Question

question 8 of 10
use a calculator to find the r-value of these data. round the value to three decimal places.

a. 0.953
b. -0.909
c. -0.953
d. 0.909

Explanation:

Step1: List data points

$x: [2,6,7,8,12],\ y: [15,13,9,8,5]$

Step2: Calculate key sums

$n=5$
$\sum x = 2+6+7+8+12 = 35$
$\sum y = 15+13+9+8+5 = 50$
$\sum xy = (2×15)+(6×13)+(7×9)+(8×8)+(12×5) = 30+78+63+64+60 = 295$
$\sum x^2 = 2^2+6^2+7^2+8^2+12^2 = 4+36+49+64+144 = 297$
$\sum y^2 = 15^2+13^2+9^2+8^2+5^2 = 225+169+81+64+25 = 564$

Step3: Apply correlation formula

$$ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} $$

Substitute values:

$$ r = \frac{5×295 - 35×50}{\sqrt{[5×297 - 35^2][5×564 - 50^2]}} $$

$$ r = \frac{1475 - 1750}{\sqrt{[1485 - 1225][2820 - 2500]}} $$

$$ r = \frac{-275}{\sqrt{[260][320]}} = \frac{-275}{\sqrt{83200}} = \frac{-275}{288.444} \approx -0.953 $$

Answer:

C. -0.953