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8.23 Sampling Theory of Regression

The regression equation of a normally distributed population is y = a + bx.

Let the regression equation of a sample drawn from it be
y = a + dx.

All you have done is, replaced the intercept ( a ) by c and the slope ( b ) by d.

To find S. E. of regression coefficient of y on x i.e. S. Ebyx use

To find S. E. of regressiojn coefficient of x on y i.e. S.Ebxy use

Also S. E. of regression estimate of y on x or S. Exy =

S. E. of regression estimate of x on y or S. Eyx =

To test b = c use t = width df = n-2

This can be used to find confidence intervals for population regression coefficient from sample values.


Example Find the regression coefficient of the prices of city A over the prices of city B given that :

                                                 A          B

Average price per kg               120        130

Standard deviation                     4            5

Coefficient of correlation           0.6

                                   n = 100

Solution: Now regression coefficient of prices of city A (x) over the prices of city B.

Thus the regression coefficient of prices of the city A over the prices of city B is 0.48 and its S.E. is 0.064.

Example For a certain group of adults, the coefficient of correlation between height and weight is 0.7. Standard deviations of height and weights are 2 inches and 10 lb respectively and the means of heights and weights for the entire group are 70 inches and 130 lbs respectively. Find out the best estimate of weight of a member who is 65 inches tall. Assign limits to this estimate, within which all probability has actual weight would lie.

Solution: Weight (y) and height (x)

Thus the best estimate of weight of an member whose weight is 112.5 lb and height 65 inches cannot deviate from the actual weight by more than 3 S. E.

\ The estimate of all probabilities of his weight lies between

= 112.5 ± 3 ( 7.41 )

= ( 112.5 - 22.23, 112.5 + 22.23 )

= ( 90. 27 lb, 134.73 lb )

**********

Index

8.1 Population
8.2 Sample
8.3 Parameters and Statistic
8.4 Sampling Distribution
8.5 Sampling Error
8.6 Central Limit Theorem
8.7 Critical Region
8.8 Testing of Hypothesis
8.9 Errors in Tesitng of Hypothesis
8.10 Power o a Hypothesis Test
8.11 Sampling of Variables
8.12 Sampling of Attributes
8.13 Estimation
8.14 Testing the Difference Between Means
8.15 Test for Difference Between Proportions
8.16 Two Tailed and one Tailed Tests
8.17 Test of Significance for Small Samples
8.18 Students t-distribution
8.19 Distribution of 't' for Comparison of Two Samples Means Independent Samples
8.20 Testing Difference Between Mens of Two Samples Dependent Samples or Matched Paired Observations
8.21 Chi-Square
8.22 Sampling Theory of Correlation
8.23 Sampling Theory of Regression

Chapter 1





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