Week 7 (19-01-26 to 23-01-26)

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. Statistics released by the National Highway Traffic Safety Administration and the National Safety Council show that on an average weekend night, 1 out of every 10 drivers on the road is drunk. If 400 drivers are randomly checked next Saturday night, what is the probability that the number of drunk drivers will be

    1. less than 32?
    2. more than 49?
    3. at least 35 but less than 47?

  2. A drug manufacturer claims that a certain drug cures a blood disease, on the average, 80% of the time. To check the claim, government testers use the drug on a sample of 100 individuals and decide to accept the claim if 75 or more are cured.

    1. What is the probability that the claim will be rejected when the cure probability is, in fact, 0.8?
    2. What is the probability that the claim will be accepted by the government when the cure probability is as low as 0.7?

  3. A manufacturing company produces metal rods whose lengths are normally distributed with a mean of 11.5 cm and a standard deviation of 1.6 cm. quality control inspector randomly selects a sample of 16 rods for inspection.

    1. What is the standard deviation of the sampling distribution of the sample mean?
    2. What is the probability that the sample mean length exceeds 12.3 cm?
    3. Between which two symmetrically located values will 68% of the sample means lie.

  4. The amount of time that a drive - through bank teller spends on a customer is a random variable with a mean μ = 3.2 minutes and a standard deviation σ = 1.6 minutes. If a random sample of 64 customers is observed, find the probability that their mean time at the teller’s window is

    1. at most 2.7 minutes;
    2. more than 3.5 minutes;
    3. at least 3.2 minutes but less than 3.4 minutes.

  5. The finished inside diameter of a piston ring is normally distributed with a mean of 10 centimeters and a standard deviation of 0.03 centimeter.

    1. What proportion of rings will have inside diameters exceeding 10.075 centimeters?
    2. What is the probability that a piston ring will have an inside diameter between 9.97 and 10.03 centimeters?
    3. Below what value of inside diameter will 15% of the piston rings fall?

Exercise

  1. An efficiency expert wishes to determine the average time that it takes to drill three holes in a certain metal clamp. How large a sample will she need to be \(95\%\) confident that her sample mean will be within 15 seconds of the true mean? Assume that it is known from previous studies that \(\sigma = 40\) seconds.

  2. 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 centimetres) 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 \(\sigma^2\) 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.

  3. The Edison Electric Institute has published figures on the number of kilowatt hours used annually by various home appliances. It is claimed that a vacuum cleaner uses an average of 46 kilowatt hours per year. If a random sample of 12 homes included in a planned study indicates that vacuum cleaners use an average of 42 kilowatt hours per year with a standard deviation of 11.9 kilowatt hours, does this suggest at the 0.05 level of significance that vacuum cleaners use, on average, less than 46 kilowatt hours annually? Assume the population of kilowatt hours to be normal.

  4. A manufacturer of sports equipment has developed a new synthetic fishing line that the company claims has a mean breaking strength of 8 kilograms with a standard deviation of 0.5 kilograms. Test the hypothesis that \(\mu = 8\) kilograms against the alternative that \(\mu = 8\) kilograms if a random sample of 50 lines is tested and found to have a mean breaking strength of 7.8 kilograms. Use a 0.01 level of significance.

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