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Name: ________________________________ AP Statistics Normal Distributions Activity 1) Professor Rogers has found that the grades on the nursing final exam are normally distributed with a mean of 64 and standard deviation of 11. a) If the passing grade is 54, what percent of the class will fail? b) If Professor Rogers wants only 85% of the class to pass, what should the passing grade be? 2) The life of a coffee pot is approximately normally distributed with a mean of 5 years and standard deviation of 1.1 years. The manufacturer will repair or replace any defective coffee pot free of charge (including labor) while the pot is under warranty. For how many years should the company guarantee its coffee pots, if the manufacturer does not wish to replace more than 5% of them? 3) A new filling machine has been installed on an assembly line that is supposed to fill each bag with 5 pounds (80 ounces) of sugar. However, the machine is not functioning properly and can be adjusted to the vendor’s specifications. It is known that the filling process is approximately normally distributed with a standard deviation of 2.4 ounces. At what level should the mean be set so that at most 5% of the time will the bags contain more than 83 ounces of sugar? 4) The drying time of one particular paint is approximately normally distributed with a mean of 45 minutes. It is also known that 15% of walls painted with this paint need more than 55 minutes to dry completely. Find the standard deviation of this distribution. 5) The Fight for Life emergency helicopter service is available for medical emergencies occurring 15 to 90 miles from the hospital. A long-term study of the service shows that the response time from receipt of the dispatch call to arrival at the scene of the emergency is normally distributed with standard deviation of 8 minutes. a) What is the mean response time (to the nearest whole minute) if only 6.7% of the calls require more than 54 minutes to respond? b) For a randomly received call, what is the probability that the response time will be less than 30 minutes? c) For a randomly received call, what is the probability that the response time will be within 1.5 standard deviations of the mean? 6) The American Veterinary Association claims that the annual cost of medical care for dogs averages $100 with standard deviation of $30, and for cats, annual cost averages $120 with standard deviation of $35. Assume that the annual veterinary expenses are independent and normally distributed. Suppose you have a dog and a cat. a) Describe the distribution of annual veterinary cost for your dog and cat. b) What is the probability that your annual cost will be more than $250? c) What is the probability that your annual cost will be within $50 of the mean? 7) Combining Random Variables: Speed Dating To save time and money, many single people have decided to try speed dating. At a speed dating event, women sit in a circle and men spend about 5 minutes getting to know a woman before moving on to the next one. Suppose that the height M of male speed daters follows a Normal distribution with a mean of 70 inches and a standard deviation of 3.5 inches and the height F of female speed daters follows a Normal distribution with a mean of 65 inches and a standard deviation of 3 inches. What is the probability that a randomly selected male speed dater is taller than the randomly selected female speed dater he is paired with? 8) The weight of adult men is approximately Normally distributed with a mean of 190 pounds and a standard deviation of 30 pounds. (a) If you randomly select three men, what are the mean and standard deviation of the sum of their weights? (b) An elevator in a small apartment building has a maximum weight capacity of 600 pounds. If three randomly selected adult men get on the elevator, what is the probability that they exceed the maximum capacity? (c) Suppose that you randomly select 10 different groups of three adult men to get on the elevator. What is the probability that at least one of the 10 groups exceeds the weight limit? (d) Suppose that the weight of the elevator car is 2300 pounds. What is the mean and standard deviation of the total weight of the elevator car and three randomly selected adult males?