Density Plots

A drug test is 95% accurate for a specific drug. This means that it will be correct for 95% of the people tested with it and incorrect for 5% of the people tested with it.

We saw that for a population of which 5% uses the drug for which this test is designed, half the people who test positive are in fact non-users. This means that if disciplinary action is taken against anyone testing positive, 50% of those people will be disciplined when they shouldn’t.

Members of a disciplinary board decide that they cannot use this drug test, and will look for a new test. They want no more than 1% error in the disciplinary actions they take with persons testing positive for this drug. How accurate must the new test be to satisfy their requirement?

A density plot shows a portion of the x-y plane. Each point in that region has an x-coordinate and a y-coordinate. A value of z is calculated when the values of any point's coordinates are entered into a formula involving x and y,

Every point in the region will be plotted in a shade of gray depending on the value of z that is calculated using that point's coordinates. The convention is that if you have two points P (with coordinates u and v) and Q (with coordinates s and t), then the point whose coordinates produce a larger value is plotted in a lighter shade than the one producing the smaller value.

So, the first density plot below shows two things. It shows that when we substitute the coordinates of the point having coordinates (0.1375,0.81875) into the formula displayed above the graph, the formula produces a value of 0.581348 (that is, z=0.581348 when x is 0.1375 and y is 0.81875. It also shows that points nearer the x and y axes produce larger values for z than do the points farther into the intererior of the region.

Subsequent graphs show values of z produced by points located at various places within the square bounded by 0<x<1 and 0<y<1.

The last graph shows that the the formula produces a value of .01 for z given that y=.05 when x is approximately .9995.