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.Q.10

Chapter 5, Problem 5.Q.10

In Exercises 6–10, refer to the accompanying table, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from “Prevalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population,” by Ohayon et al., Neurology, Vol. 78, No. 20).

Significant Events Is 4 a significantly high number of sleepwalkers in a group of 5 adults? Explain.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with determining whether 4 is a significantly high number of sleepwalkers in a group of 5 adults. To do this, we will use the probability distribution provided in the table and apply statistical concepts such as the rare event rule.

Step 2: Recall the rare event rule. According to this rule, an event is considered significantly high if its probability is very small, typically less than or equal to 0.05. We will compare the probability of having 4 sleepwalkers (P(x=4)) to this threshold.

Step 3: Locate the probability of 4 sleepwalkers in the table. From the table, P(x=4) = 0.027. This value represents the likelihood of observing 4 sleepwalkers in a group of 5 adults.

Step 4: Compare P(x=4) to the threshold of 0.05. If P(x=4) ≤ 0.05, then 4 sleepwalkers can be considered a significantly high number based on the rare event rule.

Step 5: Interpret the result. Based on the comparison, determine whether the probability of 4 sleepwalkers is rare enough to be considered significant. Provide reasoning based on the threshold and the probability value.

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Key Concepts

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

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. In this case, the table shows the probabilities of different numbers of sleepwalkers (x) in a group of five adults. Understanding this distribution helps in assessing the likelihood of observing a specific number of sleepwalkers.

Significance Level

The significance level is a threshold used to determine whether a result is statistically significant. In this context, we assess whether having 4 sleepwalkers in a group of 5 is significantly high by comparing its probability to a predefined significance level, often set at 0.05. If the probability of observing 4 or more sleepwalkers is low, it may be considered significant.

Cumulative Probability

Cumulative probability refers to the probability of a random variable being less than or equal to a certain value. To determine if 4 sleepwalkers is significantly high, we can calculate the cumulative probability of observing 0, 1, 2, 3, or 4 sleepwalkers and compare it to the probability of observing 5. This helps in understanding the rarity of the event in the context of the distribution.

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Related Practice

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

Find the mean of the random variable x described in the preceding exercise.

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Textbook Question

Kentucky Pick 4 In the Kentucky Pick 4 lottery game, you can pay \$1 for a “straight” bet in which you select four digits with repetition allowed. If you buy only one ticket and win, your prize is \$2500.

a. If you buy one ticket, what is the probability of winning?

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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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Textbook Question

Is the mean found in the preceding exercise a statistic or a parameter?

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