Answer to Question #204963 in Statistics and Probability for Waleed Abid

Question

The data set given below consists of six pairs of (x, y); (10, 70); (12, 65); (2, 96); (0, 94); (8, 75); (5, 82) I Based on a plot, determine whether the relationship between x and y is linear or not. a. b. Find the value of r to analyze the strength of the relationship.

Answer 1

a.

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & x & y & xy & x^2 & y^2 \\ \hline & 10 & 70 & 700 & 100 & 4900 \\ \hdashline & 12 & 65 & 780 & 144 & 4225 \\ \hdashline & 2 & 96 & 192 & 4 & 9216 \\ \hdashline & 0 & 94 & 0 & 0 & 8836 \\ \hdashline & 8 & 75 & 600 & 64 & 5625 \\ \hdashline & 5 & 82 & 410 & 25 & 6724 \\ \hdashline Sum=& 37 & 482 & 2682 & 337 & 39526 \\ \hdashline \end{array}$$

$$\bar{x}=\dfrac{\displaystyle\sum_{i=1}^nx_i}{n}=\dfrac{37}{6}$$

$$\bar{y}=\dfrac{\displaystyle\sum_{i=1}^ny_i}{n}=\dfrac{482}{6}$$

$$SS_{xx}=\displaystyle\sum_{i=1}^nx_i^2-\dfrac{1}{n}\big(\displaystyle\sum_{i=1}^nx_i\big)^2=\dfrac{653}{6}$$

$$SS_{yy}=\displaystyle\sum_{i=1}^ny_i^2-\dfrac{1}{n}\big(\displaystyle\sum_{i=1}^ny_i\big)^2=\dfrac{4832}{6}$$

$$SS_{xy}=\displaystyle\sum_{i=1}^ny_i^2-\dfrac{1}{n}\big(\displaystyle\sum_{i=1}^nx_i\big)\big(\displaystyle\sum_{i=1}^ny_i\big)=-\dfrac{1742}{6}$$

$$m=\dfrac{SS_{xy}}{SS_{xx}}=-\dfrac{1742}{653}\approx-2.6677$$

$$n=\bar{y}-m\bar{x}=\dfrac{482}{6}+\dfrac{37}{6}\cdot\dfrac{1742}{653}\approx96.7841$$

$$Y=96.7841-2.6667X$$

The relationship between x and y is linear. 

b.

$$r=\dfrac{SS_{xy}}{\sqrt{SS_{xx}}\sqrt{SS_{yy}}}=\dfrac{-\dfrac{1742}{6}}{\sqrt{\dfrac{653}{6}}\sqrt{\dfrac{4832}{6}}}\approx-0.9807$$

$$r^2\approx0.9617$$

$$0.7<r\leq 1$$

Strong negative correlation.