Question

question #2
use the given data set to determine the correlation coefficient r.

age (years)	height (in.)
3	37.25
4	40.25
5	43
6	45.5

• .984
• .997
• .937
• .908

Explanation:

Step1: Identify variables and n

Let \( x \) be Age (years) and \( y \) be Height (in.). \( n = 5 \).
\( x \): 2, 3, 4, 5, 6
\( y \): 33.5, 37.25, 40.25, 43, 45.5

Step2: Calculate sums

\( \sum x = 2 + 3 + 4 + 5 + 6 = 20 \)
\( \sum y = 33.5 + 37.25 + 40.25 + 43 + 45.5 = 199.5 \)
\( \sum xy = (2×33.5) + (3×37.25) + (4×40.25) + (5×43) + (6×45.5) \)
\( = 67 + 111.75 + 161 + 215 + 273 = 827.75 \)
\( \sum x^2 = 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 4 + 9 + 16 + 25 + 36 = 90 \)
\( \sum y^2 = 33.5^2 + 37.25^2 + 40.25^2 + 43^2 + 45.5^2 \)
\( = 1122.25 + 1387.5625 + 1620.0625 + 1849 + 2070.25 = 8049.125 \)

Step3: Apply correlation formula

The formula for \( r \) is:

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

Substitute values:
Numerator: \( 5×827.75 - 20×199.5 = 4138.75 - 3990 = 148.75 \)
Denominator part 1: \( 5×90 - 20^2 = 450 - 400 = 50 \)
Denominator part 2: \( 5×8049.125 - 199.5^2 = 40245.625 - 39800.25 = 445.375 \)
Denominator: \( \sqrt{50×445.375} = \sqrt{22268.75} ≈ 149.227 \)
\( r = \frac{148.75}{149.227} ≈ 0.997 \)

Answer:

\( 0.997 \) (corresponding to the option with.997)