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Ch. 6 - Normal Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 6 - Normal Probability DistributionsProblem 6.q.2
Chapter 6, Problem 6.q.2

Bone Density Test. In Exercises 1–4, assume that scores on a bone mineral density test are normally distributed with a mean of 0 and a standard deviation of 1.


Bone Density Find the bone density score that is the 90th percentile, which is the score separating the lowest 90% from the top 10%.

Verified step by step guidance
1
Step 1: Recognize that the problem involves a normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. This is a standard normal distribution (Z-distribution).
Step 2: Understand that the 90th percentile corresponds to the Z-score where 90% of the data lies below it. This means finding the Z-score such that the cumulative probability (area under the curve to the left of the Z-score) is 0.90.
Step 3: Use a Z-table, statistical software, or a calculator with inverse cumulative distribution function capabilities (often denoted as invNorm or similar) to find the Z-score corresponding to a cumulative probability of 0.90.
Step 4: Interpret the Z-score obtained. This Z-score represents the bone density score that separates the lowest 90% of the population from the top 10%.
Step 5: Verify the result by checking the cumulative probability for the obtained Z-score using a Z-table or statistical software to ensure it is approximately 0.90.

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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 probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the bone mineral density scores are assumed to follow this distribution, which is crucial for determining percentiles.
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Percentiles

A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, the 90th percentile is the score below which 90% of the data points lie. Understanding percentiles is essential for interpreting the results of the bone density test, as it helps identify how an individual's score compares to the population.

Z-scores

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. In a standard normal distribution, a Z-score of 0 corresponds to the mean, while positive and negative values indicate how many standard deviations a score is above or below the mean. To find the 90th percentile score in this context, one would typically look up the corresponding Z-score in a standard normal distribution table.
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Related Practice
Textbook Question

Seat Designs. In Exercises 7–9, assume that when seated, adult males have back-to-knee lengths that are normally distributed with a mean of 23.5 in. and a standard deviation of 1.1 in. (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats.


Find the probability that nine males have back-to-knee lengths with a mean greater than 23.0 in.

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Textbook Question

Hershey Kisses Based on Data Set 38 “Candies” in Appendix B, weights of the chocolate in Hershey Kisses are normally distributed with a mean of 4.5338 g and a standard deviation of 0.1039 g


d. What is the value of the variance?

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Textbook Question

Bone Density Test. In Exercises 1–4, assume that scores on a bone mineral density test are normally distributed with a mean of 0 and a standard deviation of 1.


Bone Density For a randomly selected subject, find the probability of a bone density score between and 2.00.

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Textbook Question

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.


Doorway Height The Boeing 757-200 ER airliner carries 200 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 68.6 in. and a standard deviation of 2.8 in. (based on Data Set 1 “Body Data” in Appendix B).


d. When considering the comfort and safety of passengers, why are women ignored in this case?

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Textbook Question

Seat Designs. In Exercises 7–9, assume that when seated, adult males have back-to-knee lengths that are normally distributed with a mean of 23.5 in. and a standard deviation of 1.1 in. (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats.


Find the probability that a male has a back-to-knee length greater than 25.0 in.

100
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Textbook Question

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.


Water Taxi Safety Passengers died when a water taxi sank in Baltimore’s Inner Harbor. Men are typically heavier than women and children, so when loading a water taxi, assume a worst-case scenario in which all passengers are men. Assume that weights of men are normally distributed with a mean of 189 lb and a standard deviation of 39 lb (based on Data Set 1 “Body Data” in Appendix B). The water taxi that sank had a stated capacity of 25 passengers, and the boat was rated for a load limit of 3500 lb.


d. Is the new capacity of 20 passengers safe?

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