Question

consider the given density curve. what is the value of the median? -10 -7 -6 -2

Explanation:

Step1: Recall the property of median for uniform distribution

For a uniform distribution (rectangular density curve), the median \(m\) satisfies \(\int_{a}^{m}f(x)dx = 0.5\), where \(a\) is the left - hand endpoint of the interval and \(f(x)\) is the probability density function. Here, \(a=-10\), \(f(x)=\frac{1}{8}\) (since the length of the interval \([- 10,-2]\) is \(b - a=-2-(-10)=8\) and for a uniform distribution \(f(x)=\frac{1}{b - a}\) on \([a,b]\)).

Step2: Set up the integral equation

We know that \(\int_{-10}^{m}\frac{1}{8}dx=0.5\). Using the integral formula \(\int_{-10}^{m}\frac{1}{8}dx=\frac{1}{8}(m + 10)\) (because \(\int kdx=kx+C\), here \(k = \frac{1}{8}\)).

Step3: Solve the equation for \(m\)

Set \(\frac{1}{8}(m + 10)=0.5\). Multiply both sides by \(8\): \(m + 10=4\). Then subtract \(10\) from both sides: \(m=-6\).

Answer:

\(-6\)