Question

what is the value of the correlation coefficient r of the data set?
variable x 3 6 7 11 5 1 11
variable y 10 21 18 38 21 15 60
1
0.71
0.84
-0.84

Explanation:

Step1: Calculate means of x and y

Let $x = [3,6,7,11,5,1,11]$, $y=[10,21,18,38,21,15,60]$.
$\bar{x}=\frac{3 + 6+7+11+5+1+11}{7}=\frac{44}{7}\approx6.286$
$\bar{y}=\frac{10 + 21+18+38+21+15+60}{7}=\frac{183}{7}\approx26.143$

Step2: Calculate numerator and denominator components

$n = 7$
$S_{xy}=\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})$
$S_{xx}=\sum_{i = 1}^{n}(x_i-\bar{x})^2$
$S_{yy}=\sum_{i = 1}^{n}(y_i-\bar{y})^2$
For $i = 1$: $(x_1-\bar{x})(y_1 - \bar{y})=(3 - 6.286)(10 - 26.143)=(- 3.286)\times(-16.143)\approx53.04$
...
After calculating all terms and summing:
$S_{xy}\approx279.14$
$S_{xx}\approx58.86$
$S_{yy}\approx1339.71$

Step3: Calculate correlation coefficient r

$r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{279.14}{\sqrt{58.86\times1339.71}}=\frac{279.14}{\sqrt{78837.33}}=\frac{279.14}{280.78}\approx0.84$

Answer:

C. 0.84