Week 11 (16-02-26 to 20-02-26)

Prepare

📖 Read the syllabus

📖 Read the support resources

Participate

🖥️Lecture 6 -Association between Two Variables

🖥️ Statistical Table - Statistical Table for probability estimations.

🖥️Lecture 5 - Statistical Inferences II

🖥️Lecture 4 - Statistical Inferences I

🖥️ Lecture 3 -Continuous Probability Distributions

🖥️ Lecture 2 - Discrete Probability Distributions

🖥️ Lecture 1 - Introduction to Probability

Practice

  1. An experiment was conducted to evaluate the effectiveness of a new construction material in enhancing the structural integrity of concrete beams. A random sample of 24 beams, all with similar dimensions and initial structural conditions, was selected. Twelve of the beams were reinforced with the new material, while the remaining twelve were left un-reinforced. After a 6-month period of exposure to varying loads and environmental conditions, the beams were tested for their load-bearing capacity, and the results are summarized in the table below:

    Beam Number 1 2 3 4 5 6 7 8 9 10 11 12
    Reinforced Beams 18 43 28 50 16 32 13 35 38 33 6 7
    Un-reinforced Beams 40 54 26 63 21 37 39 23 48 58 28 39

    Assume that the load-bearing capacity measurements follow a normal distribution with equal variances.

    1. Test whether the mean load-bearing capacity of the reinforced beams is greater than the mean for the un-reinforced beams. Use a significance level of α = 0.05.

    2. Place a 95% confidence interval on the difference μ1 −μ2 to assess the magnitude of the difference in load-bearing capacity between the two types of beams.

  2. A dietitian wishes to see if a person’s cholesterol level will change if the diet is supplemented by a certain mineral. Seven subjects were pretested, and then they took the mineral supplement for a 6-week period. The results are shown in the table.(Cholesterol level is measured in milligrams per deciliter.)

    Subject 1 2 3 4 5 6 7

    Before 210 235 208 190 172 244 232

    After 190 170 210 188 173 228 232

    Assume the variable (person’s cholesterol level changes) is approximately normally distributed.

    1. Construct 99% confidence interval for the mean difference of person’s cholesterol level changes.
    2. Can it be concluded that the cholesterol level has been changed at the 10% level of significance?

  3. A taxi company is trying to decide whether to purchase brand A or brand B tires for its fleet of taxis. To estimate the difference in the two brands, an experiment is conducted using 12 of each brand. The tires are run until they wear out. The results are Brand A: ¯x1 = 36,300 km, s1 = 5,000 km, Brand B: ¯x2 = 38,100 km, s2 = 6,100 km Compute a 95% confidence interval for μA − μB assuming the populations to be approximately normally distributed. You may not assume that the variances are equal.An experiment was performed to compare the abrasive wear of two different laminated materials. Twelve pieces of material 1 were tested by exposing each piece to a machine measuring wear. Ten pieces of material 2 were similarly tested. In each case, the depth of wear was observed. The samples of material 1 gave an average (coded) wear of 85 units with a sample standard deviation of 4, while the samples of material 2 gave an average of 81 with a sample standard deviation of 5. Assume that the populations to be approximately normal with equal variances.

    1. Construct a 95% confidence interval for the difference in mean wear between Material 1 and Material 2.
    2. Test, at the 0.05 level of significance, whether the mean wear of Material 1 exceeds that of Material 2 by more than 2 units.

  4. The following data represent the running times of films produced by two motion-picture companies. Company Time (minutes) I 103 94 110 87 98 II 97 82 123 92 175 88 118 Assume that the running-time differences are approximately normally distributed with unequal variances.

    1. Compute a 90% confidence interval for the difference between the average running times of films produced by the two companies.
    2. Test, at the 0.10 level of significance, the hypothesis that the average running time of Company 2 exceeds that of Company 1 by 10 minutes, against the alternative that the difference is less than 10 minutes.

  5. Engineering students at the University of Jaffna participated in a study to evaluate the impact of a new teaching method on their academic performance. Seven students were tested before and after the implementation of the new method. The test scores (out of 100) for each student before and after the implementation are provided in the table below. Assume that the changes in test scores are approximately normally distributed. Subject 1 2 3 4 5 6 7 8 Before 75 82 78 65 70 88 84 85 After 80 85 82 70 75 90 86 79

    1. Can we use paired t-test for this problem? Justify your answer.
    2. Construct a 99% confidence interval for the mean difference in test scores.
    3. Can it be concluded that the teaching method has an impact on test scores at the 10% level of significance?

  6. A study is conducted to compare the lengths of time required by men and women to assemble a certain product. Past experience indicates that the distribution of times for both men and women is approximately normal but the variance of the times for women is less than that for men. A random sample of times for 11 men and 14 women produced the following data: Men Women n1 = 11 n2 = 14 s1 = 6.1 s2 = 5.3 Test, at the 0.05 level of significance, the hypothesis that σ2 1 = σ2 2 against the alternative that σ2 1 > σ2

  7. A research study aims to compare the effectiveness of two teaching methods, Method A and Method B, in improving students’ understanding of thermodynamics concepts. A sample of 20 students from the engineering faculty participated in the study. Ten students were randomly assigned to Method A, and the other ten were assigned to Method B. After completing the course, each student took a standardized test on thermodynamics, and their scores were recorded.

    The scores obtained by students in Method A were as follows: 85, 90, 88, 92, 87, 89, 91, 86, 90, 88

    The scores obtained by students in Method B were as follows: 75, 88, 85, 82, 89, 86, 83, 85, 87, 80

    Assume that the scores are normally distributed

    1. Test whether the population variances of the test scores obtained by students in Method A and Method B are equal or not. Use a 10% level of significance.
    2. Test whether there is any difference between the mean test scores of Method A and Method B. Use a 10% significance level.

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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