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Mathematics and Statistics · Class 12 Commerce

Ch 11Linear Regression — Class 12 Mathematics and Statistics, concept-first.

In the Std XI chapter on correlation we learnt how to measure whether two variables are related and how strongly — Karl Pearson's coefficient tells us the direction and degree of the linear relationship.

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Key concepts

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Lines of Regression

There are two regression lines, both through the point of means . The line of on is with ; the line of on is with . Use on to estimate , and on to estimate .

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

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Meaning of Regression

In the Std XI chapter on correlation we learnt how to measure whether two variables are related and how strongly — Karl Pearson's coefficient tells us the direction and degree of the linear relationsh…

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The Two Lines of Regression and Their Equations

For a linear relationship there are two lines of regression, one for each direction of estimation. Both are straight lines and both pass through the point of means .

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Regression Coefficients and Their Properties

The two numbers and are the regression coefficients. They carry all the information the regression lines need, and they obey a set of properties that are frequently tested and are also useful checks o…

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Correlation, Estimation and Finding the Means

This section ties the regression coefficients to the correlation coefficient and shows the two standard uses of the regression equations: estimating values and recovering the means.

Exercises

+Show 4 questions4 questions
  1. Q9Obtain the two lines of regression for the data below, and estimate $y$ when $x=6$. | $x$ | 3 | 6 | 9 | 12 | 15 | |---|---|---|---|---|---|…Free
  2. Q10The two lines of regression are $x-2y+16=0$ and $5x-6y+32=0$. Find the means $\bar x,\bar y$ and the coefficient of correlation.Free
  3. Q11The coefficient of correlation between two variables is the ______ of the two regression coefficients. (a) arithmetic mean (b) geometric mea…Preview
  4. Q12State any four properties of regression coefficients.Preview

Sample & Board Papers

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

+Show 19 questions19 questions
  1. Q1$b_{XY} \cdot b_{YX} =$ ______. (a) $V(X)$ (b) $\sigma_x$ (c) $r^2$ (d) $(\sigma_y)^2$Preview
  2. Q2For certain bivariate data on 5 pairs of observations given: $\sum x = 20$, $\sum y = 20$, $\sum x^2 = 90$, $\sum y^2 = 90$, $\sum xy = 76$…Preview
  3. Q3Find the equation of the line of regression of Y on X for the following data: $n = 8$, $\sum(x_i - \bar{x})(y_i - \bar{y}) = 120$, $\bar{x}…Preview
  4. Q4For 50 students of a class, the regression equation of marks in statistics (X) on the marks in accountancy (Y) is $3y - 5x + 180 = 0$. The v…Preview
  5. Q5$b_{YX}$ is ______. (a) Regression coefficient of Y on X (b) Regression coefficient of X on Y (c) Correlation coefficient between X and Y (d…Preview
  6. Q6For a bivariate data: $\sum(x - \bar{x})^2 = 1200$, $\sum(y - \bar{y})^2 = 300$, $\sum(x - \bar{x})(y - \bar{y}) = -250$ Find: $b_{yx}$ $b_{…Preview
  7. Q7For the following data, find the regression line of Y on X | $X$ | 1 | 2 | 3 | | --- | --- | --- | --- | | $Y$ | 2 | 1 | 6 | Hence find the…Preview
  8. Q8$b_{xy}$ and $b_{yx}$ are _______. (a) Independent of change of origin and scale (b) Independent of change of origin but not of scale (c) In…Preview
  9. Q9For bivariate data, the regression coefficient of Y on X is 0.4 and the regression coefficient of X on Y is 0.9. Find the value of the varia…Preview
  10. Q10For 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: Line of re…Preview
  11. Q11State whether the following statement is True or False: The following data is not consistent: $b_{yx} + b_{xy} = 1.3$ and $r = 0.75$ (a) Tru…Preview
  12. Q12Compute the appropriate regression equation for the following data: | X | 1 | 2 | 3 | 4 | 5 | | --- | --- | --- | --- | --- | --- | | Y | 5…Preview
  13. Q13The equations of two regression lines are $10x - 4y = 80$ and $10y - 9x = -40$ Find: $\bar{x}$ and $\bar{y}$ $b_{YX}$ and $b_{XY}$ If $\text…Preview
  14. Q14If $b_{yx} > 1$ then $b_{xy}$ is ______. (a) $> 1$ (b) $< 0$ (c) $= 0$ (d) $< 1$Preview
  15. Q15Given the following information about the production (X) and demand (Y) of a commodity, obtain the regression line of X on Y. | | Production…Preview
  16. Q16For the following bivariate data obtain the equation of regression line of Y on X. | X | 1 | 2 | 3 | 4 | 5 | | --- | --- | --- | --- | --- |…Preview
  17. Q17From the following data with usual notations find correlation coefficient between X and Y: $n = 50$, $\sum (x_i - \bar{x})(y_i - \bar{y}) =…Preview
  18. Q18You are given the following information about advertising expenditure and sales: | | Advertisement expenditure (₹ in lakhs) X | Sales (₹ in…Preview
  19. Q19The equations of the two regression lines are $2x + 3y - 6 = 0$ and $5x + 7y - 12 = 0$. Find the correlation coefficient.Preview

More questions

+Show 8 questions8 questions
  1. Example 1Explain the meaning of regression and state two differences between correlation and regression.Free
  2. Example 2For the following data, obtain both lines of regression, and hence estimate $y$ when $x=7$ and estimate $x$ when $y=10$. | $x$ | 2 | 4 | 6 |…Free
  3. Example 3Find the two regression coefficients and the coefficient of correlation for the data: | $x$ | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|…Free
  4. Example 4If $b_{yx}=0.8$ and $b_{xy}=0.45$, find the coefficient of correlation between $x$ and $y$.Preview
  5. Example 5The two regression coefficients of a distribution are $b_{yx}=-0.9$ and $b_{xy}=-0.4$. Find the coefficient of correlation, and comment on i…Preview
  6. Example 6For a bivariate data set, $\bar x=40,\ \bar y=55,\ \sigma_x=5,\ \sigma_y=8$ and $r=0.8$. Obtain the two lines of regression, and estimate (i…Preview
  7. Example 7The two lines of regression are $8x-10y+66=0$ and $40x-18y-214=0$. Find (i) the means $\bar x$ and $\bar y$, (ii) which line is the regressi…Preview
  8. Example 8For a certain bivariate data, $b_{yx}=1.2$. A student claims $b_{xy}=1.3$. Is this possible? If not, what is the largest value $b_{xy}$ can…Preview