In Exercises 1–5, assume that 4.2% of workers test positive when tested for illegal drugs (based on data from Quest Diagnostics). Assume that a group of ten workers is randomly selected.

Workplace Drug Testing If none of the ten workers tests positive for illegal drugs, is that a significantly low result?

검증된 단계별 안내

Step 1: Identify the probability of a worker testing positive for illegal drugs, which is given as 4.2% or 0.042. The probability of a worker not testing positive is therefore 1 - 0.042 = 0.958.

Step 2: Recognize that the problem involves a binomial distribution, where the number of trials (n) is 10, the probability of success (testing positive) is 0.042, and the probability of failure (testing negative) is 0.958.

Step 3: Calculate the probability of none of the ten workers testing positive. This corresponds to the binomial probability P(X = 0), where X is the number of workers testing positive. Use the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k). For this case, k = 0, n = 10, and p = 0.042.

Step 4: Simplify the formula for P(X = 0). Since k = 0, the term (n choose k) simplifies to 1, and p^k simplifies to 1 as well. The formula becomes P(X = 0) = (1) * (0.042)^0 * (0.958)^10.

Step 5: Determine whether the result is significantly low by comparing the calculated probability P(X = 0) to a threshold for significance, such as 0.05 (commonly used in hypothesis testing). If P(X = 0) is less than 0.05, the result is considered significantly low.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, the probability of a worker testing positive for illegal drugs is 4.2%, which can be used to calculate the expected outcomes for a group of workers. Understanding probability helps in assessing whether the observed result (none testing positive) is unusual given the known rate.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this scenario, testing ten workers can be modeled as a binomial distribution where 'success' is defined as a worker testing positive. This distribution allows us to calculate the probability of observing zero positives among the ten workers.

Statistical Significance

Statistical significance refers to the likelihood that a result or relationship is caused by something other than mere random chance. To determine if the result of none of the ten workers testing positive is significantly low, we can compare the observed outcome to the expected distribution of results based on the known probability. If the probability of observing such a result is very low, it may indicate a significant finding.