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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblem 5.R.3
Chapter 5, Problem 5.R.3

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 mean and standard deviation for the numbers of workers in groups of ten who test positive for illegal drugs.

Verified step by step guidance
1
Step 1: Recognize that this is a binomial distribution problem because there are two possible outcomes for each worker: either they test positive for illegal drugs or they do not.
Step 2: Identify the parameters of the binomial distribution. The probability of success (testing positive) is p = 0.042, and the number of trials (workers) is n = 10.
Step 3: Use the formula for the mean of a binomial distribution, which is μ = n × p. Substitute the values of n and p into the formula to calculate the mean.
Step 4: Use the formula for the standard deviation of a binomial distribution, which is σ = √(n × p × (1 - p)). Substitute the values of n and p into the formula to calculate the standard deviation.
Step 5: Interpret the results. The mean represents the expected number of workers who test positive in a group of ten, and the standard deviation measures the variability around this mean.

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Key Concepts

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

Binomial Distribution

The scenario described involves a binomial distribution, which models the number of successes in a fixed number of independent trials, each with the same probability of success. In this case, testing positive for illegal drugs is considered a 'success,' with a probability of 0.042 (4.2%) for each of the 10 workers tested.
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Mean of a Binomial Distribution

The mean of a binomial distribution can be calculated using the formula μ = n * p, where n is the number of trials and p is the probability of success. For this question, with n = 10 and p = 0.042, the mean represents the expected number of workers testing positive in a group of ten.
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Standard Deviation of a Binomial Distribution

The standard deviation of a binomial distribution is calculated using the formula σ = √(n * p * (1 - p)). This measures the variability or spread of the number of successes around the mean. In this context, it helps to understand how much the actual number of workers testing positive may differ from the expected mean in different samples of ten workers.
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Related Practice
Textbook Question

Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).


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

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Textbook Question

Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).


c. Find the probability that on a given day, there is more than one death.


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

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Textbook Question

Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).

a. Find the mean number of deaths per day.

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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

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 exactly two of the ten workers test positive for illegal drugs.

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