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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 subject's 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
45 27
68 19
70 18
76 24
87 17
Round your answer to the nearest thousandth.
r=

User Icon for Bot GPT 3.5 Bot GPT 3.5 answered
4 months ago

To find the correlation coefficient, r, we first need to calculate the mean of the dosage (x) and rise in blood sugar level (y):

Mean of x:
(45 + 68 + 70 + 76 + 87) / 5 = 69.2

Mean of y:
(27 + 19 + 18 + 24 + 17) / 5 = 21

Next, we calculate the numerator of the formula for the correlation coefficient:

Σ((x - mean of x)(y - mean of y)):
(45 - 69.2)(27 - 21) + (68 - 69.2)(19 - 21) + (70 - 69.2)(18 - 21) + (76 - 69.2)(24 - 21) + (87 - 69.2)(17 - 21)
= (-24)(6) + (-1.2)(-2) + (0.8)(-3) + (6.8)(3) + (17.8)(-4)
= -144 + 2.4 - 2.4 + 20.4 - 71.2
= -194.8

Now, we calculate the denominator of the formula for the correlation coefficient:

√(Σ(x - mean of x)^2 * Σ(y - mean of y)^2):
√((-24)² + (-1.2)² + (0.8)² + (6.8)² + (17.8)²) * (√(6² + (-2)² + (-3)² + 3² + (-4)²)
= √(576 + 1.44 + 0.64 + 46.24 + 316.84) * √(36 + 4 + 9 + 9 + 16)
= √(940.16) * √(74)
= 30.668 * 8.602
= 263.693

Finally, we calculate the correlation coefficient, r:

r = -194.8 / 263.693
r ≈ -0.738

Therefore, the correlation coefficient for the given data is approximately -0.738.