Question

When asked an embarrassing question in a survey- such as whether the respondent has ever shoplifted individuals may be reluctant to answer truthfully. However, answers might be more truthful if the survey incorporates a random component, such as a coin toss, that prevents the questioner from determining whether any given individual is guilty (Warner 1965). For example, consider a survey of a population in which 20% of individuals really have shoplifted at least once. The survey asks every participating individual to begin by flipping a fair coin twice. If the result of the first toss is heads, then the individual is instructed to answer honestly the question "did the second toss also yield heads?" If the first coin toss yields tails, however, the respondent is instructed to answer honestly the question "have you ever shoplifted?"What is the overall probability that a randomly sampled respondent answers yes?

Answer 1

The overall probability that a randomly sampled respondent answers yes in the given survey scenario is 0.35.

Step-by-step explanation:

The overall probability that a randomly sampled respondent answers yes can be calculated by considering the different scenarios in which the respondent can answer yes. Let's break it down step by step:

1. If the first coin toss is heads, the respondent is instructed to answer honestly the question 'did the second toss also yield heads?'. The probability of this happening is 0.5 * 0.5 = 0.25.
2. If the first coin toss is tails, the respondent is instructed to answer honestly the question 'have you ever shoplifted?'. We know that 20% of the population has shoplifted, so the probability of this happening is 0.5 * 0.2 = 0.1.

Now we can calculate the overall probability by summing up the probabilities from the two scenarios: 0.25 + 0.1 = 0.35. Therefore, the overall probability that a randomly sampled respondent answers yes is 0.35.