A certain density curve looks like an interverted letter V. The first segment goes fro the point (0,0.6) to the point (0.5,1.4). The segment goes from (0.5.1.44) to (1,0.6).
(a) Sketch the curve. Verify that the area under the curve is 1, so that it is a valid density.
-- Okay, I tried graphing that on my calculator and it came out looking weird... my highest bar on the histogram is 3. How do you verfiy the area under the curve? And how am I suppose to determine the median ? Do I use ALL of those numbers?
Please help and thank you!
To sketch the curve, you can follow these steps:
1. Plot the points (0, 0.6), (0.5, 1.4), and (1, 0.6) on a graph.
2. Draw a straight line segment connecting the first two points, starting from (0, 0.6) to (0.5, 1.4). This forms the left side of the inverted V.
3. Draw another straight line segment connecting the last two points, starting from (0.5, 1.4) to (1, 0.6). This forms the right side of the inverted V.
4. Connect the two line segments at their endpoints, creating a smooth curve in the shape of an inverted letter V.
Now, to verify that the area under the curve is 1, you need to calculate the area of the shape enclosed by the curve and the x-axis. Since the shape is a triangle (in the form of an inverted V), you can use the formula for the area of a triangle:
A = (base * height) / 2
In this case, the base is the horizontal distance between the two endpoints of the triangle, which is 1 - 0 = 1. The height is the vertical distance between the highest point of the triangle and the x-axis, which is 1.4 - 0.6 = 0.8.
So, plugging these values into the formula, you get:
A = (1 * 0.8) / 2 = 0.4
Since the area of the triangle is 0.4, it means that the area under the curve is 0.4, not 1. Therefore, the given curve does not form a valid density because the area under the curve should be equal to 1.
Regarding determining the median, you first need to find the equation of the curve to determine the x-values that correspond to the median. From the given information, you can see that the curve is symmetric around x = 0.5, and as such, the median would be at x = 0.5.
Please note that the histogram with a bar at 3 is unrelated to the given density curve and the calculation of the area under the curve.