find the correlation coefficient, r, of the data described below. researchers at a pharmaceutical company…
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. \n|dosage| rise in blood sugar level|\n|----|----|\n|46|30|\n|53|29|\n|75|17|\n|77|28|\n|81|20|\nround your answer to the nearest thousandth.
Answer
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 $i$:
| $x_{i}$ | $y_{i}$ | $x_{i}-\bar{x}$ | $y_{i}-\bar{y}$ | $(x_{i}-\bar{x})(y_{i}-\bar{y})$ | $(x_{i}-\bar{x})^{2}$ | $(y_{i}-\bar{y})^{2}$ |
|---|---|---|---|---|---|---|
| 46 | 30 | $46 - 66.4=-20.4$ | $30 - 24.8 = 5.2$ | $-20.4\times5.2=-106.08$ | $(-20.4)^{2}=416.16$ | $5.2^{2}=27.04$ |
| 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:
$-0.715$