Probability and Statistics Problems PDF

1. A manufacturing company has two production lines, A and B. Line A produces 40% of the total output, and Line B produces 60%. The probability that a part from Line A is defective is 3%, and the probability that a part from Line B is defective is 5%. What is the probability that a randomly...

1. A manufacturing company has two production lines, A and B. Line A produces 40% of the total output, and Line B produces 60%. The probability that a part from Line A is defective is 3%, and the probability that a part from Line B is defective is 5%. What is the probability that a randomly selected part from either Line A or Line B is defective? 2. In a factory, Machine X and Machine Y both contribute to the production process. The probability that Machine X fails during a shift is 4%, and the probability that Machine Y fails during the same shift is 6%. If the probability that both machines fail during a shift is 1%, what is the probability that at least one of the two machines fails during the shift? 3. A manufacturing company relies on two suppliers for raw materials: Supplier A and Supplier B. The probability that Supplier A delivers materials late is 10%, and the probability that Supplier B delivers materials late is 15%. The probability that both suppliers deliver late at the same time is 4%. What is the probability that a late delivery occurs from at least one of the two suppliers? 4. A manufacturing company tracks the number of defective products produced each day. The following table shows an incomplete probability distribution for the number of defective products per day. Complete the table and calculate the mean, variance, and standard deviation. Number of Defective Products (X) Probability (P(X)) 0 0.30 1 0.25 2 ? 3 0.10 4 0.05 a. Find the missing probability value. b. Calculate the mean, variance, and standard deviation. 5. A factory records the number of machine breakdowns in a month. The incomplete probability distribution below shows the probability of each number of breakdowns. Complete the table and calculate the mean, variance, and standard deviation. Number of Breakdowns (X) Probability (P(X)) 0 0.40 1 ? 2 0.20 3 0.10 a. Find the missing probability value. b. Calculate the mean, variance, and standard deviation. 6. A manufacturing company monitors late deliveries from suppliers each week. The following table shows an incomplete probability distribution of the number of late deliveries. Complete the table and calculate the mean, variance, and standard deviation. Number of Late Deliveries (X) Probability (P(X)) 0 0.50 1 0.30 2 ? 3 0.05 a. Find the missing probability value. b. Calculate the mean, variance, and standard deviation. 7. A batch of 40 parts is produced by a manufacturing process, and the probability of a part being defective is 5% (or 0.05). a. What is the probability that exactly 2 parts are defective? b. What is the probability that at least 1 part is defective? c. What is the probability that at most 3 parts are defective? d. Find the mean, variance, and standard deviation for the number of defective parts. 8. A factory operates with 20 machines, and the probability of any machine breaking down on a given day is 4% (or 0.04). a. What is the probability that exactly 1 machine breaks down on a given day? b. What is the probability that at least 2 machines break down? c. What is the probability that at most 1 machine breaks down? d. Find the mean, variance, and standard deviation for the number of machines that break down. 9. A company receives 15 deliveries per week, and the probability of a delivery being late is 10% (or 0.10). a. What is the probability that exactly 3 deliveries are late? b. What is the probability that at least 2 deliveries are late? c. What is the probability that at most 4 deliveries are late? d. Find the mean, variance, and standard deviation for the number of late deliveries. 10. A manufacturing company records the time (in minutes) taken to produce a batch of parts. The data for the time taken (in minutes) for 15 parts is as follows: 12,15,14,16,14,13,12,18,17,15,16,14,15,16,1212, 15, 14, 16, 14, 13, 12, 18, 17, 15, 16, 14, 15, 16, 1212,15,14,16,14,13,12,18,17,15,16,14,15,16,12 a. Find the mean, median, and mode of the production times. b. Calculate the standard deviation. c. Find the quartiles (Q1, Q2, Q3), range, and interquartile range (IQR). d. Create a histogram for the production times. 11. A factory tracks the downtime (in hours) for machines over the past 10 days. The data for the downtime is as follows: 2.5,3.0,2.8,4.2,5.0,4.8,3.5,4.0,3.7,3.22.5, 3.0, 2.8, 4.2, 5.0, 4.8, 3.5, 4.0, 3.7, 3.22.5,3.0,2.8,4.2,5.0,4.8,3.5,4.0,3.7,3.2 a. Find the mean, median, and mode of the machine downtime. b. Calculate the standard deviation. c. Find the quartiles (Q1, Q2, Q3), range, and interquartile range (IQR). d. Create a histogram for the machine downtime. 12. A manufacturing company produces three types of products: A, B, and C. The number of defective products found in the last week for each type is as follows: Product A: 12 defects Product B: 9 defects Product C: 15 defects a. Create a bar plot to show the number of defects for each product type. b. Which product type has the highest number of defects? 13. A factory has three departments: Assembly, Packaging, and Quality Control. The number of machine breakdowns recorded in each department over two months is as follows: Month 1: Assembly: 4 breakdowns Packaging: 2 breakdowns Quality Control: 3 breakdowns Month 2: Assembly: 5 breakdowns Packaging: 3 breakdowns Quality Control: 2 breakdowns a. Create a side-by-side bar plot to compare the machine breakdowns in each department for the two months. b. Which department had the most machine breakdowns in Month 1 and in Month 2? 14. A manufacturing company tracks the number of defective products produced by two different machines (Machine A and Machine B) over three different shifts (Morning, Afternoon, and Night). The data is as follows: Machine Shift Defective Products A Morning 5 A Afternoon 3 A Night 7  Machine Shift Defective Products B Morning 8 B Afternoon 4 B Night 6 a. Create a cross table showing the number of defective products by machine and shift. b. Calculate the total number of defective products for each machine and each shift. 15. A company conducts a survey to measure employee satisfaction across three departments (HR, Sales, and Engineering) and two genders (Male and Female). The data is as follows: Department Gender Satisfaction Level HR Male Satisfied HR Female Not Satisfied Sales Male Satisfied Sales Female Satisfied Engineering Male Not Satisfied Engineering Female Satisfied HR Female Satisfied Engineering Male Satisfied Sales Female Not Satisfied HR Male Not Satisfied a. Create a cross table showing the relationship between Department and Gender for employee satisfaction levels. b. Calculate the total number of satisfied and not satisfied employees for each department and gender.