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STA108 (1).PDF - CONFIDENTIAL CS/OCT 2016/STA108 UNIVERSITI...

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Unformatted text preview: CONFIDENTIAL CS/OCT 2016/STA108 UNIVERSITI TEKNOLOGI MARA FINAL EXAMINATION COURSE : STATISTICS AND PROBABILITY COURSE CODE : STA108 EXAMINATION : OCTOBER 2016 TIME : 3HOURS INSTRUCTIONS TO CANDIDATES 1. This question paper consists of five (5) questions. 2. Answer ALL questions in the Answer Booklet. Start each answer on a new page. 3. Do not bring any material into the examination room unless permission is given by the invigilator. 4. Please check to make sure that this examination pack consists of : i) the Question Paper ii) a three — page Appendix 1 iii) an Answer Booklet — provided by the Faculty iv) a three — graph paper — provided by the Faculty 5. Answer ALL questions in English. DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO $0 This examination paper consists of 6 printed pages © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL CONFIDENTIAL 2 CSIOCT 2016ISTA108 QUESTION 1 a) Define each of the following terms; i) Random Sampling ii) Pilot Study (2 marks) b) For each of the following variables, classify whether it is qualitative, quantitative discrete or quantitative continuous variable. i) The number of houses owned by individuals in Perak. ii) The number of rooms available in a hotel. iii) The height of students. iv) The brand names of hand phones used by a group of lecturers. (4 marks) 0) An advertising firm is interested in determining how much television advertising to emphasize in a certain state. A sample survey is conducted to estimate the average number of hours each week households watch television. The state has three major towns (Town A, Town B and Town C). There are 300 households in Town A, 430 in Town B and 270 in Town C. A random sample of 200 households will be selected for this study. All the houses have telephone facilities. i) State the population and sampling frame for the above study. ii) Determine the variable of interest and the level of measurement for the above study. iii) The sample will be selected using stratified sampling technique. What is ONE advantage of using this technique? How many households should be selected from each town? iv) Determine the most suitable method of data collection. (9 marks) © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL CONFIDENTIAL 3 CSIOCT 2016/STA108 QUESTION 2 The table below depicts the distribution of the ages (in months) in January 2016 for 50 children in District X. Ages (in months) Number of children 30-33 10 34-37 12 38-41 13 42-45 6 46-49 6 50-53 2 54-57 1 a) Calculate the mean and standard deviation of the ages of the children. (5 marks) b) Draw a 'less than' ogive. (5 marks) 0) From the ‘less than’ogive, estimate the median and interpret the value obtained. (3 marks) d) Calculate the coefficient of skewness by using an appropriate formula. What can you conclude? (3 marks) e) The mean and variance of the ages (in months) in January 2016 for 50 children in District Y were 37.23 and 29.16, respectively. Using an appropriate measure, determine which district shows a more consistent age distribution. (4 marks) © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL CONFIDENTIAL 4 CSIOCT 2016ISTA1 08 QUESTION 3 a) Find the different number of words that can be formed from all the letters in the word MICROBIOLOGY if these words must start with M. (2 marks) b) A and B are two independent events. Given that P (A) = 0.7 and P (AflB) =O.3, calculate: i) P(B) ii) P(A|B) (4 marks) 0) Professor Z has been teaching statistics course for many years. He knows from experience that 80% of the students will complete the assigned problems. Among those who complete the assigned problems, 90% will pass the course and among those students who do not complete their assigned problems, 40% will pass the course. i) Draw a tree diagram to represent the above situation. (3 marks) ii) A student is chosen at random from Professor 23 class. Calculate the probability that he will pass the course. (3 marks) iii) A student took the course last semester from Professor Z and received a passing grade. What is the probability that he completed the assignment? (4 marks) © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL CONFIDENTIAL 5 CSIOCT 201 6ISTA108 QUESTION 4 a) The probability density function of a continuous random variable is given by x , Osx<1 f(x)= k—x, 1sx<2 O , elsewhere i) Show that the value of k = 2. (3 marks) ii) Compute the probability that X is greater than 1 (3 marks) iii) Compute the value of E(X) and E(X2). (6 marks) iv) Calculate V(5X — 25). (4 marks) b) The probability distribution of a random variable X is given by; X 0 1 2 3 4 5 P(X=x) 0.1 0.3 0.2 0.1 c 0.1 i) Find the value of c. (3 marks) ii) Construct the cumulative distribution of X. (3 marks) iii) Calculate P(1s X < 4). (2 marks) © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL CONFIDENTIAL 6 CSIOCT 2016ISTA108 QUESTION 5 a) A production process for the NH Semicon is monitored using mean and range charts. The measurements for the 20 samples with n = 5 observations are presented below. i) What are the center line and 3:: trial limits for the mean and range control charts? (6 marks) ii) Establish the mean and range control charts using the above information obtained in (|)- (8 marks) iii) Comment on the process based on the two charts. (2 marks) b) A manufacturing process produces a certain part with a mean diameter of 2 cm and a standard deviation of 0.03 cm. The lower and upper specification limits are 1.90 cm and 2.05 cm respectively. Assume that the diameter is normally distributed. i) Estimate the natural tolerance limit for the process. (2 marks) ii) Find the Cp and Cpk for the process and make a conclusion from the values obtained. (7 marks) END OF QUESTION PAPER © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL CONFIDENTIAL APPENDIX 1 (1) FORMULA LIST Sample Measurements 1. Mean, Y = g or E n n E _ 2er 2 Median, 36 = Lm + 2 C fm . A 3. Mode, X = Lmo + 1 -C A1 + A2 4. Standard Deviations, 3 = JfiIflx—flz] or s: In_1_‘]IZfiX_§)2I or 5. Coefficient of Variation, CV = §X100 CS/OCT 2016ISTA108 _1_[fo2 — (XIX)? n—1 x 6. Pearson’s Measure of Skewness = m or W Standard Deviation Standard Dev1at10n where n . total frequency Lm : lower median class boundary Lmo : lower modal class boundary wa1 ' cumulative frequencies for the classes before the median class fm : median class frequency A, : (modal class frequency) — (frequency for the class before the modal class) A2 : (modal class frequency) — (frequency for the class after the modal class) C class size © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL CONFIDENTIAL APPENDIX 1 (2) CSIOCT 2016/STA108 Probability Theory 1. Conditional Probability P(A n B) P(AlB) = P(B) 2. Multiplicative Rule P(A n B) = P(B).P(A|B) P(A n B) = P(A).P(B) if and only if A and B are independent events. 3. Bayes Theorem P(A|B)= , P(A).P(B|A) P(A).P(BIA i+ P(A' ).P(B|A'l © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL CONFIDENTIAL APPENDIX 1 (3) CSIOCT 2016/STA108 Control Chart Factors for Mean and Range (American Usage) X-bar and R charts Observations AVERAGES in sample Factors for Factors for Central Control Limits Line RANGES Factors for Control Limits ——_-_“-m ———_—-I- W“ ——__—_-_ ———_——“ “——-_—“ “—-=—_ -_ 2-97 0.3367 0-808 To drive control limits Process Average X Chan‘ Range Chart Upper Control Limit = Central Line + Azfi Upper Control Limit = 04R Lower Control Limit = Central Line — Azfi Lower COIWOI Limit = D3R Centra' Line —_— Y for trial Iimits Central LIne =§ for trIaI IImItS. Central Line = Target mean for a controlled process For achievable known process 0 . replace R by dzo. © Hak Cipta Universiti Teknologi MARA CONFIDENTIAL ...
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