6. Normal Distribution and Continuous Random Variables

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.

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%.

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?

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).

Interpreting Normal Quantile Plots Which of the following normal quantile plots appear to represent data from a population having a normal distribution? Explain.

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Curving Test Scores A professor gives a test and the scores are normally distributed with a mean of 60 and a standard deviation of 12. She plans to curve the scores.

a. If she curves by adding 15 to each grade, what is the new mean and standard deviation?

Curving Test Scores A professor gives a test and the scores are normally distributed with a mean of 60 and a standard deviation of 12. She plans to curve the scores.

c. If the grades are curved so that grades of B are given to scores above the bottom 70% and below the top 10%, find the numerical limits for a grade of B.

Outliers For the purposes of constructing modified boxplots as described in Section 3-3, outliers are defined as data values that are above Q3 by an amount greater than 1.5 x IQR or below Q1 by an amount greater than 1.5 x IQR, where IQR is the interquartile range. Using this definition of outliers, find the probability that when a value is randomly selected from a normal distribution, it is an outlier.

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?

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

Redesign of Ejection Seats When women were finally allowed to become pilots of fighter jets, engineers needed to redesign the ejection seats because they had been originally designed for men only. The ACES-II ejection seats were designed for men weighing between 140 lb and 211 lb. Weights of women are now normally distributed with a mean of 171 lb and a standard deviation of 46 lb (based on Data Set 1 “Body Data” in Appendix B).

a. If 1 woman is randomly selected, find the probability that her weight is between 140 lb and 211 lb.

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).

a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending.

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?

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

Aircraft Cockpit The overhead panel in an aircraft cockpit typically includes controls for such features as landing lights, fuel booster pumps, and oxygen. It is important for pilots to be able to reach those overhead controls while sitting. Seated adult males have overhead grip reaches that are normally distributed with a mean of 51.6 in. and a standard deviation of 2.2 in.

a. If an aircraft is designed for pilots with an overhead grip reach of 53 in., what percentage of adult males would not be able to reach the overhead controls? Is that percentage too high?

Constructing Normal Quantile Plots. In Exercises 17–20, use the given data values to identify the corresponding z scores that are used for a normal quantile plot, then identify the coordinates of each point in the normal quantile plot. Construct the normal quantile plot, then determine whether the data appear to be from a population with a normal distribution.

Earthquake Depths A sample of depths (km) of earthquakes is obtained from Data Set 24 “Earthquakes” in Appendix B: 17.3, 7.0, 7.0, 7.0, 8.1, 6.8.

Transformations The heights (in inches) of women listed in Data Set 1 “Body Data” in Appendix B have a distribution that is approximately normal, so it appears that those heights are from a normally distributed population.

a. If 2 inches is added to each height, are the new heights also normally distributed?