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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 5.5.27c
Chapter 5, Problem 5.5.27c

Daily Commute About 83% of U.S. employees drive their own vehicle to work. You randomly select a sample of U.S. employees. Find the probability that more than 100 of the employees drive their own vehicle to work. (Source: U.S. Bureau of Labor Statistics)


c. You select 150 U.S. employees.

Verified step by step guidance
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Step 1: Identify the type of probability distribution to use. Since we are dealing with a proportion (83%) and a sample size (n = 150), this problem involves a binomial distribution. However, for large sample sizes, the binomial distribution can be approximated by a normal distribution. Check if the normal approximation is valid by ensuring that both np and n(1-p) are greater than 5.
Step 2: Calculate the mean (μ) and standard deviation (σ) of the binomial distribution. The mean is given by μ = np, and the standard deviation is given by σ = √(np(1-p)), where n is the sample size and p is the proportion of employees who drive their own vehicle to work.
Step 3: Convert the problem to a normal distribution approximation. To find the probability that more than 100 employees drive their own vehicle to work, calculate the z-score using the formula z = (X - μ) / σ, where X is the value of interest (100 in this case).
Step 4: Use the z-score to find the corresponding probability. Look up the z-score in a standard normal distribution table or use statistical software to find the cumulative probability up to the z-score. Subtract this cumulative probability from 1 to find the probability of more than 100 employees driving their own vehicle to work.
Step 5: Interpret the result. The final probability represents the likelihood that more than 100 employees in the sample of 150 drive their own vehicle to work. Ensure the interpretation aligns with the context of the problem.

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Key Concepts

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

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, it helps determine the chance that more than 100 out of 150 randomly selected employees drive their own vehicle to work, given that 83% of employees do so.
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Introduction to Probability

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. Here, it applies to the scenario of selecting 150 employees, where each employee either drives their own vehicle (success) or does not (failure), allowing us to calculate the probability of observing more than 100 successes.
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Mean & Standard Deviation of Binomial Distribution

Normal Approximation

The normal approximation to the binomial distribution is used when the sample size is large, allowing for easier calculations. Since the sample size of 150 is sufficiently large, we can approximate the binomial distribution with a normal distribution to find the probability of more than 100 employees driving their own vehicle.
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Using the Normal Distribution to Approximate Binomial Probabilities
Related Practice
Textbook Question

Approximating Binomial Probabilities In Exercises 19–26, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain.


Social Media A survey of Americans found that 55% would be disappointed if Facebook disappeared. You randomly select 500 Americans and ask them whether they would be disappointed if Facebook disappeared. Find the probability that the number who say yes is (c) between 240 and 280, inclusive.

Textbook Question

Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.


Health Club Schedule The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 20 minutes and a standard deviation of 5 minutes. Find the probability that a randomly selected athlete uses a stairclimber for (c) more than 30 minutes.

Textbook Question

Employee Wellness A survey of employed U.S. adults found that only 35% believe their employer cares about their well-being. You randomly select a sample of U.S. employees. Find the probability that fewer than 100 believe their employer cares about their well-being. (Source: Gallup)


c. You select 400 U.S. employees.

Textbook Question

Approximating Binomial Probabilities In Exercises 19–26, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain.


Advancing Research In a survey of U.S. adults, 77% said are willing to share their personal health information to advance medical research. You randomly select 500 U.S. adults. Find the probability that the number who are willing to share their personal health information to advance medical research is (c) between 380 and 390 inclusive.

Textbook Question

Approximating Binomial Probabilities In Exercises 19–26, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Identify any unusual events. Explain.


Athletes on Social Issues In a survey of college athletes, 84% said they are willing to speak up and be more active in social issues. You randomly select 25 college athletes. Find the probability that the number who are willing to speak up and be more active in social issues is (c) between 18 and 22, inclusive.

Textbook Question

Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities.


MCAT Scores In a recent year, the MCAT total scores were normally distributed, with a mean of 500.9 and a standard deviation of 10.6. Find the probability that a randomly selected medical student who took the MCAT has a total score that is (c) more than 515. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges)