Question

The article "FBI Says Fewer than 25 Failed Polygraph Test" (San Luis Obispo Tribune, July 29,2001 ) states that false-positives in polygraph tests (i.e., tests in which an individual fails even though he or she is telling the truth) are relatively common and occur about \(15 \%\) of the time. Suppose that such a test is given to 10 trustworthy individuals. a. What is the probability that all 10 pass? b. What is the probability that more than 2 fail, even though all are trustworthy? c. The article indicated that 500 FBI agents were required to take a polygraph test. Consider the random variable \(x=\) number of the 500 tested who fail. If all 500 agents tested are trustworthy, what are the mean and standard deviation of \(x ?\) d. The headline indicates that fewer than 25 of the 500 agents tested failed the test. Is this a surprising result if all 500 are trustworthy? Answer based on the values of the mean and standard deviation from Part (c).

Short answer

a. The probability that all 10 pass can be calculated using the Binomial Distribution formula. b. The probability that more than 2 fail can also be calculated using the Binomial Distribution formula. c. The mean and standard deviation of the number of agents failing the test can be calculated using informations about statistical properties of binomial distribution. d. If the calculated mean value is higher than 25, then it would be surprising that fewer than 25 agents failed the test as it's lower than the expected failure rate.

Step-by-step solution

Calculate probability all 10 pass

Use the binomial distribution formula to calculate the probability that all 10 pass. The formula is \(P(x) = C(n, x) * (p^x) * ((1-p)^(n-x))\). Here, x is the number of successes (people passing), n is the number of attempts (people taking the test), and p is the probability of success (chance to pass the test). The calculation should look like this: \(P(10) = C(10, 10) * (0.85^10) * ((1-0.85)^(10-10))\).

Calculate probability more than 2 fail

To find the probability that more than 2 fail, first find the probability that 0, 1, or 2 people fail and then subtract these from 1. Calculate \(P(0)\), \(P(1)\), and \(P(2)\) using the binomial distribution formula and then subtract the sum of these probabilities from 1. The calculations should look like this: \(P(more than 2 fail) = 1 - [P(0) + P(1) + P(2)]\).

Calculate the mean and standard deviation of x

In a binomial distribution, the mean \(\mu\) is \(n*p\) and the standard deviation \(σ\) is \(\sqrt{n*p*(1-p)}\). Calculate these values using n=500 and p=0.15. The calculations should look like this: \(\mu = 500*0.15\) and \(σ = \sqrt{500*0.15*(1-0.15)}\).

Analyze whether fewer than 25 failing is surprising

Since the mean (expected value) is calculated in Step 3, compare it with 25 to analyze the expected result. If 25 is below the calculated mean value, that means it's lower than expected and thus surprising.