Ch. 6 - Normal Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 6 - Normal Probability Distributions

Problem 29

Chapter 6, Problem 29

Designing Helmets Engineers must consider the circumferences of adult heads when designing motorcycle helmets. Adult head circumferences are normally distributed with a mean of 570.0 mm and a standard deviation of 18.3 mm (based on Data Set 3 “ANSUR II 2012”). Due to financial constraints, the helmets will be designed to fit all adults except those with head circumferences that are in the smallest 5% or largest 5%. Find the minimum and maximum head circumferences that the helmets will fit.

Verified step by step guidance

Step 1: Understand the problem. The goal is to find the minimum and maximum head circumferences that the helmets will fit, excluding the smallest 5% and largest 5% of the distribution. This involves working with a normal distribution and identifying the boundaries corresponding to the middle 90% of the data.

Step 2: Recall the properties of a normal distribution. The mean (μ) is 570.0 mm, and the standard deviation (σ) is 18.3 mm. The smallest 5% and largest 5% correspond to the tails of the distribution, leaving 90% in the middle. To find the boundaries, we need the z-scores that correspond to the 5th percentile and the 95th percentile.

Step 3: Use a z-score table or statistical software to find the z-scores for the 5th percentile and 95th percentile. For a standard normal distribution, the z-score for the 5th percentile is approximately -1.645, and the z-score for the 95th percentile is approximately 1.645.

Step 4: Convert the z-scores to actual head circumferences using the formula for a normal distribution:

x = μ + z \times σ

. For the minimum circumference, use the z-score of -1.645, and for the maximum circumference, use the z-score of 1.645.

Step 5: Substitute the values into the formula. For the minimum circumference:

x = 570.0 + (- 1.645) \times 18.3

. For the maximum circumference:

x = 570.0 + (1.645) \times 18.3

. Calculate these values to find the minimum and maximum circumferences.

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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, depicting 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 adult head circumferences follow a normal distribution, which allows for the application of statistical methods to determine specific percentiles.

Percentiles

A percentile is a measure used in statistics indicating the value below which a given percentage of observations fall. For example, the 5th percentile is the value below which 5% of the data points lie. In the helmet design scenario, identifying the 5th and 95th percentiles of head circumferences helps engineers determine the minimum and maximum sizes for the helmets, ensuring they fit the majority of the adult 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. It allows for the comparison of scores from different distributions. In this case, Z-scores can be used to find the specific head circumference values that correspond to the 5th and 95th percentiles, facilitating the design of helmets that accommodate most adults.

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Related Practice

Textbook Question

In Exercises 25–28, use these parameters (based on Data Set 1 “Body Data” in Appendix B):

Men’s heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in.

Women’s heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in.

Snow White Disney World requires that women employed as a Snow White character must have a height between 64 in. and 67 in.

a. Find the percentage of women meeting the height requirement.

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Textbook Question

Aircraft Seat Width Engineers want to design seats in commercial aircraft so that they are wide enough to fit 99% of all adults. (Accommodating 100% of adults would require very wide seats that would be much too expensive.) Assume adults have hip widths that are normally distributed with a mean of 14.3 in. and a standard deviation of 0.9 in. (based on data from Applied Ergonomics). Find P99. That is, find the hip width for adults that separates the smallest 99% from the largest 1%.

139

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Textbook Question

Durations of Pregnancies The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days.

a. In a letter to “Dear Abby,” a wife claimed to have given birth 308 days after a brief visit from her husband, who was working in another country. Find the probability of a pregnancy lasting 308 days or longer. What does the result suggest?

174

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Textbook Question

Water Taxi Safety When a water taxi sank in Baltimore’s Inner Harbor, an investigation revealed that the safe passenger load for the water taxi was 3500 lb. It was also noted that the mean weight of a passenger was assumed to be 140 lb. Assume a “worst-case” scenario in which all of the passengers are adult men. Assume that weights of men are normally distributed with a mean of 188.6 lb and a standard deviation of 38.9 lb (based on Data Set 1 “Body Data” in Appendix B).

a. If one man is randomly selected, find the probability that he weighs less than 174 lb (the new value suggested by the National Transportation and Safety Board).

102

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Textbook Question

In Exercises 25–28, use these parameters (based on Data Set 1 “Body Data” in Appendix B):

Men’s heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in.

Women’s heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in.

Mickey Mouse Disney World requires that people employed as a Mickey Mouse character must have a height between 56 in. and 62 in.

a. Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as Mickey Mouse characters?

132

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Textbook Question

Correcting for a Finite Population In a study of babies born with very low birth weights, 275 children were given IQ tests at age 8, and their scores approximated a normal distribution with μ = 95.5 and σ = 16.0 (based on data from “Neurobehavioral Outcomes of School-age Children Born Extremely Low Birth Weight or Very Preterm,” by Anderson et al., Journal of the American Medical Association, Vol. 289, No. 24). Fifty of those children are to be randomly selected without replacement for a follow-up study.

b. Find the probability that the mean IQ score of the follow-up sample is between 95 and 105.

110

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