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6 September, 19:36

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?

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  1. 6 September, 22:00
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    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

    Step-by-step 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
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