Week 8 (26-01-26 to 30-01-26)

Prepare

📖 Read the syllabus

📖 Read the support resources

Participate

🖥️ Lecture 4

🖥️ Statistical Table - Statistical Table for probability estimations.

🖥️ Lecture 3 -Continuous Probability Distributions

🖥️ Lecture 2 - Discrete Probability Distributions

🖥️ Lecture 1 - Introduction to Probability

Practice

  1. A manufacturer of MP3 players conducts a set of comprehensive tests on the electrical functions of its product. All MP3 players must pass all tests prior to being sold. Out of a random sample of 500 MP3 players, 15 failed one or more tests. Find a 90% confidence interval for the proportion of MP3 players from the population that pass all tests.

  2. The production employees at a food packaging facility have been surveyed to determine the proportion of employees who prefer rotating shifts over fixed shifts. Previous surveys suggest this proportion ranges from 25% to 45%. Facility management wants to construct a 95% confidence interval estimate for the true proportion, with a margin of error no larger than 5%. What sample size should they use for the survey?

  3. A study is to be made to estimate the proportion of residents of a certain city and its suburbs who favor the construction of a nuclear power plant near the city. How large a sample is needed if one wishes to be at least 95% confident that the estimate is within 0.04 of the true proportion of residents who favor the construction of the nuclear power plant?

  4. A machine is used to manufacture cylindrical metal pieces. To evaluate the precision of the machine, a sample of metal pieces was collected, and their diameters were measured. The recorded diameters (in centimeters) are as follows: 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and 1.03. (Assuming the diameters are approximately normally distributed.)

    1. Compute the sample mean and standard deviation of the diameters.
    2. Construct a 99% confidence interval for the true mean diameter of all metal pieces produced by the machine.
    3. construct a 99% confidence interval for the population variance σ2 of the diameters of all cylindrical metal pieces produced by the machine.
    4. By using the results from part(c), construct a 99% confidence interval for the standard deviation of the diameters of all cylindrical metal pieces produced by the machine.

  5. A construction materials testing lab is investigating the compressive strength of concrete cylinders produced by a local ready-mix plant. The plant periodically tests samples of cylinders to ensure quality control. Historical data indicates that the concrete cylinders have a mean compressive strength of 11.80 MPa with a standard deviation of 0.75 MPa. Determine the number of concrete cylinders the lab must test to estimate the current average compressive strength with a 99% confidence interval and a total width of 0.50 MPa (i.e., a margin of error of 0.25 MPa).

  6. An electrical firm produces light bulbs with lifespans that are approximately normally distributed. The standard deviation of the lifespans is known to be 40 hours. A sample of 30 light bulbs is taken, and the average lifespan is found to be 780 hours. Based on this data, calculate a 96% confidence interval for the population mean lifespan of the bulbs.

  7. As part of evaluating the effectiveness of a newly designed prototype course at the Faculty of Engineering, University of Jaffna, students’ completion times for a standardized assignment were recorded. These times (in minutes) reflect the efficiency of the course structure and instructional methods. The recorded completion times for a sample of students are as follows:

  14.5, 16.6, 18.7, 16.8, 17.1, 18.3, 9.4, 10.7, 16.7, 17.7, 15.2

Assuming that these completion times follow a normal distribution, answer the following:

  1. Construct a 98% confidence interval for the mean completion time (in minutes) of the assignment, reflecting the expected performance of students in the prototype course.

  2. Construct a 95% confidence interval for the variance of completion times, which helps assess the consistency of student performance under the new course design.

  3. By using the results from part (b), construct a 95% confidence interval for the standard deviation of completion times, providing insight into the variability in student learning paces.

  1. Many cardiac patients wear an implanted pacemaker to control their heartbeat. A plastic connector module mounts on the top of the pacemaker. Assuming a standard deviation of 0.0015 inch and an approximately normal distribution, find a 95% confidence interval for the mean of the depths of all connector modules made by a certain manufacturing company. A random sample of 75 modules has an average depth of 0.310 inch.

  2. Suppose you are a civil engineer tasked with estimating the average compressive strength of a certain type of concrete used in construction projects in the Kilinochchi area. In a previous study, it was found that the standard deviation of the compressive strengths of this type of concrete is 500 psi. To ensure a high level of confidence in your estimate, you aim to conduct a survey with a 92% confidence interval. You want the estimate to be within 100 psi of the true average compressive strength. Based on this information, determine the minimum sample size needed for your survey.

Exercise

  1. A fertilizer mixing machine is set to give 12 kg of nitrate for every quintal (100 kg) bag of fertilizer. Ten 100 kg bags are randomly selected and examined. The percentages of nitrate in the sample are as follows:

     11, 14, 13, 15, 13, 11, 13, 14, 10, 12

    Assume that the sample is from a normally distributed population.

    1. Construct a 98% confidence interval estimate for the mean nitrate content for each 100 kg bag in the company.
    2. Construct a 95% confidence interval for the variance of nitrate content for each 100 kg bag in the company.
    3. By using the results from part (b), construct a 95% confidence interval for the standard deviation of nitrate content for each 100 kg bag in the company.

  2. The concentration of mercury in a lake has been monitored for a number of years. Measurements taken on a weekly basis yielded an average of 1.20 mg/m^3 (milligrams per cubic meter) with a standard deviation of 0.32 mg/m^3 . Following an accident at a smelter on the shore of the lake, 15 measurements produced the following mercury concentrations:

       1.60, 1.77, 1.61, 1.08, 1.07, 1.79, 1.34, 1.07, 1.45, 1.59, 1.43, 2.07, 1.16, 0.85, 2.11
    1. Give a point estimate (sample mean) of the mean mercury concentration after the accident.
    2. Construct a 95% confidence interval for the mean mercury concentration after the accident. Interpret this interval.
    3. Is there sufficient evidence that the mean mercury concentration has increased since the accident? Use α = 0.05.

  3. As part of their Design and Prototyping (ID3020) project, a team of students at the University of Jaffna has developed a cutting-edge smart irrigation system designed to optimize water usage in agricultural fields. The system incorporates sensors and actuators to precisely control the amount of water delivered to crops, enhancing crop yield while conserving water resources. To ensure the system’s effectiveness, the team conducts a series of tests on randomly selected plots of land. The team measures the soil moisture levels in ten randomly selected plots and records the following percentages:

     Soil Moisture Data (%) : 30, 35, 32, 38, 33, 31, 36, 34, 37, 29

    Assuming the soil moisture levels follow a normal distribution, answer the following:

    1. The team aims to estimate the mean soil moisture content per plot and construct a 98% confidence interval for each plot. The team seeks to assess the variability of soil moisture content across the plots.
    2. Construct a 95% confidence interval for the variance of soil moisture content per plot using the recorded data.
    3. Using the results obtained in part (b), construct a 95% confidence interval for the standard deviation of soil moisture content per plot, providing valuable insights into the consistency of water distribution achieved by their innovative smart irrigation system.

  4. Suppose you are a civil engineer tasked with estimating the average compressive strength of a certain type of concrete used in construction projects in the Kilinochchi area. In a previous study, it was found that the standard deviation of the compressive strengths of this type of concrete is 500 psi. To ensure a high level of confidence in your estimate, you aim to conduct a survey with a 92% confidence interval. You want the estimate to be within 100 psi of the true average compressive strength. Based on this information, determine the minimum sample size needed for your survey.

Perform

Assignment 2 Examination scheduled from 11.00 am to 11.40 am on January 30, 2026 (Friday). To ensure that the exam runs smoothly, we have arranged the exam halls as follows: 

  • Group 1 (Registration numbers 2024/E/001 to 2024/E/101) will be in Exam Hall 1 (First floor, Computer Engineering Department)  

  • Group 2 (Registration numbers 2024/E/102 to 2024/E/198and re-attempt students) will be in Exam Hall 2 (Drawing Hall 2nd floor, Admin Building).

Make sure to review all sections thoroughly to ensure you’re prepared for the exam! I hope everyone follows the exam policies and cooperates with the exam administration to make the exam run smoothly. Moreover, you can access class materials by checking out the course webpage.

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