A breath analyzer is used by the police to estimate blood alcohol content (BAC) from a breath sample. If a person’s BAC is above (or equal to) the legal limit, they can be arrested for suspicion of driving while impaired from alcohol. However, the only way to accurately measure BAC level is through taking a blood sample; a number of factors are known to influence the results of breath analyzers, such as hypoglycemia. A particular brand of breath analyzer is accurate about 80% of the time. That is, if an individual actually has BAC equal to or above the legal limit, the device indicates a positive result with probability 0.80, and if an individual actually has BAC below the legal limit, the device indicates a negative result with probability 0.80.
Suppose that on any particular Saturday night, about 5% of drivers are known to be driving under the influence.
i. Calculate the probability that a driver who tests positive actually has BAC level equal to or above the legal limit.
ii. How accurate would the device need to be for the probability in part i. to be 0.80?
iii. In language accessible to someone who has not taken a statistics course, explain why the probability in part i. is much lower than the accuracy of the breath analyzer. Limit your answer to at most five sentences.