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Ch. 5 - Normal Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 5 - Normal Probability DistributionsProblem 5.CR.1c
Chapter 5, Problem 5.CR.1c

A survey of adults in the United States found that 61% ate at a restaurant at least once in the past week. You randomly select 30 adults and ask them whether they ate at a restaurant at least once in the past week. (Source: Gallup)


c. Is it unusual for exactly 14 out of 30 adults to have eaten in a restaurant at least once in the past week? Explain your reasoning.

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Step 1: Recognize that this is a binomial probability problem. The survey indicates that the probability of success (an adult eating at a restaurant at least once in the past week) is p = 0.61, and the probability of failure is q = 1 - p = 0.39. The number of trials is n = 30, and we are interested in the probability of exactly x = 14 successes.
Step 2: Use the binomial probability formula to calculate the probability of exactly 14 successes: P(X = x) = (n choose x) * p^x * q^(n-x). Here, (n choose x) is the binomial coefficient, which can be calculated as (n! / (x! * (n - x)!)).
Step 3: Calculate the mean (μ) and standard deviation (σ) of the binomial distribution to assess whether 14 is unusual. The mean is given by μ = n * p, and the standard deviation is given by σ = sqrt(n * p * q).
Step 4: Determine the range of usual values using the rule of thumb: usual values typically fall within μ ± 2σ. Calculate the lower bound (μ - 2σ) and the upper bound (μ + 2σ).
Step 5: Compare the value of x = 14 to the range of usual values. If 14 falls outside the range, it is considered unusual. If it falls within the range, it is not unusual. Provide reasoning based on this comparison.

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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, each adult's response to whether they ate at a restaurant can be seen as a trial, with 'success' defined as having eaten at a restaurant. The parameters for this distribution are the number of trials (30 adults) and the probability of success (61% or 0.61).
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Mean & Standard Deviation of Binomial Distribution

Normal Approximation

For large sample sizes, the binomial distribution can be approximated by a normal distribution, which simplifies calculations. This approximation is valid when both np and n(1-p) are greater than 5. In this case, with n=30 and p=0.61, we can use the normal distribution to assess the likelihood of observing exactly 14 successes.
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Using the Normal Distribution to Approximate Binomial Probabilities

Unusual Events

An event is considered unusual if its probability of occurrence is less than 5%. To determine if having exactly 14 out of 30 adults eating at a restaurant is unusual, we can calculate the probability of this outcome using the binomial distribution or its normal approximation. If this probability is below 0.05, we would classify it as unusual.
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Related Practice
Textbook Question

Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.


Find the positive z-score for which 12% of the distribution’s area lies between and z.

Textbook Question

Use the probability distribution in Exercise 3 to find the probability of randomly selecting a game in which DeMar DeRozan had (a) fewer than four personal fouls,

Textbook Question

In Exercises 1–4, a population has a mean mu and a standard deviation sigma. Find the mean and standard deviation of the sampling distribution of sample means with sample size n.


Mu = 790, sigma =48, n = 250

Textbook Question

Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (a) exactly 12,

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Textbook Question

Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (b) between 8 and 11, inclusive,

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Textbook Question

Use the probability distribution in Exercise 3 to find the probability of randomly selecting a game in which DeMar DeRozan had (c) between two and four personal fouls, inclusive.