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AP Statistics – Normal Distribution Practice Exam Name:______________ DATE:______ POINTS:_____ Draw diagrams for the following. Then calculate the indicated probabilities from the Z distribution. 1. between 0.0 and 0.85 Diagram: Answer: _____________________ (2 marks) 2. between –0.33 and 0.00 Diagram: Answer: _____________________ (2 marks) 3. greater than 0.62 Diagram: Answer: _____________________ (2 marks) 4. between –1.57 and –0.89 Diagram: Answer: _____________________ (2 marks) 5. less than –0.29 Diagram: Answer: _____________________ (2 marks) 6. between –0.33 and 2.40 Answer: _____________________ (2 marks) Diagram: Given that Z is normally distributed with =12 and =4, what is the probability of the following? Please show all work and include a diagram. Give your Answer in decimal and percent form. 7. Z is between 14.0 and 16.0 Diagram: Answer: ______________________ (3 marks) 8. Z is between 10.0 and 12.0 Diagram: Answer: ______________________ (3 marks) 9. Z is less than 11.0 Diagram: Answer: ______________________ (2 marks) 10. Z is greater than 12.5 Diagram: Answer: ______________________ (2 marks) 11. Z is between 11.0 and 14.0 Answer: _______________________ (3 marks) Diagram: 12. Find z so that 5% of the area under the standard normal curve lies to the right of z. Answer: _______________________ (2 marks) 13. Find z so that 1% of the area under the standard normal curve lies to the left of z. Answer: _______________________ (2 marks) 14. Find z so that 95% of the area under the standard normal curve lies between -z and z. Answer: _______________________ (3 marks) 15. Find z so that 99% of the area under the standard normal curve lies between -z and z. Answer: _______________________ (3 marks) 16. On a practical nursing licensing exam, the mean score is 79 and the standard deviation is 9 points. a) What is the standardized score of a student with a raw score of 87? Answer: _______________________ (2 marks) b) What is the standardized score of a student with a raw score of 79? Answer: _______________________ (2 marks) c) Assuming the scores follow a normal distribution, what is the probability that a score selected at random is above 85? Answer: _______________________ (2 marks) 17. In the town of Rockwood, a survey found that the number of hours grade school children watch TV per week is normally distributed with mean 20 hours and standard deviation 2 hours. If a child is chosen at random, what is the probability that he or she watches TV a) Less than 14 hours per week? Answer: _______________________ (2 marks) b) More than 22 hours per week? Answer: _______________________ (2 marks) c) Between 18 and 20 hours per week? Answer: _______________________ (3 marks) 18. Express Courier Service has found that the delivery time for packages is normally distributed with mean 14 hours and standard deviation 2 hours. a) For a package selected at random, what is the probability that it will be delivered in 18 hours or less? Answer: _______________________ (2 marks) b) What should be the guaranteed delivery time on all packages in order for be 95% sure that the package will be delivered before this time? (Hint: Note that 5% of the packages will be delivered at a time beyond the guarantee time period.) Answer: _______________________ (2 marks) 19. A Flight for Life helicopter service is available for emergencies occurring 15 to 90 miles from the hospital. Other emergencies are serviced by ambulance. A study of the service shows that the response time is normally distributed with a mean of 42 minutes and standard deviation of 8 minutes. For a random call, what is the probability that the response time will be a) Between 30 and 45 minutes? Answer: _______________________ (3 marks) b) Less than 30 minutes? Answer: _______________________ (2 marks) c) More than 60 minutes? Answer: _______________________ (2 marks) 20. A factory uses two machines to can fruits. Machine A puts fruit into the cans and Machine B adds some syrup. The amount of syrup is independent of the amount of fruit in a can. Machine A is set so that the mean mass of fruit put into each can is 305 grams with a standard deviation of 2 grams. (a) Calculate the probability that a can contains less than 300 grams of fruit. [1 mark] Answer:___________ Machine B is set so that the mean mass of syrup put into each can is 154 grams with a standard deviation of 1.5 grams. (b) Calculate the probability that a can contains less than 150 grams of syrup. [1 mark] Answer:___________ The factory processes 10 000 cans each day. (c) (i) How many cans, each day, are expected to contain less than 300 grams of mark] fruit? [1 Answer:___________ (ii) How many cans, each day, are expected to contain less than 150 grams of mark] syrup? [1 Answer:___________ (iii) Calculate the probability that a can will contain less than 300 grams of fruit and less than 150 grams of syrup. [2 marks] Answer:___________ (iv) How many cans containing less than 300 grams of fruit and less than 150 grams of syrup are expected to be produced each day? [2 marks] Answer:___________ One year there is a shortage of fruit so the manager decides that in order to cut costs he will reduce the mean mass of fruit to 304 grams and risk a probability that 2.5% of the cans have less than 300 grams of fruit when using Machine A. (d) What Z-value corresponds to a probability of 2.5% on a standard normal distribution? [2 marks] Answer:___________ (e) Calculate the new standard deviation for Machine A. [2 marks] Answer:___________ Practice Exam Answers: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 0.3023 0.1293 0.2676 0.1285 0.3859 0.6211 0.1498 0.1915 0.4013 0.4503 0.2902 1.645 -2.33 1.96 2.575 a) 0.89 b) 0 c) 0.2514 a) 0.0013 b) 0.1587 c) 0.3413 a) 0.9772 b) 17.3 hours a) 0.5812 b) 0.0668 c) 0.0122 a) 0.0062 or 0.62% b) 0.0038 or 0.38% c) i. 62 cans ii. 38 cans iii. 0.00002356 iv. less than one can d) -1.96 e) 2.04