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17 June, 06:05

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.

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  1. 17 June, 08:58
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    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
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