A vendor converts the weights on the packages she sends out from pounds to kilograms (1kg approximately equals 2.2 pounds).

a. How does this affect the mean weight of the packages?

b. How does this affect the standard deviation of the weights?

c. If the vendor begins using heavier packages, which increases the weight of each package by 50 g; how much does this affect the mean weight of the packages?

a) b) the standard deviation and the mean is affected by the conversion factor as well

c) the mean is displaced by b units

Step-by-step explanation:

for a new variable

Y=a*X, where a = constant (conversion factor = 1 kg/2.2 pounds)

then

p (y) = p (a*X) = p (X)

a) mean = μ=E (Y) = ∑ a*X*p (y) = a ∑ X*p (x) = a * E (X)

mean = μ=a*μₓ

b) σ² = ∑ (Y-μ) ² * p (y) = ∑ (a*X-a μₓ) ² * p (y) = a²*∑ (X-μₓ) ² * p (x) = a²*σₓ²

then

standard deviation = σ = √σ²=√ (a²*σₓ²) = a*σₓ

standard deviation = σ = a*σₓ

then the standard deviation and the mean is affected by the conversion factor as well

c) nevertheless for a displacement b

Y₂=X + b (b = constant = 50 gr)

p (Y₂) = p (X + b) = p (X)

then

mean = μ=∑ (X-b) * p (y) = ∑ X*p (x) - b ∑ p (x) = E (X) -

mean = μ=μₓ - b

then the mean is displaced by b units