Question

find the correlation coefficient, r, of the data described below. 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	rise in blood sugar level
53	29
75	17
77	28
81	20

round your answer to the nearest thousandth.

Explanation:

Step1: Calculate the means

Let $x$ be the dosage and $y$ be the rise in blood - sugar level.
$n = 5$
$\bar{x}=\frac{46 + 53+75+77+81}{5}=\frac{332}{5}=66.4$
$\bar{y}=\frac{30 + 29+17+28+20}{5}=\frac{124}{5}=24.8$

Step2: Calculate the numerator and denominators

The formula for the correlation coefficient $r$ is $r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}$
Calculate $(x_{i}-\bar{x})(y_{i}-\bar{y})$, $(x_{i}-\bar{x})^{2}$ and $(y_{i}-\bar{y})^{2}$ for each $i$:

$x_{i}$	$y_{i}$	$x_{i}-\bar{x}$	$y_{i}-\bar{y}$	$(x_{i}-\bar{x})(y_{i}-\bar{y})$	$(x_{i}-\bar{x})^{2}$	$(y_{i}-\bar{y})^{2}$
53	29	$53 - 66.4=-13.4$	$29 - 24.8 = 4.2$	$-13.4\times4.2=-56.28$	$(-13.4)^{2}=179.56$	$4.2^{2}=17.64$
75	17	$75 - 66.4 = 8.6$	$17 - 24.8=-7.8$	$8.6\times(-7.8)=-67.08$	$8.6^{2}=73.96$	$(-7.8)^{2}=60.84$
77	28	$77 - 66.4 = 10.6$	$28 - 24.8 = 3.2$	$10.6\times3.2 = 33.92$	$10.6^{2}=112.36$	$3.2^{2}=10.24$
81	20	$81 - 66.4 = 14.6$	$20 - 24.8=-4.8$	$14.6\times(-4.8)=-70.08$	$14.6^{2}=213.16$	$(-4.8)^{2}=23.04$

$\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=-106.08-56.28 - 67.08+33.92-70.08=-265.6$
$\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=416.16+179.56+73.96+112.36+213.16=995.2$
$\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}=27.04+17.64+60.84+10.24+23.04=138.8$

Step3: Calculate the correlation coefficient

$r=\frac{-265.6}{\sqrt{995.2\times138.8}}=\frac{-265.6}{\sqrt{138133.76}}=\frac{-265.6}{371.663}$
$r\approx - 0.715$

Answer:

$-0.715$