Question

find the correlation coefficient, r, of the data described below. a medical student reads that with enough sun exposure, the human body can manufacture vitamin d3. this sounds too amazing to be true, so he decides to test it out. over the next few weeks, he spends a varying number of hours in the sun, x. at the end of each week, he measures the concentration of vitamin d3 in his blood (in nanograms per milliliter), y. hours spent in sun concentration of vitamin d3 (in ng per ml) 3 32 4 36 5 48 8 38 9 77 round your answer to the nearest thousandth. r =

Explanation:

Step1: Calcular sumatorias

Sean $x_i$ las horas en el sol y $y_i$ la concentración de vitamina D3.
$n = 5$
$\sum_{i = 1}^{n}x_i=3 + 4+5 + 8+9=29$
$\sum_{i = 1}^{n}y_i=32 + 36+48 + 38+77=231$
$\sum_{i = 1}^{n}x_i^2=3^2 + 4^2+5^2 + 8^2+9^2=9 + 16+25 + 64+81=195$
$\sum_{i = 1}^{n}y_i^2=32^2 + 36^2+48^2 + 38^2+77^2=1024+1296 + 2304+1444+5929=12001$
$\sum_{i = 1}^{n}x_iy_i=3\times32 + 4\times36+5\times48 + 8\times38+9\times77=96+144 + 240+304+693=1477$

Step2: Aplicar fórmula de coeficiente de correlación

La fórmula para el coeficiente de correlación $r$ es:
\[r=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{\sqrt{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}\sqrt{n\sum_{i = 1}^{n}y_i^2-(\sum_{i = 1}^{n}y_i)^2}}\]
Sustituir valores:
\[n\sum_{i = 1}^{n}x_iy_i=5\times1477 = 7385\]
\[\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i=29\times231 = 6699\]
\[n\sum_{i = 1}^{n}x_i^2=5\times195 = 975\]
\[(\sum_{i = 1}^{n}x_i)^2=29^2 = 841\]
\[n\sum_{i = 1}^{n}y_i^2=5\times12001 = 60005\]
\[(\sum_{i = 1}^{n}y_i)^2=231^2 = 53361\]

\[r=\frac{7385 - 6699}{\sqrt{975-841}\sqrt{60005 - 53361}}\]
\[r=\frac{686}{\sqrt{134}\sqrt{6644}}\]
\[r=\frac{686}{\sqrt{134\times6644}}\]
\[r=\frac{686}{\sqrt{889296}}\]
\[r=\frac{686}{943.0249}\]
\[r\approx0.727\]

Answer:

$r\approx0.727$