# AE-December-2015-04-BS-2

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NATIONAL EXAMINATIONS DECEMBER 2015 04 -BS -2 PROBABILITY AND STATISTICS 2 HOURS DURATION NOTES: I- If doubt exists as to the interpretation of any question, the candidate is urged to submit with the answer paper a clear statement of any assumption made. 2. "Closed Book" - no -aids other than (0 A Casio or Sharp approved calculator (ii) ONE hand-written information sheet (8.5"xl 1"), filled on both sides. 3. Any 5 questions constitute a complete paper. Only 5 questions will be marked. if 4. All questions are of equal value. 5. Statistical tables of the normal, t, chi-square and F distributions are provided. 6. Questions involving hypothesis testing must be clearly formulated. Marking Scheme 1. (a) 5 marks (b) 5 marks (c) 5 marks (d) 5 marks 2. (A) (a) 5 marks (b) 5 marks (c) 5 marks ;(B) 5 marks 3. (A) (a) 5 marks (b) 5 marks (B) (a) 5 marks (b) 5 marks 4. (a) 5 marks (b) 5 marks (c) 5 marks (d) 5 marks 5. (a) 7 marks (b) 7 ;narks (c) 6 marks 6. (A) (a) 5 marks (b) 5 marks ; (B) (a) 5 marks (b) 5 marks 7. (a) 10 marks (b) 10 marks 8. (a) 5 marks (b) 5 marks (c) 5 marks (d) 5 marks Page 1 of 5
NATIONAL EXAMINATIONS DECEMBER 2015 04 -BS -2 1.A review of the extensive data available in the files of Reliable Tires revealed that the life X of the all-weather tires manufactured by the company is a normally distributed random variable with mean and standard deviation equal to 175,000kms and 20,000kms respectively. (a) Find the probability that the life of a randomly selected tire lasts less than 190,0001(ms. Write down the probability density function of X. Then draw the probability density function of X, neatly and clearly, and indicate the area that corresponds to this probability. (b) Compute the probability that the life of a selected tire differs from the mean by less than 10,000kms. Then cipm the probability density function of X, neatly and clearly, and indicate the area (that corresponds to this probability. (c) Let M represent the average life of a random sample of four tires. (i) Find the mean and standard deviationiiof the probability distribution of M. (ii) Write down the probability density function of M. (iii) Draw, neatly and clearly, the probability density function of X and M on the same diagram. (iv) Compute the probability that M exceeds 170,000 kms. (d) Mr. Pinky, the owner of a fleet of nine limousines, bought a set of 36 tires. Let T be the sum of the lives of the 36 tires. (i) Write down the probability density function of T. (ii) Compute Ole probability that T exceeds 6,300,000kms 2.(A) A city-wide survey carried out on behalf of the Metropolitan Council of a large urban centre revealed that 60% of tkie adult inhabitants of that centre were in favour of extending the two subway lines currently available. (a) What is the probability that in a random sample of 15 adult inhabitants more than five but fewer than nine would be in favour of that extension? (b) What is the probability that in a random sample of 12 adult inhabitants fewer than three would not be in favour of that extension? (c) A leading newspaper carried,an additional survey on a random sample of 4,000 adult inhabitants. Use an appropriate approximation to compute the probability that fewer than 2,450 were in favour of the extension under consideration. 2.(B) The probability that a member of a large professional association is sued for malpractice is 0.0015. Use an appropriate approximation to compute the probability that in a random sample of 2,000 members more than two were sued for malpractice. Explain, briefly and clearly, why the approximation used is appropriate. Page 2 of 5
NATIONAL EXAMINATIONS DECEMBER 2015 04 -BS -2 3.(A) Information gathered by the transportation engineer of a large municipality indicates that the number of buses that need maintenance during any day follows the Poisson law with an average of four buses per day. (a) Compute the probability that on any given day more than three buses will need maintenance . (b) What is the probability that more than four but fewer than eight buses will need maintenance in any two-day period? 3.(B) In the month of May the owner of Friendly Hardware received a lot of twelve Fresh -Air portable air conditioners from the manufacturer. Unknown to the owner of Friendly Hardware, five of the air conditioners were substandard. (a) In July of the same year, during a major heat wave, the owner of the store sold eight air conditioners from the lot he received in May. What is the probability that at most three of the eight sold were substandard? (b) Let X denote the number of substandard air conditioners in a random sample of eight air conditioners. Find the probability distribution of X. Then compute E(X). 4. The probability density function of the random variable Y is defined as follows Kly 1<y 0 otherwise (a) Find the value of K. Then graph f(y) against y clearly and neatly. (b) Find E(Y). (c) Find the variance of Y. (d) Find the cumulative distribution function F(y). Then graph F(y) against y. 5. Twenty-one measurements of Young's modulus of a certain type of hard rubber, in MPa (MegaPascals), yielded the following information: EX = 630.0 EX 2 =18,980.0 (a) Find the 99% confidence limits of (i) the true mean and (ii) the true standard deviation of the probability distribution of X. Assume that X is a normally distributed random variable. (b) Test the hypothesis that the mean value of the probability distribution of X is not significantly different from 31.0 MPa. Let a = 0.05. (c) Test the hypothesis that the true standard deviation of the probability distribution of X is not significantly different from 1.6 MPa. Let a = 0.05. Page 3 of 5 3