Question

NASA performed aerodynamic and acoustic tests of two and three-dimensional airfoil blade sections. These tests were conducted in an anechoic wind tunnel. One input (X) measured was the free-stream velocity, in meters per second. The output (Y ) was scaled sound pressure level, in decibels. From this data, we found the following values: µX = 50.86; µY = 124.8; σXY = 13.44; σX = 15.57; σY = 6.90. (a) Find the value of the correlation coefficient to 3 decimals. (b) Is the linear relationship / association between X and Y strong, weak, or is there no relationship? (c) Is there a positive linear relationship, negative linear relationship, or no linear relationship?

Answer 1

a) ρ (X,Y) = 0.125

b) The linear relationship is weak because ρ (X,Y) is almost 0

c) There is a positive linear relationship because ρ (X,Y) > 0

Explanation:

The Pearson correlation coefficient is calculate as :

ρ (X,Y) = (σXY) / (σX)(σY)

-1 ≤ ρ ≤ 1

When ρ = 0 there is no linear relationship between the variables. This does not mean that the variables are independent.

When ρ = 1 the linear relationship is perfect positive. ρ = 1 means a direct relation between X and Y : When X increases also Y increases in constant proportion. When Y increases also X increases in constant proportion.

When ρ = -1 the linear relationship is perfect negative. ρ = -1 means a inverse relation between X and Y : When X increases Y decreases in constant proportion. When Y increases X decreases in constant proportion.

In our exercise ρ (X,Y) = 13.44 / (15.57)(6.90) = 0.125

a) ρ (X,Y) = 0.125

b) The linear relationship is weak because ρ (X,Y) is almost 0

c) There is a positive linear relationship because ρ (X,Y) > 0