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Statistics · Class 12 Commerce

Ch 3Linear Regression — Class 12 Statistics, concept-first.

Linear regression is a statistical technique that studies the nature and extent of the relationship between two variables and uses that relationship to estimate the probable value of one variable when the value of the other is known.

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Key concepts

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Regression Lines (Y on X and X on Y)

There are two regression lines for any bivariate data set. The line of on , , minimizes vertical deviations and estimates from . The line of on , , minimizes horizontal deviations and estimates from .

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Chapter contents

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Meaning and Concept of Linear Regression

Linear regression is a statistical technique that studies the nature and extent of the relationship between two variables and uses that relationship to estimate the probable value of one variable when…

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Regression Lines: Y on X and X on Y

There are two regression lines for any set of bivariate data:

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Regression Coefficients: byx and bxy

The regression coefficient of on , denoted , measures the amount of change in for a unit change in :

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Computing Regression Equations from Raw Data

To find the two regression equations from a set of paired observations , follow these steps:

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Prediction Using Regression Equations

Once the regression equation is known, prediction is simple substitution:

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Relationship between Correlation and Regression Coefficients

Regression and correlation are closely linked — regression coefficients and the correlation coefficient are computed from the very same covariance and standard-deviation values.

Exercises

+Show 6 questions6 questions
  1. Q6The following data relate to advertising expenditure ($X$, in ₹ lakh) and sales ($Y$, in ₹ lakh) of a firm for 5 years: | X | 3 | 5 | 6 | 8…Free
  2. Q7Given the data: $X$: 1, 2, 3, 4, 5 and $Y$: 6, 8, 10, 12, 14, find both regression equations and the coefficient of correlation. Comment on…Free
  3. Q8The two regression lines for a bivariate distribution are $3X+2Y=26$ and $6X+Y=31$. Find (i) the means of $X$ and $Y$, and (ii) the coeffici…Preview
  4. Q9Why should predictions made using a regression equation be treated cautiously when the given value of the independent variable lies far outs…Preview
  5. Q10Which of the following statements about the regression coefficients $b_{yx}$ and $b_{xy}$ is correct? (a) They always have opposite signs. (…Preview
  6. Q11A researcher reports the regression coefficients of a bivariate data set as $b_{yx}=1.6$ and $b_{xy}=0.9$. Is this result possible? Justify…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 31 questions31 questions
  1. Q1What is the error $e$ in estimation in case of regression line of $Y$ on $X$? (a) $y-\hat{y}$ (b) $\hat{x}-\hat{y}$ (c) $x-\hat{x}$ (d) $\ha…Preview
  2. Q2What are the constants $a$ and $b$ in the regression line $\hat{y} = a + bx$?Preview
  3. Q3If $b_{yx} = 0.75$, $u = 6(x - 20)$ and $v = 2(y - 15)$ for the data in the study of a regression line then find the value of $b_{vu}$.Preview
  4. Q4Average rain and yield of a crop per acre in the year of an orid region is given: | Particulars | Rainfall (cm) | Yield of crop (kg) | |---|…Preview
  5. Q5From the following information, obtain the regression line of monthly expenditure on monthly income. If Namrata's monthly income is ₹75 thou…Preview
  6. Q6The regression line always passes through which point? (a) $(\bar{x}, \bar{y})$ (b) $(0, \bar{y})$ (c) $(\bar{x}, 0)$ (d) $(0, 0)$Preview
  7. Q7What is coefficient of determination in the study of regression for two variables? (a) Product of two standard deviation (b) Square of corre…Preview
  8. Q8Give the name of a method to obtain the best fitted regression line.Preview
  9. Q9The fitted regression line of $Y$ on $X$ is $\hat{y} = 23.2 - 1.2x$ and one of the observations used in fitting of the line is $(6, 17)$. Fi…Preview
  10. Q10The following information is obtained for monthly-advertisement cost and the sales of the last year for a company providing online shopping.…Preview
  11. Q11If $b_{yx} = 0.8$ then find the value of $b_{vu}$ for the following $u$ and $v$. (i) $u = x - 105$ and $v = y - 90$ (ii) $u = \dfrac{x - 140…Preview
  12. Q12The monthly sale of different types of laptops (in hundred units) and its profit (in lakh ₹) for the last six months for a company is given…Preview
  13. Q13What is coefficient of determination in the study of regression for two variables? (a) (A) Product of two standard deviations (b) (B) Square…Preview
  14. Q14Interpret Regression coefficient $b > 0$.Preview
  15. Q15If $\bar{x} = 60, \bar{y} = 75$, and $S_x^2 : \text{Cov}(x, y) = 5 : 3$, obtain the regression line of $Y$ on $X$.Preview
  16. Q16The information of investment (in lakh ₹) and its market price (In lakh ₹) after six months in share market in last seven years for a Mutual…Preview
  17. Q17In order to study relationship between the repairing time of accident damaged cars and the cost of repair, the following information is coll…Preview
  18. Q18The regression line always passes through which point? (a) $(\bar{x}, \bar{y})$ (b) $(0, \bar{y})$ (c) $(\bar{x}, 0)$ (d) $(0, 0)$Preview
  19. Q19What is coefficient of determination in the study of regression for two variables? (a) Product of two standard deviations (b) Square of corr…Preview
  20. Q20Will the regression coefficient change if the values of both the variables $x$ and $y$ are doubled with the help of transformation of scale?Preview
  21. Q21If $\bar{x} = 30$, $\bar{y} = 20$ and $b = 0.5$, find the intercept of the regression line of $Y$ on $X$ and write equation of the line.Preview
  22. Q22The information of price (in ₹) of a ball pen and the supply of ball pen (in units) at the end of each month of a year for a company making…Preview
  23. Q23If the regression line of $Y$ on $X$ is $\hat{y} = 11 + 3x$ and $S_x : S_y = 3 : 10$, find the coefficient of determination.Preview
  24. Q24To study the relationship between the time of usage of cars and its average annual maintenance cost of a car manufacturing company, the foll…Preview
  25. Q25The best fitted line of regression can be obtained by which method? (a) Least Square Method (b) Karl Pearson's Method (c) Maximum Square Met…Preview
  26. Q26The regression line of Y on X is $\hat{y} = 30 - 1.5x$. What is the value of $\bar{y}$ if $\bar{x} = 10$? (a) $28.5$ (b) $20$ (c) $15$ (d) $…Preview
  27. Q27State the Linear Regression model.Preview
  28. Q28The fitted regression line of Y on X is $\hat{y} = 23.2 - 1.2x$ and one of the observations used in fitting of the line is $(6, 17)$. Find t…Preview
  29. Q29The following measures obtained to study the relation between rainfall in cm(X) and yield of Bajri in Quintal per hectare (Y) in ten differe…Preview
  30. Q30The following information is obtained to study the relationship between average rainfall (in cm) and the yield of Maize (in quintal per hect…Preview
  31. Q31Obtain the regression line of the demand on the price using the following information collected for the demand and the price of a commodity.…Preview

More questions

+Show 5 questions5 questions
  1. Example 1From the following data of 5 pairs of observations, find the two regression equations of $Y$ on $X$ and $X$ on $Y$. Also estimate the value…Free
  2. Example 2For a bivariate distribution, $r = 0.6$, $\sigma_x = 5$, $\sigma_y = 4$, $\bar X = 25$ and $\bar Y = 30$. Find the two regression equations,…Free
  3. Example 3If $b_{yx}=0.8$ and $b_{xy}=0.2$ for a set of bivariate data, find the coefficient of correlation $r$, and verify that the arithmetic mean o…Preview
  4. Example 4Distinguish between correlation and regression.Preview
  5. Example 5A student calculates the regression coefficients from a set of data as $b_{yx} = 2.4$ and $b_{xy} = 0.9$. Comment on the correctness of thes…Preview