Page 347 - DMTH404_STATISTICS

Unit 23: Regression Analysis

5. Write a note on the standard error of the estimate. Notes

6. " The regression line gives only a 'best estimate' of the quantity in question. We may
assess the degree of uncertainty in this estimate by calculating its standard error ". Explain.
7. Given a scatter diagram of a bivariate data involving two variables X and Y. Find the
2
å
conditions of minimisation of (Y Y- C ) and hence derive the normal equations for the
linear regression of Y on X. What sum is to be minimised when X is regressed on Y? Write
down the normal equation in this case.
8. Explain, fully, the meaning of regression of one variable Y on another variable X. Discuss
the method of least squares for fitting a linear regression of the form Y = a + bX. Write

s
down the normal equations and show that b = × Y , where the symbols have their usual
r
s X
meaning.
9. Show that the coefficient of correlation is the geometric mean of the two regression
coefficients.
10. What is the method of least squares ? Show that the two lines of regression obtained by
this method are irreversible except when r = ± 1. Explain.
11. Show that, in principle, there are always two lines of regression for a bivariate data. Prove
that the coefficient of correlation between two variables is either + 1 or - 1 when the two
lines are identical and is zero when they are perpendicular.
12. Fit a linear regression of Y on X to the following data :

X : 1 2 3 4 5 6 7 8
Y : 65 80 45 86 178 205 200 250

13. Obtain the two lines of regression from the following data and show them on a graph.
Also construct a scatter diagram of the data.

(
Age of husband X)
: 23 27 28 28 28 30 30 33 35 38
( in years)
(
Age of wife Y)
: 18 20 22 27 21 29 27 29 28 29
( in years)
14. The following table gives the information on the years of education (X) of nine farmers
and annual yields per acre (Y) on their farms :

X : 0 2 4 6 8 10 12 14 16
Y : 4 4 6 10 10 8 12 8 6
(a) Find the regression equation of yield per acre on education and give an economic
interpretation to it.
(b) What is the magnitude of the 'explained variation' in the dependent variable? Find
the coefficient of correlation from it.

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