The variable x has median 20 and interquartile range 10. The variable y is related to x by y = ax - b. Find the relationship between a and b so that the median of y equals the interquartile range of y.

The relationship between a and b ⇒ b = 10a

Step-by-step explanation:

The variable y is related to x by y = ax - b

When x has median 20

∴ y = 20a - b ⇒ (1)

interquartile range of y is the difference between two values of y

IQR of y = (ax₂ - b) - (ax₁ - b)

= ax₂ - b - ax₁ + b

= ax₂ - ax₁

= a (x₂ - x₁)

Where (x₂ - x₁) = interquartile range of x

∴ IQR of y = ax

When interquartile range of x = 10

y = 10a ⇒ (2)

median of y equals the interquartile range of y

From (1) and (2)

∴ 20a - b = 10a

Combine like terms

∴ 20a - 10a = b

∴ b = 10a

So, The relationship between a and b ⇒ b = 10a