We are in the year 2050 and Aliens (the Radmen) coming from planet Radar in the galaxy RF, are invading planet Earth. The Global World Government (GWG) makes the decision to eradicate the invasion and to detect all the invaders. The problem is that Radar aliens look like Humans and only very sophisticated (and expensive) medical exams allow to surely identify them. However, due to a different gravity on their planet, the Radmen have a higher rate of heartbeat than Humans.
As it is not possible to examine all the world population, it was decided that the detection of the aliens will be based on a heartbeat criteria easily applicable to the whole population.
A human heartbeat data base (HHDB) was establish as the rate of heartbeat density function of the population (measured in beats per minute (beats/min)) (table1).
Beats per min |
< 45 |
45 50 |
50 55 |
55 60 |
60 65 |
65 70 |
70 75 |
75 80 |
80 85 |
85 90 |
90 95 |
95 100 |
100 105 |
> 105 |
% |
0.18 |
0.70 |
1.45 |
3.9 |
10.1 |
16.9 |
21.0 |
18.9 |
13.3 |
7.3 |
3.6 |
1.7 |
0.73 |
0.24 |
The first wave of invaders was composed only of males resulting from a severe selection (based on physical and intellectual criteria) and gave a quite uniform (non fluctuating) population whose characteristics were determined from a set of 200 identified elements (table 2).
Beats per min |
< 85 |
85 90 |
90 95 |
95 100 |
100 105 |
105 110 |
110 115 |
115 120 |
120 125 |
125 130 |
130 135 |
135 140 |
140 145 |
> 145 |
number |
0 |
1 |
2 |
10 |
18 |
32 |
46 |
43 |
23 |
17 |
6 |
1 |
0 |
1 |
The information in the two tables is plotted below in histogram fashion. To calculate the probability density, convert the number of individuals in each group into the fraction that the group represents of the whole population. In the histograms plotted below, the probability density is based on the discrete group intervals in the tables. In a more formal determination of probability density for this problem, one would present the information as discrete points. We present the data continuously because it is easier to visualize the meaning behind the probability density curve.
The GWG wants to set a threshold heart rate to decide whether the test subject is human or alien. If the subject’s heart rate is above the threshold, then the GWG will say that they are aliens; if the subject’s heart rate is below the threshold, the GWG will say the subject is human. You will need to help the GWG set the threshold and calculate the effectiveness of the test.
This selection process is subject to two kinds of errors :
 | Humans can be selected as Aliens : this error is called False Alarm , the probability of occurrence is the false alarm probability (Pfa). This would occur if a tested human has a high heart rate |
| Aliens are not detected whose probability of occurrence is 1-Pd (Pd being the probability of detection). This would occur if a tested alien had a lower heart rate. |
Question
Economic considerations (cost of the full medical exam) determine that the acceptable Pfa is 0.01 (1%) .
What is the Pd (percentage of detected aliens) for the pioneer wave of invaders ?