Ch. 5 - Normal Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 5 - Normal Probability Distributions

Problem 5.2.12b

Chapter 5, Problem 5.2.12b

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 (b) between 20 and 28 minutes.

Verified step by step guidance

Step 1: Identify the key parameters of the normal distribution. The mean (μ) is 20 minutes, and the standard deviation (σ) is 5 minutes. The problem asks for the probability that the time spent on the stairclimber is between 20 and 28 minutes.

Step 2: Standardize the values 20 and 28 using the z-score formula: z = (x - μ) / σ. For x = 20, calculate z₁ = (20 - 20) / 5. For x = 28, calculate z₂ = (28 - 20) / 5.

Step 3: Use the standard normal distribution table or technology to find the cumulative probabilities corresponding to z₁ and z₂. The cumulative probability for z₁ represents the area under the curve to the left of z₁, and similarly for z₂.

Step 4: Subtract the cumulative probability for z₁ from the cumulative probability for z₂ to find the probability that the time spent is between 20 and 28 minutes. This is because the area between z₁ and z₂ represents the desired probability.

Step 5: Interpret the result. The final probability represents the likelihood that a randomly selected athlete spends between 20 and 28 minutes on the stairclimber, based on the given normal distribution.

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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, it describes how the workout times of athletes are distributed around the average time of 20 minutes, with most athletes working out close to this mean and fewer athletes working out significantly longer or shorter.

Mean and Standard Deviation

The mean is the average value of a dataset, while the standard deviation measures the amount of variation or dispersion from the mean. In this scenario, the mean of 20 minutes indicates the typical workout time, and the standard deviation of 5 minutes shows how much individual workout times vary from this average, helping to understand the spread of workout durations.

Finding Probabilities

Finding probabilities in a normal distribution involves calculating the area under the curve between two values, which can be done using z-scores or statistical software. For the given problem, we need to determine the probability that an athlete works out between 20 and 28 minutes, which requires integrating the normal distribution function or using a standard normal table to find the corresponding probabilities.

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Textbook Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Water Footprint A water footprint is a measure of the appropriation of fresh water. The water footprint (in cubic meters) for a kilogram of wheat can be approximated by a normal distribution, as shown in the figure. (Source: Ecological Indicators)

b. What water footprint represents the 29th percentile?

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.

b. a final with a mean of 72, standard deviation of 9, and 90th percentile score of 93

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 (b) less than 23

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 (b) between 490 and 510. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges)

Textbook Question

Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.

Billboard Hot 100 The length (in seconds) of the 100 most popular songs during the week of May 5, 2021, can be approximated by a normal distribution, as shown in the figure. (Source: Spotify)

b. What song length represents the 17th percentile?

Textbook Question

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 (b) at least 300