \frac{Cov(x, y)}{\sigma_x^2} \)
\( b_{yx} = \frac{(\frac{\sum(x - \bar{x})(y - \bar{y})}{n})}{\sigma_x^2} \)
\( = \frac{\frac{120}{8}}{2^2} \)
\( = \frac{15}{4} \)
\( = 3.75 \)
Regression equation of Y on X is
\( (y - \bar{y}) = b_{yx} (x - \bar{x}) \)
\( (y - 36) = 3.75(x - 20) \)
\( y - 36 = 3.75x - 75 \)
\( y = 3.75x - 39 \)
In simple words: To find the regression equation of Y on X, we first calculate the regression coefficient \( b_{yx} \) using the covariance and variance of X. Then, we substitute this coefficient along with the means into the point-slope form of the regression line equation.

🎯 Exam Tip: Ensure you use the correct variance (in this case, variance of X for \( b_{yx} \)) in the denominator when calculating the regression coefficient. Double-check your arithmetic when substituting values into the equation.

Question 10. The following result was obtained from records of age (X) and systolic blood pressure (Y) of a group of 10 men.

and ∑(xᵢ - x)(yᵢ - x) = 1120. Find the Prediction of blood pressure of a man of age 40 years.
Solution:
Given, n = 10, \( \bar{x} = 50, \bar{y} = 140, \sigma_x^2 = 160, \sigma_y^2 = 165 \)
\( b_{yx} = \frac{Cov(x, y)}{\sigma_x^2} \)
\( b_{yx} = \frac{(\frac{\sum(x - \bar{x})(y - \bar{y})}{n})}{\sigma_x^2} \)
\( = \frac{\frac{1120}{10}}{160} \)
\( = \frac{112}{160} \)
\( = 0.7 \)
Regression equation of Y on X is
\( (y - \bar{y}) = b_{yx} (x - \bar{x}) \)
\( (y - 140) = 0.7(x - 50) \)
For a man of age X = 40 years:
\( y - 140 = 0.7(40 - 50) \)
\( y - 140 = 0.7(-10) \)
\( y - 140 = -7 \)
\( \implies y = 133 \)
In simple words: We first calculate the regression coefficient of Y on X using the given covariance and variance of X. Then, we use the regression equation to predict the systolic blood pressure (Y) for a man with an age (X) of 40 years.

🎯 Exam Tip: When predicting a value, ensure you use the correct regression equation (Y on X if predicting Y, X on Y if predicting X). Always clearly state your knowns and the final predicted value.

Question 11. The equations of two regression lines are 10x - 4y = 80 and 10y - 9x = -40 Find:
(i) x and ỹ
(ii) byx and bxy
(iii) If var(Y) = 36, obtain var(X)
(iv) r
Solution:
(i) Since (x, ỹ) is the point of intersection of regression
10x - 4y = 80 ......(i)
-9x + 10y = -40 ........(ii)
Multiply (i) by 5: 50x - 20y = 400
Multiply (ii) by 2: -18x + 20y = -80
Adding the new equations:
(50x - 20y) + (-18x + 20y) = 400 - 80
32x = 320
x = 10
Substituting x = 10 in equation (i)
10(10) - 4y = 80
100 - 4y = 80
-4y = 80 - 100
-4y = -20
y = 5
\( \implies \bar{x} = 10, \bar{y} = 5 \)
(ii) From 10x - 4y = 80 \( \implies \) 10x = 4y + 80 \( \implies x = \frac{4}{10}y + \frac{80}{10} \)
\( \implies x = \frac{2}{5}y + 8 \)
Comparing with \( x - \bar{x} = b_{xy}(y - \bar{y}) \), we get \( b_{xy} = \frac{2}{5} = 0.4 \)

From 10y - 9x = -40 \( \implies \) 10y = 9x - 40 \( \implies y = \frac{9}{10}x - \frac{40}{10} \)
\( \implies y = \frac{9}{10}x - 4 \)
Comparing with \( y - \bar{y} = b_{yx}(x - \bar{x}) \), we get \( b_{yx} = \frac{9}{10} = 0.9 \)
Our assumption is correct since \( b_{yx} \cdot b_{xy} = 0.9 \cdot 0.4 = 0.36 \in [0, 1] \).
(iii) Given var(Y) = 36 \( \implies \sigma_y = 6 \)
We know \( b_{yx} = r \frac{\sigma_y}{\sigma_x} \)
And \( r^2 = b_{yx} \cdot b_{xy} \)
\( r^2 = 0.9 \cdot 0.4 = 0.36 \)
\( r = \sqrt{0.36} = \pm 0.6 \)
Since \( b_{yx} \) and \( b_{xy} \) are positive, r must be positive. So, \( r = 0.6 \)
Now, using \( b_{yx} = r \frac{\sigma_y}{\sigma_x} \):
\( 0.9 = 0.6 \cdot \frac{6}{\sigma_x} \)
\( 0.9 \sigma_x = 3.6 \)
\( \sigma_x = \frac{3.6}{0.9} = 4 \)
var(X) = \( \sigma_x^2 = 4^2 = 16 \)
(iv) r = 0.6 (Calculated in part iii)
In simple words: This problem involves finding the means of X and Y by solving the system of two regression equations. Then, the regression coefficients \( b_{yx} \) and \( b_{xy} \) are derived from these equations. Finally, using the variance of Y and the calculated regression coefficients, the variance of X and the correlation coefficient 'r' are determined.

🎯 Exam Tip: When given two regression lines, the point of intersection gives the means (\( \bar{x}, \bar{y} \)). Be careful to correctly identify which equation is for Y on X and which is for X on Y to extract the correct regression coefficients. Remember \( r^2 = b_{yx} \cdot b_{xy} \) and its value must be between 0 and 1.

Question 12. If byx = -0.6 and bxy = -0.216 then find correlation coefficient between X and Y comment on it.
Solution:
\( r^2 = b_{yx} \cdot b_{xy} \)
\( r^2 = (-0.6) \times (-0.216) \)
\( r^2 = 0.1296 \)
\( r = \pm \sqrt{0.1296} \)
\( r = \pm 0.36 \)
Since \( b_{yx} \) and \( b_{xy} \) are negative, 'r' must also be negative.
\( r = -0.36 \)
In simple words: The correlation coefficient 'r' is found by taking the square root of the product of the two regression coefficients. Since both regression coefficients are negative, the correlation coefficient must also be negative.

🎯 Exam Tip: The sign of the correlation coefficient 'r' must always match the sign of the regression coefficients (\( b_{yx} \) and \( b_{xy} \)). If \( b_{yx} \cdot b_{xy} \) results in a negative value, it indicates a calculation error, as \( r^2 \) cannot be negative.