Question

researchers at a pharmaceutical company are testing a new drug that regulates blood sugar. in one test, subjects were prescribed a random and safe dose of the drug. once the drugs were administered, the researchers measured each subjects blood sugar levels before and after a meal. for each subject, the company recorded the given dose (in milligrams), x, and the rise in blood sugar (in milligrams per deciliter), y.

dosage\trise in blood sugar level
32\t20
52\t30
66\t15
69\t15
82\t17

find the correlation coefficient, r, of the data described below. round your answer to the nearest thousandth.

Explanation:

Step1: Recall correlation - coefficient formula

The formula for the correlation coefficient $r$ is $r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$. Let $x$ be the dosage and $y$ be the rise in blood - sugar level.
We have $n = 5$ data points.
First, calculate the sums:
Let $x_1 = 32,x_2 = 52,x_3 = 66,x_4 = 69,x_5 = 82$ and $y_1 = 20,y_2 = 30,y_3 = 15,y_4 = 15,y_5 = 17$.
$\sum x=32 + 52+66+69+82=301$
$\sum y=20 + 30+15+15+17=97$
$\sum xy=(32\times20)+(52\times30)+(66\times15)+(69\times15)+(82\times17)$
$=640+1560 + 990+1035+1394=5619$
$\sum x^{2}=32^{2}+52^{2}+66^{2}+69^{2}+82^{2}$
$=1024+2704+4356+4761+6724=19569$
$\sum y^{2}=20^{2}+30^{2}+15^{2}+15^{2}+17^{2}$
$=400+900+225+225+289=2039$

Step2: Substitute values into the formula

$n\sum xy=5\times5619 = 28095$
$\sum x\sum y=301\times97=29197$
$n\sum x^{2}=5\times19569=97845$
$(\sum x)^{2}=301^{2}=90601$
$n\sum y^{2}=5\times2039 = 10195$
$(\sum y)^{2}=97^{2}=9409$
The denominator is:
$\sqrt{(97845 - 90601)(10195 - 9409)}$
$=\sqrt{7244\times786}$
$=\sqrt{5693784}\approx2386.17$
The numerator is $28095-29197=-1102$
$r=\frac{-1102}{2386.17}\approx - 0.462$

Answer:

$-0.462$