Sample-Stat-353-Midterm-Exam

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Sample Stat 353 Midterm Exam SZ 8/22/2023 Question 1: Conditional Probability (10 points) In an engineering project, there are two machines, Machine A and Machine B. Machine A produces 60% of the total output, while Machine B produces the remaining 40%. The probability of a defective product coming from Machine A is 0.05, and from Machine B is 0.12. Calculate the probability that a randomly selected defective product came from Machine A. Question 2: Independence Probability (10 points) In a manufacturing process, two quality control tests are conducted: Test 1 and Test 2. The probability of a randomly selected component passing Test 1 is 0.85, and the probability of passing Test 2 is 0.75. If the two tests are independent, calculate the probability that a randomly selected component passes both Test 1 and Test 2. Question 3: Total Probability Formula (10 points) In a factory, there are three production lines: Line 1, Line 2, and Line 3. The probability that a product is defective on Line 1 is 0.05, on Line 2 is 0.08, and on Line 3 is 0.12. The probability that a randomly selected product is from Line 1, Line 2, or Line 3 is 0.30, 0.40, and 0.30, respectively. Calculate the probability that a randomly selected product is defective. Question 4: Bayes’ Formula (10 points) In a quality control process, components are produced by two machines: Machine X and Machine Y. Machine X produces 70% of the components, and Machine Y produces 30%. The probability of a defective component coming from Machine X is 0.08, and from Machine Y is 0.12. If a randomly selected component is defective, calculate the probability that it was produced by Machine Y. Question 5: Correlation based on Joint Discrete Distribution (10 points) Engineers are analyzing the relationship between the temperature and pressure in a chemical reaction. The joint probability distribution is given below: Temperature (x) Pressure (y) P(x, y) 50 10 0.15 50 20 0.10 60 10 0.05 60 20 0.20 70 10 0.20 70 20 0.30 Calculate the correlation coefficient between temperature (X) and pressure (Y). 1

Question 6: Probability based on Normal Distribution (10 points) The weight of a certain product follows a normal distribution with a mean of 150 kg and a standard deviation of 10 kg. Calculate the probability that a randomly selected product weighs less than 140 kg. Question 7: Probability based on Exponential Distribution (10 points) The time between arrivals at a service center follows an exponential distribution with a mean of 5 minutes. Calculate the probability that a customer arrives within 3 minutes after the previous customer. Question 8: Probability based on Poisson Distribution (10 points) In a manufacturing process, the number of defects per unit follows a Poisson distribution with a mean of 2 defects. Calculate the probability of having exactly 3 defects in a randomly selected unit. Question 9: Mean of Binomial Distribution (10 points) In a quality control process, the probability of a product passing a test is 0.90. If 100 products are tested, calculate the mean number of products that will pass the test. Question 10: Properties of Probability Density Function (10 points) The probability density function (PDF) of a random variable X is given by: f ( x ) = 0 . 08 x, 0 < x < 5 a) Verify that the PDF is properly defined. b) Calculate the probability that X is between 2 and 4. Question 11: Descriptive Statistics (10 points) A set of data represents the temperatures (in Celsius) recorded during a week in an engineering laboratory: 23, 24, 25, 22, 21, 23, 25. Calculate: a) The mean temperature. b) The median temperature. c) The range of temperatures. d) The variance of temperatures. e) The standard deviation of temperatures. Provide your answers with appropriate units and round your final answers to two decimal places. Another set of 11 questions: Question 1: Conditional Probability (10 points) In a communication network, there are two routers: Router A and Router B. Router A has a failure rate of 0.03, while Router B has a failure rate of 0.06. The probability of a data packet being lost due to a router failure in Router A is 0.08, and in Router B is 0.15. Calculate the probability that a randomly lost data packet was lost due to Router A. 2

Question 2: Independence Probability (10 points) In a manufacturing process, two sensors are used to monitor a system: Sensor X and Sensor Y. The probability of Sensor X detecting a malfunction is 0.90, and the probability of Sensor Y detecting a malfunction is 0.85. If the two sensors are independent, calculate the probability that a malfunction is detected by both Sensor X and Sensor Y. Question 3: Total Probability Formula (10 points) A production line produces three types of components: Type A, Type B, and Type C. The probability of a defect in Type A is 0.06, in Type B is 0.10, and in Type C is 0.08. The proportion of Type A, Type B, and Type C components in the production line is 0.40, 0.30, and 0.30, respectively. Calculate the probability that a randomly selected component is defective. Question 4: Bayes’ Formula (10 points) In a manufacturing facility, two machines are used to produce components: Machine P and Machine Q. Machine P produces 60% of the components, and Machine Q produces 40%. The probability of a defective component being produced by Machine P is 0.05, and by Machine Q is 0.08. If a randomly selected component is defective, calculate the probability that it was produced by Machine Q. Question 5: Correlation based on Joint Discrete Distribution (10 points) Engineers are studying the relationship between the pressure and volume of a gas in a container. The joint probability distribution is given below: Pressure (x) Volume (y) P(x, y) 10 100 0.12 10 200 0.18 20 100 0.08 20 200 0.22 30 100 0.15 30 200 0.25 Calculate the correlation coefficient between pressure (X) and volume (Y). Question 6: Probability based on Normal Distribution (10 points) The diameter of bolts produced in a factory follows a normal distribution with a mean of 10 mm and a standard deviation of 0.5 mm. Calculate the probability that a randomly selected bolt has a diameter between 9.5 mm and 11 mm. Question 7: Probability based on Exponential Distribution (10 points) The time between equipment failures in a manufacturing plant follows an exponential distribution with a mean of 800 hours. Calculate the probability that the time between failures is less than 600 hours. Question 8: Probability based on Poisson Distribution (10 points) In a call center, the average number of incoming calls per hour is 20. Calculate the probability of receiving exactly 25 calls in a randomly selected hour. 3

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Question 9: Mean of Binomial Distribution (10 points) In a quality control process, the probability of a product passing a test is 0.95. If 50 products are tested, calculate the mean number of products that will pass the test. Question 10: Properties of Probability Density Function (10 points) The probability density function (PDF) of a continuous random variable X is given by: f ( x ) = 0 . 375 x 2 , 0 < x < 2 a) Verify that the PDF is properly defined. b) Calculate the probability that X is between 1 and 1.5. Question 11: Probability Mass Function (10 points) Consider a discrete random variable X representing the number of defective components in a batch. The probability mass function (PMF) of X is given by: P(X = 0) = 0.4 P(X = 1) = 0.3 P(X = 2) = 0.2 P(X = 3) = 0.1 a) Calculate the probability that a randomly selected batch has at least one defective component. b) Calculate the mean number of defective components in a batch. c) Calculate the standard deviation of the number of defective components in a batch. 4

Sample-Stat-353-Midterm-Exam

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