Question

consider the given density curve. what is the value of the median? 8 12 14 20

Explanation:

Step1: Recall the property of median in density curve

The median \(m\) of a density curve satisfies \(\int_{-\infty}^{m}f(x)dx = 0.5\). For a uniform density curve \(f(x)=\frac{1}{b - a}\) (here \(a = 8\), \(b=20\), \(f(x)=\frac{1}{12}\)), the area of the rectangle formed by the density curve from \(a\) to \(m\) is \(A=(m - a)\times f(x)\).

Step2: Set up the equation for the median

We know that \(A = 0.5\), \(f(x)=\frac{1}{12}\). Substitute into the area formula \(A=(m - a)\times f(x)\), we get \(0.5=(m - 8)\times\frac{1}{12}\).

Step3: Solve the equation for \(m\)

Multiply both sides of the equation \(0.5=(m - 8)\times\frac{1}{12}\) by \(12\): \(0.5\times12=m - 8\). Then \(6=m - 8\). Add \(8\) to both sides: \(m=6 + 8=14\).

Answer:

14