Question

6 observations on (X,Y) yielded the following data:
X  30, Y 180, X Y 1000,
i i i i
X 200, Y 5642. 2
i
2  i   
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.

Answer 1

Answer on Question #80960 – Math – Statistics and Probability Question

6 observations on $(X,Y)$ yielded the following data:

$$\sum_{i} X_{i} = 30, \sum_{i} Y_{i} = 180, \sum_{i} X_{i} Y_{i} = 1000,$$ $$\sum_{i} X_{i}^{2} = 200, \sum_{i} Y_{i}^{2} = 5642.2$$

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 $X$.

Solution

i)

$$r = \frac{n \sum_{i} X_{i} Y_{i} - \sum_{i} X_{i} \sum_{i} Y_{i}}{\sqrt{[n \sum_{i} X_{i}^{2} - (\sum_{i} X_{i})^{2}][n \sum_{i} Y_{i}^{2} - (\sum_{i} Y_{i})^{2}]}}$$ $$r = \frac{6(1000) - 30(180)}{\sqrt{[6(200) - (30)^{2}][6(5642.2) - (180)^{2}]}} \approx 0.90871548437$$

ii)

$$a = \frac{(\sum_{i} Y_{i})(\sum_{i} X_{i}^{2}) - (\sum_{i} X_{i})(\sum_{i} X_{i} Y_{i})}{n(\sum_{i} X_{i}^{2}) - (\sum_{i} X_{i})^{2}}$$ $$b = \frac{n(\sum_{i} X_{i} Y_{i}) - (\sum_{i} X_{i})(\sum_{i} Y_{i})}{n(\sum_{i} X_{i}^{2}) - (\sum_{i} X_{i})^{2}}$$ $$a = \frac{180(200) - 30(1000)}{6(200) - (30)^{2}} = 20$$ $$b = \frac{6(1000) - 30(180)}{6(200) - (30)^{2}} = 2$$ $$Y = a + bX$$ $$Y = 20 + 2X$$ $$X = 10: Y = 20 + 2(10) = 40$$

iii)

$$Y = 20 + 2X = 15 \Rightarrow 2X = -5 \Rightarrow X = -2.5$$

Answer: i) 0.90871548437; ii) 40; iii) -2.5.

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