Answer to Question #349267 in Statistics and Probability for gabriela

Question

Solve r and interpret the result

X 2,4,6,7,10

Y 8,10,12,6,16

In the given table on the below, solve for Pearson r and interpret the result

X 80,84,86,87,89,90,91,93,94,96

Y 78,83,80,84,89,90,88,91,93,96

Answer 1

a) In order to compute the regression coefficients, the following table needs to be used:

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2 \\ \hline & 2 & 8 & 16 & 4 & 64 \\ \hdashline & 4 & 10 & 40 & 16 & 100 \\ \hdashline & 6 & 12 & 72 & 36 & 144 \\ \hdashline & 7 & 6 & 42 & 49 & 36 \\ \hdashline & 10 & 16 & 160 & 100 & 256 \\ \hdashline Sum= & 29 & 52 & 330 & 205 & 600 \\ \hdashline \end{array}$$

$$\bar{X}=\dfrac{1}{n}\sum _{i}X_i=\dfrac{29}{5}=5.8$$

$$\bar{Y}=\dfrac{1}{n}\sum _{i}Y_i=\dfrac{52}{5}=10.4$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum _{i}X_i)^2$$ $$=205-\dfrac{29^2}{5}=36.8$$

$$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum _{i}Y_i)^2$$ $$=600-\dfrac{(52)^2}{5}=59.2$$

$$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum _{i}X_i)(\sum _{i}Y_i)$$ $$=330-\dfrac{29(52)}{5}=28.4$$

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}SS_{YY}}}=\dfrac{28.4}{\sqrt{36.8(59.2)}}$$

$$=0.608462$$

$$0.4<r<0.7$$

Moderate positive correlation

$$m=slope=\dfrac{SS_{XY}}{SS_{XX}}=\dfrac{28.4}{36.8}=0.7717$$ $$n=\bar{Y}-m\bar{X}=10.4-\dfrac{28.4}{36.8}(5.8)=5.9239$$

The regression equation is:

$$y=5.9239+0.7717x$$

(b) In order to compute the regression coefficients, the following table needs to be used:

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2 \\ \hline & 80 & 78 & 6240 & 6400 & 6084 \\ \hdashline & 84 & 83 & 6972 & 7056 & 6889 \\ \hdashline & 86 & 80 & 6880 & 7396 & 6400 \\ \hdashline & 87 & 84 & 7308 & 7569 & 7056 \\ \hdashline & 89 & 89 & 7921 & 7921 & 7921 \\ \hdashline & 90 & 90 & 8100 & 8100 & 8100 \\ \hdashline & 91 & 88 & 8008 & 8281 & 7744 \\ \hdashline & 93 & 91 & 8463 & 8649 & 8281 \\ \hdashline & 94 & 93 & 8742 & 8836 & 8649 \\ \hdashline & 96 & 96& 9216 & 9216 & 9216 \\ \hdashline Sum= & 890 & 872 & 77850 & 79424 & 76340 \\ \end{array}$$

$$\bar{X}=\dfrac{1}{n}\sum _{i}X_i=\dfrac{890}{10}=89$$

$$\bar{Y}=\dfrac{1}{n}\sum _{i}Y_i=\dfrac{872}{10}=87.2$$

$$SS_{XX}=\sum_iX_i^2-\dfrac{1}{n}(\sum _{i}X_i)^2$$ $$=79424-\dfrac{890^2}{10}=214$$

$$SS_{YY}=\sum_iY_i^2-\dfrac{1}{n}(\sum _{i}Y_i)^2$$ $$=76340-\dfrac{(872)^2}{10}=301.6$$

$$SS_{XY}=\sum_iX_iY_i-\dfrac{1}{n}(\sum _{i}X_i)(\sum _{i}Y_i)$$ $$=77850-\dfrac{890(872)}{10}=242$$

$$r=\dfrac{SS_{XY}}{\sqrt{SS_{XX}SS_{YY}}}=\dfrac{242}{\sqrt{214(301.6)}}$$

$$=0.952561$$

Strong positive correlation