When confronted with an in-flight medical emergency, pilots and crew can consult staff physicians at MedAire, an emergency facility in Tempe, Arizona. If MedAire is called, there is a 4.8 percent chance that the flight will be diverted for an immediate landing.

(a) If MedAire is called 8,465 times (as it was in 2002), what is the expected number of diversions?

(b) What is the probability of at least 400 diversions?

(c) Fewer than 450 diversions? Show your work carefully. (Data are from Popular Science 263, no. 5 [November 2003], p. 70.)

The cabin of a business jet has a cabin height 5 feet 9 inches high. If a business traveler’s height is N(5'10'', 2.7''), what percentage of the business travelers will have to stoop?

For a large Internet service provider (ISP), web virus attacks occur at a mean rate of 150 per day.

(a) Estimate the probability of at least 175 attacks in a given day.

(b) Estimate the probability of fewer than 125 attacks.

(c) Is the normal approximation justified? Show all calculations.

(d) Use Excel to calculate the actual Poisson probabilities. How close were your approximations?

The Johnson family uses a propane gas grill for cooking outdoors. During the summer they need to replace their tank on average every 30 days. At a randomly chosen moment, what is the probability that they can grill out

(a) at least 40 days before they need to replace their tank;

(b) no more than 20 days?

Find the associated z-score or scores that represent the following standard normal areas.

a. Middle 50 percent b. Lowest 5 percent c. Middle 90 percent

The time it takes a ski patroller to respond to an accident call has an exponential distribution with an average equal to 5 minutes. (a) In what time will 90 percent of all ski accident calls be responded to? (b) If the ski patrol would like to be able to respond to 90 percent of the accident calls within 10 minutes, what does the average response time need to be?

Passengers using New York’s MetroCard system must swipe the card at a rate between 10 and 40 inches per second, or else the card must be re-swiped through the card reader. Research shows that actual swipe rates by subway riders are uniformly distributed between 5 and 50 inches per second.

(a) What is the mean swipe rate?

(b) What is the standard deviation of the swipe rate?

(c) What are the quartiles?

(d) What percentage of subway riders must re-swipe the card because they were outside the acceptable range? (Data are from The NewYork Times, July 18, 2004, p. 23.)

Passengers using New York’s MetroCard system must swipe the card at a rate between 10 and 40 inches per second, or else the card must be re-swiped through the card reader. Research shows that actual swipe rates by subway riders are uniformly distributed between 5 and 50 inches per second.

(a) What is the mean swipe rate?

(b) What is the standard deviation of the swipe rate?

(c) What are the quartiles?

(d) What percentage of subway riders must re-swipe the card because they were outside the acceptable range?

The length of a Colorado brook trout is normally distributed.

(a) What is the probability that a brook trout’s length exceeds the mean?

(b) Exceeds the mean by at least 1 standard deviation?

(c) Exceeds the mean by at least 2 standard deviations?

(d) Is within 2 standard deviations?

The length of a Colorado brook trout is normally distributed. (a) What is the probability that a brook trout’s length exceeds the mean? (b) Exceeds the mean by at least 1 standard deviation? (c) Exceeds the mean by at least 2 standard deviations? (d) Is within 2 standard deviations?

A statistics textbook chapter contains 60 exercises, 6 of which are essay questions. A student is assigned 10 problems. Define X to be the number of essay questions the student receives.

(a) Use Excel to calculate the entire hypergeometric probability distribution for X.

(b) What is the prob- ability that none of the questions are essay?

(c) That at least one is essay?

(d) That two or more are essay?

(e) Make an Excel graph of the PDF of the hypergeometric distribution and describe its appearance.

Shower temperature at the Oxnard Health Club showers is regulated automatically. The heater kicks in when the temperature falls to $99^{\circ} \mathrm{F}$, and shuts off when the temperature reaches $107^{\circ} \mathrm{F}$. Water temperature then falls slowly until the heater kicks in again. At a given moment, the water temperature is a uniformly distributed random variable $U(99,107$ ). (a) Find the mean temperature. (b) Find the standard deviation of the temperature. (c) Find the 75 th percentile for water temperature.

The distribution of scores on a statistics exam is $T(50,60,95)$. (a) Find the mean. (b) Find the standard deviation. (c) Find the probability that a score will be less than 75. (d) Sketch the distribution and shade the area for the event in part (c).

The average number of items (such as a drink or dessert) ordered by a Noodles & Company customer in addition to the meal is 1.4. These items are called add-ons. Define $X$ to be the number of add-ons ordered by a randomly selected customer. (a) Justify the use of the Poisson model. (b) What is the probability that a randomly selected customer orders at least 2 add-ons? (c) No add-ons? (d) Construct the probability distribution using Excel or Appendix B, make a graph of its PDF, and describe its shape.