Statistics Problem Set: Colds and Exercise

18. Last year, at Northern Manufacturing Company, 200 people had colds during the year. One hundred fifty-five people who did no exercising had colds, and the remainder of the people with colds were involved in a weekly exercise program. Half of the 1,000 employees were involved in some type of exercise.
(a) What is the probability that an employee will have a cold next year?
(b) Given that an employee is involved in an exercise program, what is the probability that he or
she will get a cold next year?
(c) What is the probability that an employee who is not involved in an exercise program will get a cold next year?
(d) Are exercising and getting a cold independent events? Explain your answer.

35. Trowbridge Manufacturing produces cases for personal computers and other electronic equipment. The quality control inspector for this company believes that a particular process is out of control. Normally, only 5% of all cases are deemed defective due to discolorations. If 6 such cases are sampled, what is the probability that there will be 0 defective cases if the process is operating correctly? What is the probability that there will be exactly 1 defective case?

38. Steve Goodman, production foreman for the Florida Gold Fruit Company, estimates that the average sale of oranges is 4,700 and the standard deviation is 500 oranges. Sales follow a normal distribution.
(a) What is the probability that sales will be greater than 5,500 oranges?
(b) What is the probability that sales will be greater than 4,500 oranges?
(c) What is the probability that sales will be less than 4,900 oranges?
(d) What is the probability that sales will be less than 4,300 oranges?

40. Armstrong Faber produces a standard number-two pencil called Ultra-Lite. Since Chuck Armstrong started Armstrong Faber, sales have grown steadily. With the increase in the price of wood products, however, Chuck has been forced to increase the price of the Ultra-Lite pencils. As a result, the demand for Ultra-Lite has been fairly stable over the past 6 years. On the average, Armstrong Faber has sold 457,000 pencils each year. Furthermore, 90% of the time sales have been between 454,000 and 460,000 pencils. It is expected that the sales follow a normal distribution with a mean of 457,000 pencils. Estimate the standard deviation of this distribution. (Hint: Work backward from the normal table to find Z. Then apply Equation 1-15.)

42. A new integrated computer system is to be installed worldwide for a major corporation. Bids on this project are being solicited, and the contract will be awarded to one of the bidders. As a part of the proposal for this project, bidders must specify how long the project will take. There will be a significant penalty for finishing late. One potential contractor determines that the average time to complete a project of this type is 40 weeks with a standard deviation of 5 weeks. The time required to complete this project is assumed to be normally distributed.
(a) If the due date of this project is set at 40 weeks, what is the probability that the contractor will have to pay a penalty (i.e., the project will not be finished on schedule)?
(b) If the due date of this project is set at 43 weeks, what is the probability that the contractor will have to pay a penalty (i.e., the project will not be finished on schedule)?
(c) If the bidder wishes to set the due date in the proposal so that there is only a 5% chance of being late (and consequently only a 5% chance of having to pay a penalty), what due date should be set?

43. Patients arrive at the emergency room of Costa Valley Hospital at an average of 5 per day. The demand for emergency room treatment at Costa Valley follows a Poisson distribution.
(a) Using Appendix C, compute the probability of exactly 0, 1, 2, 3, 4, and 5 arrivals per day.
(b) What is the sum of these probabilities, and why is the number less than 1?

49. Burger City is a large chain of fast-food restaurants specializing in gourmet hamburgers. A mathematical model is now used to predict the success of new restaurants based on location and demographic information for that area. In the past, 70% of all restaurants that were opened were successful. The mathematical model has been tested in the existing restaurants to determine how effective it is. For the restaurants that were successful, 90% of the time the model predicted they would be, while 10% of the time the model predicted a failure. For the restaurants that were not successful, when the mathematical model was applied, 20% of the time it incorrectly predicted a successful restaurant while 80% of the time it was accurate and predicted an unsuccessful restaurant. If the model is used on a new location and predicts the restaurant will be successful, what is the probability that it actually is successful?