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

Larson 8th Edition

Ch. 5 - Normal Probability Distributions

Problem 5.2.12c

Chapter 5, Problem 5.2.12c

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.

Verified step by step guidance

Step 1: Identify the key parameters of the normal distribution. The problem states that the mean (μ) is 20 minutes and the standard deviation (σ) is 5 minutes. We are tasked with finding the probability that a randomly selected athlete uses the stairclimber for more than 30 minutes.

Step 2: Standardize the value of 30 minutes to a z-score using the z-score formula: z = (X - μ) / σ. Here, X is the value of interest (30 minutes), μ is the mean (20 minutes), and σ is the standard deviation (5 minutes). Substitute the values into the formula.

Step 3: Once the z-score is calculated, use a z-table or technology (e.g., a statistical calculator or software) to find the cumulative probability associated with this z-score. The cumulative probability represents the area under the normal curve to the left of the z-score.

Step 4: Since the problem asks for the probability of using the stairclimber for more than 30 minutes, subtract the cumulative probability from 1. This is because the total area under the normal curve is 1, and the area to the right of the z-score represents the desired probability.

Step 5: Interpret the result. The final value represents the probability that a randomly selected athlete uses the stairclimber for more than 30 minutes. Ensure the result is reasonable given 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.

Normal Distribution

Normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the mean represents the average time spent on the stairclimber, while the standard deviation indicates the variability of workout times. Understanding this distribution is crucial for calculating probabilities related to workout durations.

Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It is calculated by subtracting the mean from the value and dividing by the standard deviation. In this problem, the Z-score will help determine how many standard deviations the time of 30 minutes is from the mean of 20 minutes, allowing for the calculation of the corresponding probability.

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring within a defined set of outcomes. For normally distributed data, this often requires using Z-scores and standard normal distribution tables or technology to find the area under the curve that corresponds to the desired outcome. In this case, we need to find the probability that an athlete uses the stairclimber for more than 30 minutes.

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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.

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Textbook Question

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.

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Textbook Question

Uniform Distribution A uniform distribution is a continuous probability distribution for a random variable x between two values a and b (a<b), where (a ≤ x ≤ b) and all of the values of x are equally likely to occur. The graph of a uniform distribution is shown below.

The probability density function of a uniform distribution is

on the interval from (x=a) to (x=b). For any value of x less than a or greater than b, y=0 . In Exercises 59 and 60, use this information.

For two values c and d, where a ≤ c < d ≤ b, the probability that x lies between c and d is equal to the area under the curve between c and d, as shown below.

So, the area of the red region equals the probability that x lies between c and d. For a uniform distribution from (a=1) to (b=25) , find the probability that

d. x lies between 8 and 14.

248

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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.

106

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Textbook Question

History Grades In a history class, the grades for various assessments are all positive numbers and have different distributions. Determine whether the grades for each assessment could be normally distributed. Explain your reasoning.

e. an extra credit assignment with a mean of 2.25 and a standard deviation of 2.49

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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)

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