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Question

6 observations on (X,Y) yielded the following data:
$\sum X_{_{i}} = 30, \sum Y_{i} = 180, \sum X_{i}Y_{i} = 1000,$

$\sum X{_{i}}^{2} = 200,\sum Y{_{i}}^{2} = 5642.$

i) Determine the correlation coefficient between X and Y.
ii) Given X =10 , what will be the predicated value of Y?
iii) Given Y = 15, predict X.

14 Feb 2018

Answer :

Word Count : 351

We are given: $$ n = 6, \quad \sum X_i = 30, \quad \sum Y_i = 180, \quad \sum X_iY_i = 1000, \quad \sum X_i^2 = 200, \quad \sum Y_i^2 = 5642 $$ Step 1: Compute means of X and Y $$ \bar{X} = \frac{\sum X_i}{n} = \frac{30}{6} = 5 $$ $$ \bar{Y} = \frac{\sum Y_i}{n} = \frac{180}{6} = 30 $$ Step 2: Compute correlation coefficient $r$ The formula for $r$ is: $$ r = \frac{\sum X_i Y_i - n \bar{X} \bar{Y}}{\sqrt{\left(\sum X_i^2 - n \bar{X}^2\right)\left(\sum Y_i^2 - n \bar{Y}^2\right)}} $$ Compute numerator: $$ \sum X_i Y_i - n \bar{X} \bar{Y} = 1000 - 6 \cdot 5 \cdot 30 = 1000 - 900 = 100 $$ Compute denominator terms: $$ \sum X_i^2 - n \bar{X}^2 =

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