Samacheer Kalvi Class 11 Business Maths Solutions Chapter 9 Correlation and Regression Analysis Ex 9.3

Question 1. An example of a positive correlation is:
(a) Income and expenditure
(b) Price and demand
(c) Repayment period and EMI
(d) Weight and Income
Answer: (a) Income and expenditure
Income and expenditure usually move in the same direction; as your income goes up, your spending tends to go up too. This shows a direct relationship where one variable increases with the other.
In simple words: When a person earns more money, they usually spend more. This is a positive link where both things rise together.

🎯 Exam Tip: Remember, a positive correlation means two variables increase or decrease together, while a negative correlation means one increases as the other decreases.

Question 2. If the values of two variables move in the same direction then the correlation is said to be:
(a) Negative
(b) Positive
(c) Perfect positive
(d) No correlation
Answer: (b) Positive
When two variables, like height and weight in children, tend to increase together, they are said to have a positive correlation. This indicates a direct relationship between their movements.
In simple words: If two things go up together or down together, they have a positive correlation. They move in the same way.

🎯 Exam Tip: Visualizing a scatter plot can help: positive correlation points generally slope upwards from left to right.

Question 3. If the values of two variables move in the opposite direction then the correlation is said to be:
(a) Negative
(b) Positive
(c) Perfect positive
(d) No correlation
Answer: (a) Negative
A negative correlation occurs when one variable's values increase while the other's decrease, showing an inverse relationship. For example, as the price of a product increases, its demand often decreases.
In simple words: If one thing goes up while the other goes down, they have a negative correlation. They move against each other.

🎯 Exam Tip: Think of speed and travel time: faster speed means less travel time, a classic example of negative correlation.

Question 4. Correlation co-efficient lies between:
(a) 0 to \( \infty \)
(b) -1 to +1
(c) -1 to 0
(d) -1 to \( \infty \)
Answer: (b) -1 to +1
The correlation coefficient, often denoted by 'r', always falls within the range of -1 and +1, inclusive. A value of +1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 means no linear correlation.
In simple words: The number that tells us how two things are linked is always between -1 and +1. It can never be smaller or bigger than these numbers.

🎯 Exam Tip: Values close to 1 or -1 show a strong relationship, while values near 0 show a weak or no linear relationship.

Question 5. If r(X, Y) = 0 the variables X and Y are said to be:
(a) Positive correlation
(b) Negative correlation
(c) No correlation
(d) Perfect positive correlation
Answer: (c) No correlation
When the correlation coefficient \( r \) is 0, it indicates that there is no linear relationship between the variables X and Y. This means knowing the value of one variable does not help predict the value of the other in a straight-line manner.
In simple words: If the correlation number is zero, it means the two things are not related in a simple, straight way. They move independently.

🎯 Exam Tip: A correlation of 0 only indicates no *linear* relationship; there might still be a non-linear relationship between the variables.

Question 6. The correlation coefficient from the following data N = 25, \( \Sigma X = 125 \), \( \Sigma Y = 100 \), \( \Sigma X^2 = 650 \), \( \Sigma Y^2 = 436 \), \( \Sigma XY = 520 \):
(a) 0.667
(b) -0.006
(c) -0.667
(d) 0.70
Answer: (a) 0.667
To find the correlation coefficient \( r \), we use the formula: \[ r = \frac{N\Sigma XY - (\Sigma X)(\Sigma Y)}{\sqrt{N\Sigma X^2 - (\Sigma X)^2} \sqrt{N\Sigma Y^2 - (\Sigma Y)^2}} \] Now, substitute the given values into the formula: \[ r = \frac{25(520) - 125 \times 100}{\sqrt{25 \times 650 - (125)^2} \sqrt{25 \times 436 - (100)^2}} \] First, calculate the numerator: \( 25 \times 520 = 13000 \) \( 125 \times 100 = 12500 \) Numerator \( = 13000 - 12500 = 500 \) Next, calculate the first part of the denominator: \( 25 \times 650 = 16250 \) \( (125)^2 = 15625 \) \( 16250 - 15625 = 625 \) \( \sqrt{625} = 25 \) Then, calculate the second part of the denominator: \( 25 \times 436 = 10900 \) \( (100)^2 = 10000 \) \( 10900 - 10000 = 900 \) \( \sqrt{900} = 30 \) Now, combine the denominator parts: Denominator \( = 25 \times 30 = 750 \) Finally, calculate \( r \): \( r = \frac{500}{750} = \frac{2}{3} \approx 0.667 \) The calculated correlation coefficient is approximately 0.667. This positive value suggests a moderate to strong positive linear relationship between X and Y.
In simple words: We use a special formula to find how X and Y are connected. We put all the given numbers into the formula and do the math step by step. The final answer, 0.667, shows a good positive connection.

🎯 Exam Tip: Always double-check your calculations, especially the squares and square roots, as small errors can lead to a completely different correlation coefficient.

Question 7. From the following data, N = 11, \( \Sigma X = 117 \), \( \Sigma Y = 260 \), \( \Sigma X^2 = 1313 \), \( \Sigma Y^2 = 6580 \), \( \Sigma XY = 2827 \). the correlation coefficient is:
(a) 0.3566
(b) -0.3566
(c) 0
(d) 0.4566
Answer: (a) 0.3566
To find the correlation coefficient \( r \), we use the same formula as before: \[ r = \frac{N\Sigma XY - (\Sigma X)(\Sigma Y)}{\sqrt{N\Sigma X^2 - (\Sigma X)^2} \sqrt{N\Sigma Y^2 - (\Sigma Y)^2}} \] Substitute the given values into the formula: \[ r = \frac{11 \times 2827 - 117 \times 260}{\sqrt{11 \times 1313 - (117)^2} \sqrt{11 \times 6580 - (260)^2}} \] First, calculate the numerator: \( 11 \times 2827 = 31097 \) \( 117 \times 260 = 30420 \) Numerator \( = 31097 - 30420 = 677 \) Next, calculate the first part of the denominator: \( 11 \times 1313 = 14443 \) \( (117)^2 = 13689 \) \( 14443 - 13689 = 754 \) \( \sqrt{754} \approx 27.459 \) Then, calculate the second part of the denominator: \( 11 \times 6580 = 72380 \) \( (260)^2 = 67600 \) \( 72380 - 67600 = 4780 \) \( \sqrt{4780} \approx 69.137 \) Now, multiply the denominator parts: Denominator \( = \sqrt{754} \times \sqrt{4780} \approx 27.459 \times 69.137 \approx 1898.45 \) Finally, calculate \( r \): \( r = \frac{677}{1898.45} \approx 0.3566 \) The correlation coefficient is approximately 0.3566. This positive value indicates a weak to moderate positive linear relationship between X and Y.
In simple words: We use the correlation formula with the given numbers to find the relationship between X and Y. After doing all the steps carefully, we get 0.3566, which shows a positive but not very strong connection.

🎯 Exam Tip: Pay close attention to the order of operations and make sure to square values before subtracting, especially in the denominator, to avoid errors.

Question 8. The correlation coefficient is:
(a) \( r(X,Y) = \frac{\sigma_X \sigma_Y}{\mathrm{cov}(x,y)} \)
(b) \( r(X,Y) = \frac{\mathrm{cov}(x, y)}{\sigma_X \sigma_Y} \)
(c) \( r(X,Y) = \frac{\mathrm{cov}(x, y)}{\sigma_Y} \)
(d) \( r(X,Y) = \frac{\mathrm{cov}(x, y)}{\sigma_X} \)
Answer: (b) \( r(X,Y) = \frac{\mathrm{cov}(x, y)}{\sigma_X \sigma_Y} \)
The correlation coefficient \( r(X,Y) \) is defined as the covariance between X and Y divided by the product of their standard deviations. This formula normalizes the covariance, making it a dimensionless measure between -1 and 1. It helps understand the strength and direction of the linear relationship.
In simple words: The correlation coefficient is found by dividing the covariance (how X and Y change together) by the standard deviation of X multiplied by the standard deviation of Y.

🎯 Exam Tip: Remember this fundamental formula, as it links covariance and standard deviation to the correlation coefficient, showing how they define the relationship.

Question 9. The variable whose value is influenced or is to be predicted is called:
(a) dependent variable
(b) independent variable
(c) regressor
(d) explanatory variable
Answer: (a) dependent variable
In statistics, the dependent variable is the one that is being measured or tested in an experiment. Its value is thought to depend on, or be influenced by, changes in the independent variable. This variable is usually plotted on the y-axis of a graph.
In simple words: The variable that changes because of another variable, or the one we try to guess, is called the dependent variable. It "depends" on something else.

🎯 Exam Tip: Think of "cause and effect": the independent variable is the "cause," and the dependent variable is the "effect" that we observe.

Question 10. The variable which influences the values or is used for prediction is called:
(a) Dependent variable
(b) Independent variable
(c) Explained variable
(d) Regressed
Answer: (b) Independent variable
The independent variable is the one that is changed or controlled in an experiment. It is assumed to cause a change in the dependent variable and is often plotted on the x-axis. It is also known as a predictor or explanatory variable.
In simple words: The variable that we change or that causes a change in another variable is called the independent variable. It acts on its own.

🎯 Exam Tip: Distinguish clearly between dependent (effect) and independent (cause) variables, as this is crucial for setting up any regression analysis correctly.

Question 11. The correlation coefficient:
(a) \( r = \pm \sqrt{b_{xy} \times b_{yx}} \)
(b) \( r = \frac{1}{b_{xy} \times b_{yx}} \)
(c) \( r = b_{xy} \times b_{yx} \)
(d) \( r = \pm \sqrt{\frac{1}{b_{xy} \times b_{yx}}} \)
Answer: (a) \( r = \pm \sqrt{b_{xy} \times b_{yx}} \)
The correlation coefficient \( r \) can be found as the geometric mean of the two regression coefficients, \( b_{xy} \) (regression of X on Y) and \( b_{yx} \) (regression of Y on X). The sign of \( r \) is the same as the sign of the regression coefficients. This relationship is important for checking consistency between correlation and regression.
In simple words: The correlation coefficient is the square root of multiplying the two regression coefficients together. You add a plus or minus sign, matching the sign of the regression coefficients.

🎯 Exam Tip: Remember that \( b_{xy} \) and \( b_{yx} \) must always have the same sign. If they don't, there's a calculation error.

Question 12. The regression coefficient of X on Y:
(a) \( b_{xy} = \frac{N\Sigma dx dy - (\Sigma dx)(\Sigma dy)}{N \Sigma dy^2 - (\Sigma dy)^2} \)
(b) \( b_{yx} = \frac{N\Sigma dx dy - (\Sigma dx)(\Sigma dy)}{N \Sigma dy^2 - (\Sigma dy)^2} \)
(c) \( b_{xy} = \frac{N\Sigma dx dy - (\Sigma dx)(\Sigma dy)}{N \Sigma dx^2 - (\Sigma dx)^2} \)
(d) \( b_y = \frac{N\Sigma xy - (\Sigma x)(\Sigma y)}{\sqrt{N\Sigma x^2 - (\Sigma x)^2} \times \sqrt{N\Sigma y^2 - (\Sigma y)^2}} \)
Answer: (a) \( b_{x y}=\frac{\mathrm{N} \Sigma d x d y-(\Sigma d x)(\Sigma d y)}{\mathrm{N} \Sigma d y^{2}-(\Sigma d y)^{2}} \)
The regression coefficient of X on Y, denoted as \( b_{xy} \), measures the average change in X for a unit change in Y. The denominator uses the sum of squares of Y deviations because Y is the independent variable when predicting X. This helps in understanding how Y influences X.
In simple words: When we want to see how X changes for every unit change in Y, we use this specific formula for \( b_{xy} \). The bottom part of the formula uses the values related to Y.

🎯 Exam Tip: For \( b_{xy} \), the denominator always contains the sums of squares for the *independent* variable (Y in this case), while the numerator is common for both regression coefficients.

Question 13. The regression coefficient of Y on X:
(a) \( b_{xy} = \frac{N\Sigma dx dy - (\Sigma dx)(\Sigma dy)}{N \Sigma dy^2 - (\Sigma dy)^2} \)
(b) \( b_{yx} = \frac{N\Sigma dx dy - (\Sigma dx)(\Sigma dy)}{N \Sigma dy^2 - (\Sigma dy)^2} \)
(c) \( b_{yx} = \frac{N\Sigma dx dy - (\Sigma dx)(\Sigma dy)}{N \Sigma dx^2 - (\Sigma dx)^2} \)
(d) \( b_{xy} = \frac{N\Sigma xy - (\Sigma x)(\Sigma y)}{\sqrt{N\Sigma x^2 - (\Sigma x)^2} \times \sqrt{N\Sigma y^2 - (\Sigma y)^2}} \)
Answer: (c) \( b_{y x}=\frac{\mathrm{N\Sigma} d x d y-(\Sigma d x)(\Sigma d y)}{\mathrm{N} \Sigma d x^{2}-(\Sigma d x)^{2}} \)
The regression coefficient of Y on X, denoted as \( b_{yx} \), measures the average change in Y for a unit change in X. In this formula, the denominator uses the sum of squares of X deviations because X is the independent variable when predicting Y. This helps us predict Y's value based on X.
In simple words: When we want to see how Y changes for every unit change in X, we use this specific formula for \( b_{yx} \). The bottom part of the formula uses the values related to X.

🎯 Exam Tip: Notice the difference in the denominator between \( b_{xy} \) and \( b_{yx} \): it always reflects the independent variable's sum of squares.

Question 14. When one regression coefficient is negative, the other would be:
(a) Negative
(b) Positive
(c) Zero
(d) None of the options
Answer: (a) Negative
Both regression coefficients, \( b_{xy} \) and \( b_{yx} \), must always have the same sign. If one is negative, the other must also be negative, indicating an inverse relationship between the variables. This ensures consistency in the direction of the relationship, regardless of which variable is predicted.
In simple words: The two regression numbers (for X on Y and Y on X) must always have the same sign. So, if one is negative, the other must also be negative.

🎯 Exam Tip: This rule is a quick way to check your calculations. If you find one regression coefficient positive and the other negative, you've made an error.

Question 15. If X and Y are two variates, there can be at most:
(a) one regression line
(b) two regression lines
(c) three regression lines
(d) more regression lines
Answer: (b) two regression lines
When dealing with two variables (variates) X and Y, there are two possible regression lines: one for predicting X from Y (X on Y) and another for predicting Y from X (Y on X). These two lines are distinct unless there is a perfect correlation (\( r = \pm 1 \)), in which case they coincide. This allows for predictions in both directions.
In simple words: For any two variables, we can draw two main prediction lines: one to guess X using Y, and another to guess Y using X.

🎯 Exam Tip: Understand that the line of regression of X on Y is generally different from the line of regression of Y on X, unless perfect correlation exists.

Question 16. The lines of regression of X on Y estimates:
(a) X for a given value of Y
(b) Y for a given value of X
(c) X from Y and Y from X
(d) none of these
Answer: (a) X for a given value of Y
The line of regression of X on Y is used to predict the value of X when the value of Y is known. In this model, Y is considered the independent variable and X is the dependent variable. This is useful in scenarios where Y is more easily observable or controllable.
In simple words: The regression line for X on Y helps us guess the value of X if we already know the value of Y.

🎯 Exam Tip: Always remember that "X on Y" means X is being predicted based on Y, and "Y on X" means Y is being predicted based on X.

Question 17. Scatter diagram of the variate values (X, Y) give the idea about:
(a) functional relationship
(b) regression model
(c) distribution of errors
(d) no relation
Answer: (a) functional relationship
A scatter diagram is a graph that displays the relationship between two variables, X and Y. By plotting points for each pair of data, it visually indicates the type of correlation (positive, negative, or no correlation) and the strength of the linear functional relationship between them. It is a first step in understanding bivariate data.
In simple words: A scatter diagram shows how two sets of numbers, X and Y, are linked together. It helps us see if there is a pattern or a connection between them.

🎯 Exam Tip: Look at the overall trend of the points on a scatter diagram to quickly estimate the direction and strength of the correlation before calculating any coefficients.

Question 18. If regression co-efficient of Y on X is 2, then the regression co-efficient of X on Y is:
(a) \( \le \frac{1}{2} \)
(b) 2
(c) \( > \frac{1}{2} \)
(d) 1
Answer: (a) \( \le \frac{1}{2} \)
We know that the product of the two regression coefficients, \( b_{yx} \times b_{xy} \), must be less than or equal to 1. This is because \( r^2 = b_{yx} \times b_{xy} \), and the correlation coefficient \( r \) lies between -1 and 1, so \( r^2 \) must be between 0 and 1. Given \( b_{yx} = 2 \), let \( b_{xy} \) be the regression coefficient of X on Y. So, \( 2 \times b_{xy} \le 1 \) This implies \( b_{xy} \le \frac{1}{2} \). The regression coefficient of X on Y must be less than or equal to 0.5. This property holds true in all linear regression models, ensuring that the square of the correlation coefficient does not exceed 1.
In simple words: The multiplication of the two regression numbers cannot be more than 1. If one number is 2, then the other number must be 1 divided by 2 (or less) so that their product does not go above 1.

🎯 Exam Tip: Always remember the property that the product of regression coefficients (\( b_{yx} \times b_{xy} \)) is equal to \( r^2 \) and thus cannot exceed 1.

Question 19. If two variables move in a decreasing direction then the correlation is:
(a) positive
(b) negative
(c) perfect negative
(d) no correlation
Answer: (a) positive
If two variables move in a "decreasing direction," it means that as one variable's value goes down, the other variable's value also goes down. When two variables consistently decrease together, they still exhibit a positive correlation because they are moving in the same relative direction. For example, if both the number of hours studied and exam scores decrease for all students, it would still show a positive relationship (as opposed to one decreasing while the other increases).
In simple words: If both things go down at the same time, they are still moving in the same direction. This means they have a positive connection.

🎯 Exam Tip: The key to positive correlation is that the variables move *together*, whether both increasing or both decreasing. Direction (up or down) refers to the *movement* of each variable, not their absolute values.

Question 20. The person suggested a mathematical method for measuring the magnitude of the linear relationship between two variables say X and Y is:
(a) Karl Pearson
(b) Spearman
(c) Croxton and Cowden
(d) Ya Lun Chou
Answer: (a) Karl Pearson
Karl Pearson developed the most widely used method for measuring the linear correlation between two variables, known as Pearson's product-moment correlation coefficient. This method is fundamental in statistics for quantifying the strength and direction of a linear relationship. His work laid the groundwork for modern correlation analysis.
In simple words: Karl Pearson was the person who came up with the main mathematical way to measure how strongly two things are related in a straight line.

🎯 Exam Tip: While Spearman's rank correlation is also important, Karl Pearson's coefficient specifically measures linear relationships and is often the default method in many applications.

Question 21. The lines of regression intersect at the point:
(a) (X, Y)
(b) \( (\overline{X}, \overline{Y}) \)
(c) (0, 0)
(d) \( (\sigma_X, \sigma_Y) \)
Answer: (b) \( (\overline{\mathrm{X}}, \overline{\mathrm{Y}}) \)
Both regression lines (the line of regression of X on Y and the line of regression of Y on X) always intersect at the point represented by the mean of X and the mean of Y, i.e., \( (\overline{X}, \overline{Y}) \). This point is known as the centroid of the data and represents the average values of the two variables. It is a crucial reference point for understanding the overall trend.
In simple words: The two lines that predict how variables are linked always cross each other at one special point. This point is where the average value of X meets the average value of Y.

🎯 Exam Tip: This is a fundamental property of regression lines; knowing this point helps in plotting the lines accurately and understanding their central tendency.

Question 22. The term regression was introduced by:
(a) R.A Fisher
(b) Sir Francis Galton
(c) Karl Pearson
(d) Croxton and Cowden
Answer: (b) Sir Francis Galton
Sir Francis Galton, a British polymath, introduced the concept of "regression towards the mean" in his studies of heredity, particularly height. He observed that children of very tall or very short parents tended to have heights closer to the average. This observation led to the statistical method of regression, which predicts one variable's value based on another.
In simple words: Sir Francis Galton was the first person to use the word "regression" when he noticed that children's heights tend to be closer to the average height than their parents' extreme heights.

🎯 Exam Tip: While many scientists contributed to statistics, remember Galton for introducing the foundational concept and term "regression."

Question 23. If r = -1, then correlation between the variables:
(a) perfect positive
(b) perfect negative
(c) negative
(d) no correlation
Answer: (b) perfect negative
A correlation coefficient \( r = -1 \) signifies a perfect negative linear correlation. This means that as one variable increases, the other decreases in a perfectly predictable straight-line manner. All data points would lie exactly on a downward-sloping straight line, showing a completely inverse relationship.
In simple words: If the correlation number is exactly -1, it means the two things are perfectly linked but move in exact opposite directions. When one goes up, the other goes down perfectly.

🎯 Exam Tip: Remember that "perfect" implies all data points fall exactly on the regression line, with no scatter.

Question 24. The coefficient of correlation describes:
(a) the magnitude and direction
(b) only magnitude
(c) only direction
(d) no magnitude and no direction
Answer: (a) the magnitude and direction
The correlation coefficient provides two key pieces of information about the linear relationship between variables: its magnitude (strength) and its direction. The absolute value of the coefficient indicates strength (closer to 1 is stronger), while its sign (positive or negative) indicates the direction of the relationship. For example, a coefficient of 0.8 is strong positive, and -0.8 is strong negative.
In simple words: The correlation coefficient tells us two things: how strong the link is between two variables (its size), and if they move in the same way (positive) or opposite ways (negative).

🎯 Exam Tip: Never forget that both the sign and the numerical value of 'r' are essential for a complete interpretation of correlation.

Question 25. If \( \mathrm{Cov}(x, y) = -16.5 \), \( \sigma_{x}^{2} = 2.89 \), \( \sigma_{y}^{2} = 100 \). Find correlation coeffient.
(a) -0.12
(b) 0.001
(c) -1
(d) -0.97
Answer: (d) -0.97
To find the correlation coefficient \( r \), we use the formula: \[ r = \frac{\mathrm{Cov}(x, y)}{\sigma_x \sigma_y} \] First, we need to find the standard deviations \( \sigma_x \) and \( \sigma_y \) from the given variances: \( \sigma_x = \sqrt{\sigma_x^2} = \sqrt{2.89} = 1.7 \) \( \sigma_y = \sqrt{\sigma_y^2} = \sqrt{100} = 10 \) Now, substitute these values along with the covariance into the formula for \( r \): \( r = \frac{-16.5}{1.7 \times 10} \) \( r = \frac{-16.5}{17} \) \( r \approx -0.970588... \) Rounding to two decimal places, the correlation coefficient is approximately -0.97. This value indicates a very strong negative linear relationship between the variables X and Y.
In simple words: To find the correlation, we divide the covariance by the standard deviations of X and Y multiplied together. First, we find the square root of the given variances to get the standard deviations, then we put all the numbers into the formula to get -0.97.

🎯 Exam Tip: Always remember that standard deviation is the square root of variance. Ensure you calculate this correctly before proceeding with the correlation coefficient formula.