Continuous Distributions

1.   The strength of an aluminium alloy is normally distributed with mean 10 GPa and standard deviation 1.4 GPa.

a.   What is the probability that a specimen of this alloy will have strength greater than 12 GPa?

b.   What is the probability that a specimen of this alloy will have strength smaller than 9 GPa?

2.   The Young’s modulus of steel has a mean of 210 GPa and a standard deviation of

30 GPa. It is assumed that the Young’s modulus of steel is lognormally distributed.

a.   What is the probability of Young’s modulus being less than 180 GPa?

b.   What is the probability of Young’s modulus having a value between 190 GPa and

230 GPa?

Discrete Distributions

3.   In a large shipment of printed circuit boards (PCBs), 2% have a short circuit. Seven PCBs have been chosen for installation into a piece of factory equipment .

a.   What is the probability that two of the boards have a short circuit?

b.   What is the probability that one or more boards have a short circuit?

4.   Joseph works in the construction industry. In the past ten years, Joseph has had tools or equipment stolen on seven occasions . Assume that these theft events are independent.

a.   What is the probability that Joseph will experience ten thefts within the next five years?

b.   What is the probability that Joseph will experience fewer than three thefts within the next five years?

Confidence Intervals

5.   Find the value of Z that corresponds to the following confidence levels:

a.   99%

b.   90%

6.   Find the confidence levels of the confidence intervals with the following values of Za/2:

a.   1.28

b.   3.28

7.   Find the value of t that corresponds to the following confidence levels:

a.   99% for 10 samples

b.   75% for 14 samples

8.   Find the confidence level of a two-sided confidence interval that is based on the given value of tn−1, a/2  and the given sample size:

a.   t = 3.106 with sample size 12

b.   t = 1.746 with sample size 17

9.   In a sample of 100 boxes of a certain type, the average compressive strength was 6230 N and the standard deviation was 221 N.

a.   Find a 95% confidence interval for the mean compressive strength of boxes of this type.

b.   How many boxes must be sampled so that a 95% confidence interval will specify the mean to within ±25 N?

10. Six measurements were made of the concentration (in percent) of ash in a certain         variety of spinach. The sample mean was 19.35 and the sample standard deviation was 0.577. Find a 95% confidence interval for the concentration of ash .

11. A model of heat transfer from a cylinder immersed in a liquid predicts that the heat-  transfer coefficient for the cylinder will become constant at very low flow rates of the fluid. A sample of 10 measurements is taken. The results, in W/m2 K, are

13.7     12.0     13.1     14.1     13.1     14.1     14.4     12.2     11.9     11.8

Find a 95% confidence interval for the heat-transfer coefficient.

12. The melting points of two alloys are being compared. Thirty-five specimens of alloy 1      were melted. The average melting temperature was 269.4°C and the standard deviation was 1.3°C. Forty-seven specimens of alloy 2 were melted. The average melting                 temperature was 265.6°C and the standard deviation was 1.2°C. Find a 99% confidence  interval for the difference between the melting points.

Hypothesis Testing

13. In an experiment to measure the lifetimes of parts manufactured from a certain    aluminium alloy, 73 parts were loaded cyclically until failure. The mean number of kilocycles to failure was 783, and the standard deviation was 120.

Let μ represent the mean number of kilocycles to failure for parts of this type. A test is made of H0 : μ ≤ 750 versus H1 : μ > 750.

Can you conclude that the mean number of kilocycles is greater than 750 based on a 5% level of significance?

14. Twenty-five pieces of timber were drawn at random from a large shipment. Their mean length was found to be 3.481 m with a standard deviation of 0.055 m.

Can you conclude that the mean length of pieces of timber in the shipment is less than 3.5 m to a 5% level of significance?

15. 75 samples were selected from a new type of power supply for home computers. The mean lifetime of the sample of new power supplies is 4387 h with standard deviation

252 h.

Additionally, 80 samples were selected from old power supplies . The mean lifetime of the sample of old power supplies is 4260 h with standard deviation of 231 h.

Can you conclude that the mean lifetime of the new type of power supply is greater than that of the old power supplies?

16. In a study of the relationship of the shape of a tablet to its dissolution time, 6 disk-       shaped ibuprofen tablets and 8 oval-shaped ibuprofen tablets were dissolved in water. The dissolution times, in seconds, were as follows:

Disk: 269.1  249.3  255.2  252.7  247.0  261.6

Oval: 268.8  260.0  273.5  253.9  278.5  289.4  261.6  280.2              Can you conclude that the mean dissolve times differ between the two shapes?

17. Two microprocessors are compared on a set of six benchmark tests to determine          whether there is a difference in speed. The times (in seconds) taken by each processor to run each test are:

Can you conclude that processor A is faster than processor B?

Power and Sample Size

18. Find the power of a test of H0 : u ≥ 50000 against H1 : u < 50000 with a sample size of 100, standard deviation of 5000 and significance level of 5%. Assume that the alternative        mean, uA, is 49500.

19. A type of low-capacity battery is undergoing testing to ensure that the mean lifetime of the batteries under the test load is no less than 4 h. A random sample of 50 packs is       selected and tested. The average life of these batteries is 4.05 h with a standard             deviation of 0.2 h.

a.   Compute the power of the test if the true mean battery life is 4.2 h.

b.   What sample size would be required to detect a true mean battery life of 4.2 h if we wanted the power of the test to be at least 90%?

Linear Regression

20. Inertial weight (in tons) and fuel economy (in kilometres per litre) were measured for a sample of seven diesel trucks:

a.   Compute the correlation coefficient between inertial weight and fuel economy.

b.   Compute the least-squares line for predicting fuel economy from inertial weight.

c.    Predict the fuel economy for a truck with an inertial weight of 15 t.

Engineering Reliability

21. A simply supported beam of length l = 4 m is loaded with a uniformly distributed load w with u = 3 kN/m and σ = 0.5 kN/m. The bending strength of similar beams has been        found to have a mean strength uR = 8 kNm with a coefficient of variation (COV) of 0.2.

Assuming that the beam self-weight and any variation in the length of beam can be ignored, evaluate the probability of failure.

l 2 w

Note: The applied moment is given by S =

22. A simply supported steel beam W14x61 (capacity µ R = 360.7 kNm, σR = 72.9 kNm) with span l = 6 m has been designed to carry a dead load (µD = 2.6 kN/m2, σD = 0.35 kN/m2 ) and a live load (µL = 2.75 kN/m2, σL = 1 kN/m2 ). The height a is 3 m.

Assuming dead load (D), live load (L) and beam capacity (R) are statistically independent normal variables, and the total load T = D + 0.8L, evaluate the probability of failure.

Note: The applied moment is given by S =

Answers