Polygraph lie-detector machines are commonly used in criminal investigations. The device measures nervous excitement, operating on the idea that if a person is telling the truth they will remain calm. The American Polygraph Association claims that polygraphs accurately identify liars 90% of the time. In other words, if a person lied the test results will be positive 90% of the time. Although the polygraph test is good at identifying liars, there is a 50% chance that the polygraph test will say a truthful person is lying. A police force wants to predict the accuracy of lie-detector tests for 1,000 suspected criminals. Assume that 700 per 1,000 suspected criminals tell the truth during polygraph tests. Lie Truth Row Totals Test result: negative P Test result: postive Q Column Totals 1,000 Barber, N. (2013). Do Lie Detectors Work? Psychology Today . What is the value of Q?

Answer:

The value of Q is 350.

Step-by-step explanation:

Denote the events as follows:

X = the test is positive

Y = the test is negative

A = the person is telling the truth

B = the person is telling a lie.

It is provided that:

P (B ∩ X) = 0.90

P (A ∩ Y) = 0.50

Then:

P (B ∩ Y) = 1 - P (B ∩ X) = 1 - 0.90 = 0.10

P (A ∩ X) = 1 - P (A ∩ Y) = 1 - 0.50 = 0.50

To test the accuracy of lie-detector a police officer tests 1000 suspected criminals.

It is assumed that 700 per 1,000 suspected criminals tell the truth during polygraph tests.

The table needed to be completed is:

                       Truth (A)        Lie (B)         TOTAL

Positive (X)           Q                 __               __

Negative (Y)         __                 P                __

TOTAL                   __                __               __

For the 1000 suspected criminals selected complete the table as follows:

Given:

                       Truth (A)        Lie (B)         TOTAL

Positive (X)           Q                 __               __

Negative (Y)         __                 P                __

TOTAL                  700              300           1000

n (A ∩ X) = P (A ∩ X) × n (A)

               = 0.50 × 700

               = 350

n (A ∩ Y) = P (A ∩ Y) × n (A)

               = 0.50 × 700

               = 350

n (B ∩ X) = P (B ∩ X) × n (B)

               = 0.90 × 300

               = 270

n (B ∩ Y) = P (B ∩ Y) × n (B)

               = 0.10 × 300

               = 30

n (X) = n (B ∩ X) + n (A ∩ X)

        =  270 + 350

        = 620

n (Y) = n (B ∩ Y) + n (A ∩ Y)

        =  30 + 350

        = 380

Complete table is:

                         Truth (A)        Lie (B)        TOTAL

Positive (X)       Q = 350           270            620

Negative (Y)        350             P = 30          380

TOTAL                  700               300            1000

Thus, the value of Q is 350.