Answer to Question #197671 in Statistics and Probability for Jack lakos

Question

1) The following data is obtained from a research

a) Find the best equation fitting this data by using regression analyses method

b) Find the correlation coefficient of the equation

c) Find the y value for the x=5.5

X Y

5 784

10 1043

15 1666

20 2478

25 4060

30 6069

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c} X & Y & XY & X^2 & Y^2 \\ \hline 5 & 784 & 3920 & 25 & 614656 \\ \hdashline 10 & 1043 & 10430 & 100 & 1087849 \\ \hdashline 15 & 1666 & 24990 & 225 & 2775556 \\ \hdashline 20 & 2478 & 49560 & 400 & 6140484 \\ \hdashline 25 & 4060 & 101500 & 625 & 16483600 \\ \hdashline 30 & 6069 & 182070 & 900 & 36832761 \\ \hline \sum=\ \ \ 105 & 16100 & 372470 & 2275 & 63934906 \\ \end{array}$$

$$\bar{X}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nX_i=\dfrac{105}{6}=17.5$$

$$\bar{Y}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nY_i=\dfrac{16100}{6}=2683.333333$$

$$SS_{XX}=\displaystyle\sum_{i=1}^nX_i^2-\dfrac{1}{n}(\displaystyle\sum_{i=1}^nX_i)^2$$

$$=2275-\dfrac{105^2}{6}=437.5$$

$$SS_{YY}=\displaystyle\sum_{i=1}^nY_i^2-\dfrac{1}{n}(\displaystyle\sum_{i=1}^nY_i)^2$$

$$=63934906-\dfrac{16100^2}{6}=20733239.333333$$

$$SS_{XY}=\displaystyle\sum_{i=1}^nX_iY_i-\dfrac{1}{n}(\displaystyle\sum_{i=1}^nX_i)(\displaystyle\sum_{i=1}^nY_i)$$

$$=372470-\dfrac{105\times16100}{6}=90720$$

$$m=\dfrac{SS_{XY}}{SS_{XX}}=\dfrac{90720}{437.5}=207.36$$

$$n=\bar{Y}-m\cdot \bar{X}=2683.333333.2-207.36\cdot17.5$$

$$=−945.466667$$

The regression equation is:

$$Y=−945.466667+207.36X$$

b)

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}$$

$$=\dfrac{90720}{\sqrt{437.5}\sqrt{20733239.333333}}\approx0.952534$$

$$0.7<0.952534\leq1$$

Strong positive correlation,

c)

$X=5.5$

$$Y=−945.466667+207.36(5.5)=195$$