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# Chapter : 6 REGRESSION ANALYSIS

Regression analysis is a method of predicting or estimating one variable knowing
the value of the other variable. Estimation is required in different fields in every day
life. A businessman wants to know the effect of increase in advertising expenditure
on sales or a doctor wishes to observe the effect of a new drug on patients. An
economist is interested in finding the effect of change in demand pattern of some
commodities on prices.
We observe, different pairs of variables related to each other like saving depends
upon income, cost of production depends upon the number of units produced, the
production depends on the number of workers present on a particular day etc. The
relationship between two variables can be established with the help of any measure
of correlation, as we have seen in the previous chapter. When it is observed that
two variables are highly correlated, it leads to interdependence of the variables. We
can study the cause and effect relationship between them and then we can apply
the regression analysis. The analysis helps in finding a mathematical model of the
relationship. In this chapter we will discuss linear regression only.
There are two types of regression lines viz., regression equation of x on y and
regression equation of y on x. The first one is used to estimate the value of x when
the corresponding value of y is given and the second used to find the value of y
when the corresponding value of x is given. They are given by
Regression equation of x on y is

x  x  bxy  y  y 

Regression equation of y on x is

y  y  byx x  x

and
where bxy and byx are known as

regression coefficients.
The regression coefficients are given by

bxy  r  x
y

byx  r  y
x

 

 bxy  byx   r  x  r  y

 y   x

i.e., bxy  byx  r 2
i.e., r   bxy  byx

Thus, when the regression coefficients are known to us we can determine the coefficient of correlation.
The regression coefficients can be found independently as

1

32 & 11. x : y : 1 4 1 0 1 5 1 1 9 1 2 6 8 6 4 3 7 5 9 and hence [ Answer: 5. find the regression equations and further estimate y if x = 16 and x if y = 18. x : y : 7 5 8 2 8 0 7 8 9 3 8 6 6 5 7 2 8 7 9 1 7 1 8 0 9 8 9 5 6 8 7 2 8 9 8 9 7 7 7 4 [ Answer: 80.394 ~ 80 and 94. From the following data find the two regression equations estimate y when x = 13 and estimate x when y = 10. Find the two regression equations and also estimate y when x = 13 and estimate x when y = 10 x : y : 1 1 1 6 7 9 5 8 6 1 4 1 2 1 1 1 5 1 4 1 0 1 7 [ Answer: 17. From the following data.5359 & 5.0693 ] 3. The following data represents the marks in Algebra (x) and Geometry (y) of a group of 10 students. x : y : 3 4 6 1 2 1 1 1 5 1 0 1 6 1 2 1 9 1 3 1 7 [ Answer: 20. x : y : 1 0 2 0 1 2 2 4 1 4 2 5 1 9 2 1 1 1 2 2 8 1 6 1 7 2 0 [ Answer: 21. Find the regression equations for the following data and hence estimate y when x = 15 and x when y = 18.2858 & 8.9337 ~ 95 ] 4.1428 ] 2.bxy   xy    x   y n y 2 y   n   and byx  2  xy    x   y n x 2 x   n   2 Problems: 1. Find both regression equations and hence estimate y if x = 78 and x if y = 94.64 & 11.54 ] 5.8 ] 2 .

Find the two regression equations and hence estimate the expenditure on advertising when expenditure on office staff is Rs. The following data gives the marks obtained at the preliminary examination (x) and the final examination (y) for a group of 10 students. (ii) the most probable demand for B when demand for A is 115 kgs. Height of mother (x): Height of daughter (y): 15 7 16 2 16 0 15 9 16 3 16 5 16 5 16 7 16 7 17 2 16 8 17 0 17 0 16 8 16 4 16 6 [ Answer: 164. x : y : 10 7 10 5 11 3 11 0 10 9 10 3 10 3 10 0 11 0 11 0 11 7 11 1 11 4 10 8 [ Answer: 110.0996 ] 10.13746 and Rs.97. x : y : 5 4 5 8 6 5 6 7 7 5 7 6 8 2 8 0 5 7 6 0 5 9 6 4 6 0 6 5 6 4 6 5 5 8 6 0 6 2 7 0 [ Answer: 71. expressed in hundreds of Rs. x : y : 12 9 13 7 13 8 13 5 13 9 13 4 98 94 93 91 96 95 14 5 10 0 [ Answer: Rs. The following data gives the expenditure on advertising. The following data represents the figures in kgs. for demands of two commodities A and B.21 ] 8. Also obtain the most likely expenditure on office staff when the advertising expenditure is Rs. expressed in thousands of Rs. x : y : 14 0 12 2 15 5 13 5 16 3 14 0 16 7 13 9 14 5 12 5 15 0 13 0 14 8 12 5 3 . Also obtain the estimate of the height of a daughter when mother’s height is 162 cms.94780 ] 9. Obtain the regression equation of y on x. From the following data. (y) for 7 different companies.13. The heights in cms of a group of mothers and daughters are given below.6.000. estimate y when x = 142 and x when y = 126.7069 & 110. (x) and the expenditure on office staff. Find (i) the most probable demand for A when demand for B is 107 kgs. Find the regression equations and hence find the most likely height of a mother when the daughter’s height is 164 cms. Hence find the most probable marks at the final examination of a student who has scored 70 marks at the preliminary examination.200.2942 & 162.4556 ] 7.

  x  x  2  414.9 Find the regression equation of y on x and then estimate y when x = 78.4495 ] 13.5774 ] 19. [ Answer: 81. if the regression equations are 2 x  y  3  0 and 2 x  3 y  5  0 [ 1.7 cms and 850 kgs respectively. [ Average price [ 40. 1 and -0. find (i) mean values of x and y (ii) coefficient of correlation where the equation are 2 x  3 y  5  0 and 5 x  8 y  13  0 [ 1. Mean values of x and y ii. ] 14. [1.  x  5. [ Estimated yield is 839.  x  x   y  y   326.000.8165 ] 18.2 cms and 14.15. Find the regression equation and hence estimate y when x = 56 and x when y = 45. r  0.7 Standard deviation = 5.  y  2.  x  88. The regression equation of supply in thousands of Rs.1 kgs. The most probable value of y when x = 63 ii. The average supply is Rs. The ratio of standard deviation of supply and price is 2 .85 16. 1 and 0. The averages for rainfall and yield of a crop are 42. For a bivariate distribution. y  89. find i. The corresponding standard deviations are 3.04 ] Given the two regression equations. 1 and -0.274 & 62. The coefficient of correlation is 0.3 ] 11.4.65. is Rs. Mean value of x = 65 Mean value of y = 53 Standard deviation = 4. Estimate the yield when the rainfall is 39.2 Coefficient of correlation = 0. The most probable value of x when y = 50 [ Answer: 51.885 ] 12.99 kgs.(x) is 2 x  5 y  60  0 . x  54. Find the average price 3 and the coefficient of correlation between supply and price.6 ] 15. the following results are obtained.6. Given that for 10 pair of observations. Given the regression equations 3 x  y  25  0 and 2 x  3 y  30  0 .78 Find the two regression equations and hence obtain i. y  40.9682 ] 17.3.2 cms.8. Find the means of x and y and the coefficient of correlation.000 & r = 0.18.25 & 146. Coefficient of correlation 4 .9 & 62. Given the regression equations 2 x  3 y  5  0 and x  y  2  0 .1.(y) on price in thousands of Rs. Find the mean values of x and y and correlation coefficient.[ Answer: 123.

From i. r  0. iv.1. Find the two regression equations for the following: x  30. iii. 0.54 ] 21.4714.33 and 16.2.67 ] 20.1800.5 ] 22.8164. 0.562. Estimate of y when x = 17 Estimate of x when y = 25 [ 15. Find the coefficient of correlation between income and expenditure. [ 25.iii. The ratio of the standard deviation of income and expenditure is 4 : 3. 21. Also find the average income if the average expenditure is Rs. x  0.0146 x  19. 20. find Mean values of x and y Coefficient of correlation Most probable value of y when x = 28 Most probable value of x when y = 35. iv. 0.  y  5. ii. the following regression equations 2 x  y  17  0 and 4 x  3 y  1  0 .6308 y  1. 33. 37.8 [ y  1. 26 ] 5 . y  50. The regression equation of income (x) on expenditure (y) is 3 x  2 y  3900  0 .  x  4. [2500.
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