Name: Student #: DCAD 7350 3.0 Test (October 24, 2008) Time allowed: 180 minutes Aids permitted: Calculator(s), One 2-sided 8.5”x11” formula sheet Instructions: 1. Check that you have all 9 pages (including this cover page) of the test. 2. Answer the questions in the spaces provided. If you need more space, be sure to indicate clearly where the rest of your answer can be found. 3. Marks for each question are as indicated. You should allocate your time accordingly. 4. Your work must justify the answer you give. ∼ GOOD LUCK ∼ FOR INSTRUCTOR’S USE ONLY 1 / 10 2 / 20 3 / 10 4 / 10 5 / 10 6 / 10 7 / 10 8 / 10 Total / 80 Question 1: (10 marks) A new ﬁlter system for swimming pools is designed to ﬁlter out certain harmful particles that can get into the water. A study shows that the number of particles per gallon of water is normally distributed with a mean of 20000 and a standard deviation of 3000. The ﬁlter is designed to catch 25000 particles per gallon. a. (5 marks) Determine the probability that the ﬁlter will allow some particles to escape back into the pool. b. (5 marks) The manufacturer of the ﬁlter claims that the ﬁlter removes 90% of the particles from the water. How many particles per gallon of water would the ﬁlter remove? Question 2: (10 marks) Rolling Hills is a semiprivate golf club in rural North Carolina. Like most golf courses, Rolling Hills constantly battle the slow-play issue. The manager timed 45 players playing an 18-hole round. Let x be the time (in minutes) it took these 45 players to complete the round. The manager reported the data histogram is roughly symmetrical with n xi = 11774 and n (xi − x)2 = 19292. Moreover the fastest and the i=1 i=1 ¯ slowest players to complete the round took 231 minutes, and 345 minutes respectively. a. (3 marks) Calculate the mean time it took these 45 players to complete the round. Also report the standard deviation. b. (4 marks) Check if the fastest and the slowest players are outliers? c. (3 marks) The upper quartile of this data set is 267 minutes. How many standard deviation it is away from the mean? Question 3: (10 marks) The Slimy Soap Company has two production facilities, one in Ontario and one in Alberta. The Ontario plant makes 60% of the company’s total soap output, and the Alberta plant makes the rest of it. The quality assurance manager deter- mined that 5% of the soap produced in Ontario and 10% of the soap produced in Alberta is unusable due to some quality problems. a. (5 marks) If a bar of soap is randomly selected from the display in Loblaws, what is the chance that the soap is unusable? b. (5 marks) If a bar of soap is randomly selected from the display in Loblaws and is known to be unusable, what is the probability that it was produced by the Ontario plant? Question 4: (10 marks) The personnel manager of the Cumberland Pig Iron Company is studying the number of on-the-job accidents over a period of one month. He developed the following probability distribution. Number of Accidents 0 1 2 3 4 Probability 0.40 0.20 0.20 0.10 0.10 a. (4 marks) Compute the mean, and standard deviation of the number of accidents in a month. b. (3 marks) Let Xi be the number of on-the-job accidents in the i-th month. Assume Xi and Xj are independent random variables for all i = j. Furthermore let T be the total number of on-the-job accidents over a 12 months period (i.e. T = X1 + · · · X12 ). What is the mean and standard deviation of T ? c. (3 marks) Continue from part (b), What is the mean and variance for T /12, which is the average number of on-the-job accidents. Question 5: (10 marks) Information from the American Institute of Insurance indicates the amount of life insurance per household in US is normally distributed with mean $100000 and standard deviation of $40000. a. (2 marks) A household is randomly selected. What is the probability that the life insurance policy for this household is exactly $100000? b. (4 marks) A random sample of 25 households are randomly selected. What is the probability that the average life insurance policy for these household is at least $112000? c. (4 marks) What is the average life insurance policy of 25 household such that it will be exceeded by 2.5% of all the life insurance policy? Question 6: (10 marks) The manager of the grocery store, Dawdle, collected information on the waiting time at the checkout counter. Part of his result is reported in the following stem-and-leaf plot where the leaf unit is 0.1 minutes. 5 1 5 5 6 44 6 677 7 1344 7 56778 8 022344 8 5667789 9 01123 9 57 a. (1 marks) Comment on the overall shape of the given plot. b. (5 marks) Calculate the 5-number summary. c. (4 marks) Identify all the outliers (if exists). Question 7: (10 marks) The Wood County sheriﬀ classiﬁes crimes by age (in years) of the criminal and whether the crime is violent or nonviolent. As shown below, a total of 150 crimes were reported by the sheriﬀ last year. Age (in years) Type of Crime Uner 20 20 to 40 Over 40 Violent 27 41 14 Non violent 12 34 22 a. (3 marks) A case from last year is randomly selected. What is the probability that the selected case involved a violent crime or an oﬀender less than 20 years old? b. (3 marks) A case from last year is randomly selected. It is reported that the case involved a violent crime. What is the probability that the crime was committed by a person under 20 years old? c. (4 marks) Two cases are randomly selected. What is the probability that exactly one involved a violent crime? Question 8: (10 marks) Past sales records indicate that sales at the store are skewed with a long right tail. Assume the population mean sales is $12.50 per customer with a standard deviation of $5.50. The store manager has selected a random sample of 100 sales receipts. a. (5 marks) What is the probability that the mean sales is between $12.25 and $13.00? b. (5 marks) What is the probability that the mean sales exceed $14.00? Is it a rare event? Why?