Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.RE.5

Chapter 5, Problem 5.RE.5

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 four of the ten workers test positive for illegal drugs, is that a significantly high result?

Verified step by step guidance

Step 1: Define the problem in terms of a binomial distribution. The number of workers testing positive (X) follows a binomial distribution with parameters n = 10 (number of trials) and p = 0.042 (probability of success, i.e., testing positive). The probability mass function for a binomial distribution is given by: P(X = k) = (n choose k) * p^k * (1-p)^(n-k).

Step 2: Calculate the mean (μ) and standard deviation (σ) of the binomial distribution. The mean is given by μ = n * p, and the standard deviation is given by σ = sqrt(n * p * (1-p)).

Step 3: Determine the threshold for a 'significantly high' result. A common rule of thumb is to consider results significantly high if they are greater than μ + 2σ. Compute this threshold using the values of μ and σ calculated in Step 2.

Step 4: Compare the observed value (4 workers testing positive) to the threshold calculated in Step 3. If the observed value exceeds the threshold, it is considered significantly high.

Step 5: Conclude whether the result is significantly high based on the comparison in Step 4. If the observed value is not greater than the threshold, it is not significantly high; otherwise, it is.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 context, testing positive for illegal drugs can be seen as a 'success,' and the distribution helps determine the likelihood of observing a certain number of positives among the ten workers.

Significance Level

The significance level, often denoted as alpha (α), is the threshold used to determine whether a result is statistically significant. Commonly set at 0.05, it indicates the probability of rejecting the null hypothesis when it is true. In this scenario, it helps assess whether the observed four positive tests are unusually high compared to what would be expected under the null hypothesis.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating a null hypothesis (e.g., the proportion of positive tests is 4.2%) and an alternative hypothesis (e.g., the proportion is greater than 4.2%). The results from the binomial distribution can be used to determine if the observed data provides enough evidence to reject the null hypothesis.

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06:21

Step 1: Write Hypotheses

Related Practice

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b. Find the probability that on a given day, there are no deaths.

Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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 Find the probability that at least one of the ten workers tests positive for illegal drugs.

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Textbook Question

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