Question

what is the value of the correlation coefficient and its interpretation?
0.76: there is a strong, positive association between discharge and speed.
0.87: there is a strong, positive association between discharge and speed.
-0.76: there is a strong, negative association between discharge and speed.
-0.87: there is a strong, negative association between discharge and speed.
(chart with discharge (ft³) values: 1.3, 22, 5.8, 11, 12, 14, 16, 21, 49, 51 and speed values: 2.3, 0.99, 3.5, 5, 16, 22, 27, 14, 33, 35)

Explanation:

Step1: Define variables and count data

Let discharge = $x$, speed = $y$. $n=10$ (number of data pairs)

Step2: Calculate sum of $x$, $y$, $xy$, $x^2$, $y^2$

$\sum x = 1.3+2.2+5.8+11+12+14+16+21+49+51 = 183.3$
$\sum y = 2.3+0.99+3.5+5+16+22+27+14+33+35 = 168.79$
$\sum xy = (1.3×2.3)+(2.2×0.99)+(5.8×3.5)+(11×5)+(12×16)+(14×22)+(16×27)+(21×14)+(49×33)+(51×35) = 2.99+2.178+20.3+55+192+308+432+294+1617+1785 = 4708.468$
$\sum x^2 = 1.3^2+2.2^2+5.8^2+11^2+12^2+14^2+16^2+21^2+49^2+51^2 = 1.69+4.84+33.64+121+144+196+256+441+2401+2601 = 6199.17$
$\sum y^2 = 2.3^2+0.99^2+3.5^2+5^2+16^2+22^2+27^2+14^2+33^2+35^2 = 5.29+0.9801+12.25+25+256+484+729+196+1089+1225 = 4022.5201$

Step3: Apply correlation formula

Pearson correlation coefficient:
$$r=\frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$$
Substitute values:
$$r=\frac{10×4708.468 - 183.3×168.79}{\sqrt{[10×6199.17 - (183.3)^2][10×4022.5201 - (168.79)^2]}}$$
Calculate numerator: $47084.68 - 30949.207 = 16135.473$
Calculate denominator terms:
$10×6199.17 - 183.3^2 = 61991.7 - 33598.89 = 28392.81$
$10×4022.5201 - 168.79^2 = 40225.201 - 28489.0641 = 11736.1369$
Denominator: $\sqrt{28392.81×11736.1369}=\sqrt{333221000.1}≈18254.34$
$$r=\frac{16135.473}{18254.34}≈0.87$$

Step4: Interpret the coefficient

$r≈0.87$, which is close to 1, so there is a strong positive association.

Answer:

0.87; There is a strong, positive association between discharge and speed.