Problem 19

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. b. If half of the 200 passengers are men, find the probability that the mean height of the 100 men is less than 72 in. c. When considering the comfort and safety of passengers, which result is more relevant: the probability from part (a) or the probability from part (b)? Why? d. When considering the comfort and safety of passengers, why are women ignored in this case?

Problem 19

Use the data in the table below for sitting adult males and females (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. (Hint: Draw a graph in each case.) $$\begin{array}{|l|l|l|l|} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \mathrm{in} . & 1.1 \mathrm{in} . & \text { Normal } \\ \hline \text { Females } & 22.7 \mathrm{in} . & 1.0 \mathrm{in} . & \text { Normal } \\ \hline \end{array}$$ Significance Instead of using 0.05 for identifying significant values, use the criteria that a value \(x\) is significantly high if \(P(x \text { or greater) } \leq 0.01\) and a value is significantly low if \(P(x \text { or less }) \leq 0.01 .\) Find the back-to-knee lengths for males, separating significant values from those that are not significant. Using these criteria, is a male back-to- knee length of 26 in. significantly high?

Problem 19

In a survey of 1002 people, 701 said that they voted in a recent presidential election (based on data from ICR Research Group). Voting records showed that \(61 \%\) of eligible voters actually did vote. a. Given that \(61 \%\) of eligible voters actually did vote, find the probability that among 1002 randomly selected eligible voters, at least 701 actually did vote. b. What does the result suggest?

Problem 20

In a study of 420,095 cell phone users in Denmark, it was found that 135 developed cancer of the brain or nervous system. For those not using cell phones, there is a 0.000340 probability of a person developing cancer of the brain or nervous system. We therefore expect about 143 cases of such cancers in a group of 420,095 randomly selected people. a. Find the probability of 135 or fewer cases of such cancers in a group of 420,095 people. b. What do these results suggest about media reports that suggest cell phones cause cancer of the brain or nervous system?

Problem 20

Use the data in the table below for sitting adult males and females (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. (Hint: Draw a graph in each case.) $$\begin{array}{|l|l|l|l|} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \mathrm{in} . & 1.1 \mathrm{in} . & \text { Normal } \\ \hline \text { Females } & 22.7 \mathrm{in} . & 1.0 \mathrm{in} . & \text { Normal } \\ \hline \end{array}$$ Significance Instead of using 0.05 for identifying significant values, use the criteria that a value \(x\) is significantly high if \(P(x \text { or greater) } \leq 0.025\) and a value is significantly low if \(P(x \text { or less }) \leq 0.025 .\) Find the female back-to-knee length, separating significant values from those that are not significant. Using these criteria, is a female back-to-knee length of 20 in. significantly low?

Problem 20

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of \(1 .\) In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table \(A-2,\) round answers to four decimal places. Less than 2.56

Problem 20

Loading Aircraft Before every flight, the pilot must verify that the total weight of the load is less than the maximum allowable load for the aircraft. The Bombardier Dash 8 aircraft can carry 37 passengers, and a flight has fuel and baggage that allows for a total passenger load of 6200 lb. The pilot sees that the plane is full and all passengers are men. The aircraft will be overloaded if the mean weight of the passengers is greater than \(6200 \mathrm{lb} / 37=167.6 \mathrm{lb}\). What is the probability that the aircraft is overloaded? Should the pilot take any action to correct for an overloaded aircraft? 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 I "Body Data" in Appendix B).

Problem 20

Constructing Normal Quantile Plots.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. McDonald's Dinner Service Times A sample of drive-through service times (seconds) at McDonald's during dinner hours, as listed in Data Set 25 "Fast Food" in Appendix B: 84,121,119,146,266,181,123,152,162

Problem 21

The probability of a baby being born a boy is \(0.512 .\) Consider the problem of finding the probability of exactly 7 boys in 11 births. Solve that problem using (1) normal approximation to the binomial using Table \(A-2 ;\) (2) normal approximation to the binomial using technology instead of Table \(A-2 ;\) (3) using technology with the binomial distribution instead of using a normal approximation. Compare the results. Given that the requirements for using the normal approximation are just barely met, are the approximations off by very much?

Problem 22

A Boeing 767-300 aircraft has 213 seats. When someone buys a ticket for a flight, there is a 0.0995 probability that the person will not show up for the flight (based on data from an IBM research paper by Lawrence, Hong, and Cherrier). How many reservations could be accepted for a Boeing \(767-300\) for there to be at least a 0.95 probability that all reservation holders who show will be accommodated?