Question

which graph represents a density curve, and why?
graph a only, because the area under the curve equals 1, and the curve is above the horizontal axis
graph b only, because the area under the curve equals 2, and the curve is above the horizontal axis
both graph a and graph b, because both curves are above the horizontal axis, and their areas are positive
neither graph a nor graph b, because, even though both curves are above the horizontal axis, their areas are not the same value

Explanation:

Step1: Recall density curve properties

A density curve must satisfy two conditions: \( y\geq0 \) (the curve is above or on the horizontal axis) and the total area under the curve is \( 1 \).

Step2: Calculate area for Graph A

Assume the base of the rectangle in Graph A: \( b = 6 - 3=3 \), height \( h=\frac{1}{3} \). Using the area formula for a rectangle \( A = b\times h \), we have \( A=3\times\frac{1}{3}=1 \), and \( y = \frac{1}{3}\geq0 \).

Step3: Analyze other options

For Graph B (not shown in full details, but from the reasoning in the options): If the area is \( 2
eq1 \), it does not satisfy the area - equal - to - 1 condition. For the option of both graphs: Since Graph B (if area is not 1) fails. For the option of neither: Graph A satisfies the conditions.

Answer:

graph A only, because the area under the curve equals 1, and the curve is above the horizontal axis.