Answer to Question #124577 in Statistics and Probability for Mishalgul

Question

No.3. A researcher conducted an experiment on rats and found the below given data of initial weights X of 15 four weeks-old rats and their weight gains Y, after 9 weeks on a special diet. (2+2+1)

Initial weight(X) 55,68,72,43,56,54,62,59,74,73,64,51,70,49,63
Weight gain(Y) 112,95,120,121,122,105,85,89,98,95,104,135,125,118,90

The researcher wish to know
(a) Whether or not the weight gain depends upon birth weight
(b) Is there any relationship between birth weight and weight gain after 9 weeks of special diet.
(c) Interpret your finding in part (a) and part(b). write down the relationship between correlation and regression coefficients.

Answer 1

$$\displaystyle\sum_{i=1}^nx_i=913, \bar{x}={1\over 15}\cdot913=60.866667$$

$$\displaystyle\sum_{i=1}^ny_i=1614, \bar{y}={1\over 15}\cdot1614=107.6$$

$$SS_{xx}=\displaystyle\sum_{i=1}^nx_i^2-{1\over n}(\displaystyle\sum_{i=1}^nx_i )^2=1259.733333$$

$$SS_{yy}=\displaystyle\sum_{i=1}^ny_i^2-{1\over n}(\displaystyle\sum_{i=1}^ny_i )^2=3317.6$$

$$SS_{xy}=\displaystyle\sum_{i=1}^nx_iy_i-{1\over n}(\displaystyle\sum_{i=1}^ny_i )(\displaystyle\sum_{i=1}^nx_i)=-822.8$$

$$m=\dfrac{S_{xy}}{S_{xx}}=-0.6532$$

$$n=\bar{y}-m\bar{x}=147.3553$$

Therefore, we find that the regression equation is:

$$\hat{Y}=147.3553-0.6532X$$

$$r=\dfrac{S_{xy}}{\sqrt{S_{xx}}\sqrt{S_{yy}}}=-0.4025$$

$0.4<|r|<0.7$ moderate corelation

(a) The weight gain depends upon birth weight.

(b) There is relationship between birth weight and weight gain after 9 weeks of special diet.

(c) There is a negative moderate relationship between birth weight and weight gain after 9 weeks of special diet. 

$$\hat{Y}=147.3553-0.6532X$$ $$r=-0.4025$$