Probability and Statistics

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CALEDONIAN COLLEGE OF ENGINEERING DEPARTMENT OF MATHEMATICS AND STATISTICS Probability and Statistics (MAT0202) ASSIGNMENT – SEMESTER B (2017 - 2018) ANSWER ALL THE QUESTIONS TOTAL MARKS: 100 NOTE: P is the last digit of your student ID. ____________________________________________________________________________ Q1 A packing plant fills bags with cement. The weight X kilogram of bag cement can be modeled by a normal distribution with mean 50kg and standard deviation 2 kg : [15] a) Find . b) Find the weight that exceed by of the bags. c) Three bags chosen at random, find the probability that two weigh more than kg and one weighs less than kg. d) How many bags must be sampled to establish with 99% confidence that the true mean is within ±0.5% of the sample mean, regardless of what the population mean is? e) If X has unknown mean but known standard deviation of 2 kg , what is the sample size required to construct a 95% confidence interval on the mean, that has total width of 1.0. Q2 A discrete random variable has the probability function of { [15] } a) b) c) d) Find the value of k Compute . Find . If two independent observations are made of X, Find the probability that the sum between is a prime number. e) If Y is a random variable such that , then find . Q3 After production, a component is given a score A, B or C. 70% of the components were given an A score, 18% of the components were given a quality score B and 12% of the components were given a quality score C. Furthermore it was found the 2% of components given a quality score of A eventually failed and the failure was 10% for components given a quality score of B, and 18% for components given quality score C. If a component failed, find the probability that it had received a B score? Probability& Statistics MATO202 ASSIGNMENT Page 1 [6] Q4 a chemical engineer is interested in determining whether a certain trace impurity is present in a product. An experiment has a probability of 0.80 of detecting the impurity if it is present. The probability of not detecting the impurity if it is absent is 0.90. The prior probabilities of impurities being present and being absent are 0.40 and 0.60 respectively. Three separate result in only two detections. What is the posterior probability that the impurity is present. [9] Q5 Data collected at Muscat International Airport suggest that an exponential distribution with mean value of 2.7P hours is a good model for rainfall duration. a) What is the probability that the duration of particular rainfall event at Muscat is: i. At least 2 hours? ii. At most three hours? iii. Between 2 and 3 hours b) What is the probability that rainfall exceeds the mean value by more than two standard deviations? What is the probability that it is less than the mean value by one standard deviation? [10] Q6 Bacteria are distributed independently of each other in a solution and it is known that the number of bacteria per milliliter follows a Poisson distribution with mean 2.9P. a) Find the probability that a sample of 1 ml of solution contains i. More than 3 bacteria. ii. At most three bacteria? [10] b) If five samples, each of 1 ml of liquid, are taken, then, find the mean number of bacteria for the five samples, taken together. Q7 A polymer is manufactured in a batch chemical process. Viscosity measurements are normally made on each batch, along experience with the process has indicated that the variability in the process is fairly stable with . Fourteen batch viscosity measurements are given as follows : 72P, 718, 776, 760, 745, 759, 756, 742, 740, 740, 76P, 749, 739, 747 A process change is made which involves switching the type of catalyst used in the process. Following the process change , eight batch viscosity measurements are taken : 73P, 775, 729, 755, 783 759, 756, 742 Assume that process variability is unaffected by the catalyst change. If the difference in the mean batch viscosity is 10 or less , the manufacturer would like to detect it with high probability. a) What distribution (test statistic) is to be used? Justify your answer. b) Formulate and test an appropriate hypothesis using α=0.05. c) What are your conclusions d) Find the 90% confidence interval on the difference in mean batch viscosity resulting from the process change. e) Compare the results of part c) and part d) and discuss your findings. f) What are the implications of making the wrong decision? Probability& Statistics MATO202 ASSIGNMENT Page 2 [20] Q8 A chemical engineer student wants to compare the hardness of four blends of paint. Six samples of each paint blend were applied to a piece of metal. The pieces of metal were cured. Then each sample was measured for hardness. Blend 1 13.4P 13.84 16.83 15.86 13.01 15.45 Blend 2 8.1P 7.64 9.58 6.03 8.40 8.81 Blend 3 12.3P 14.26 13.58 13.35 13.50 13.27 Blend 4 16.6P 18.58 17.60 18.96 17.65 19.40 a) Construct a comparative boxplot for the four types of blends and comment on interesting features. b) What is the appropriate statistical method that the analyst has to use in order to test for the equality of means and to assess the differences between pairs of means: Justify your answer. c) Perform the appropriate analyses using R and interpret your results. d) What test can be used to compare whether the difference between a pair of groups is statistically significant? Apply the appropriate test and comment on your results. ***** For detailed study of pedagogical things concerning this part of coursework, refer to the following Ebrary resources. 1. DeCoursey, William. Statistics and Probability for Engineering Applications, Elsevier Science, 2003. 2. Shanmugam, Ramalingam, and Rajan Chattamvelli. Statistics for Scientists and Engineers, John Wiley & Sons, Incorporated, 2015. Probability& Statistics MATO202 ASSIGNMENT Page 3 [15]

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School: Carnegie Mellon University

HiSince the time is 10 mi...

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