Question 1 - The size y (in millimetres) of a crack in a Pontiac Trans Am's front sub-frame weld is described by a random variable X with the following PDF:
(a) Sketch the PDF and CDF.
(b) Determine the mean crack size.
(c) What is the probability that a crack will be smaller than 3 mm?
(d) Determine the median crack size.
(e) Suppose there are four cracks in the weld. What is the probability that only one of the four cracks is larger than 3 mm?
Question 2 - The average speed of vehicles on a freeway is being studied. Assume that the standard deviation of vehicle speed is known to be 8 km/h.
(a) Suppose observations on 120 vehicles yielded a sample mean of 105 km/h. Determine two-sided 99.5% confidence intervals of the mean speed. (Assume a normal distribution).
(b) In part (a), how many additional vehicles' speed should be observed such that the mean speed can be estimated to within ±1 km/h with 99.5% confidence?
(c) Suppose Jason and Britney are assigned to collect data on the speed of vehicles on this highway. After each person has separately observed 60 vehicles, what is the probability that Jason's sample mean will be equal to Britney's sample mean ±0.5 km/h?
(d) Repeat part (c) if each person has separately observed 120 vehicles instead.
Question 3- The fuel consumption of a certain make of car may not be exactly that rated by the manufacturer. Suppose ten cars of the same model were tested for combined city and highway fuel consumption, with the following results:
|
Car No.
|
Observed fuel consumption
|
|
1
|
7.1 litres per 100km
|
|
2
|
6.4 litres per 100km
|
|
3
|
6.8 litres per 100km
|
|
4
|
6.2 litres per 100km
|
|
5
|
7.8 litres per 100km
|
|
6
|
6.1 litres per 100km
|
|
7
|
6.6 litres per 100km
|
|
8
|
7.8 litres per 100km
|
|
9
|
6.4 litres per 100km
|
|
10
|
7.4 litres per 100km
|
(a) Estimate the sample mean and sample standard deviation of the actual fuel consumption of this particular model of car.
(b) Suppose that the manufacturer's stated fuel consumption of this particular model of car is 7 litres per 100km; perform a hypothesis test to verify the stated fuel consumption with a significance level of 2%.
Question 4- The occurrence of bushfires in the Port Stephens area may be modelled by a Poisson process. The average occurrences of bushfires v is assumed to be either 15 (event A1), 20 (event A2) or 25 (event A3) times a year. The probability of A2 is the same as the probability A1 and the probability of A3 is half of A1.
(a) Determine the probability that there will be 20 occurrences of bushfire in the next year.
(b) If there are exactly 20 bushfires in the next year (event B), what will be P(A1|B)? Determine P(A2|B) and P(A3|B).
Question 5- (Additional Question - No need for submission)
Measurements of the velocity in a fully developed flow in a circular pipe whose wall radius is R, gave the following data:
|
r/R
|
0.0
|
0.102
|
0.206
|
0.412
|
0.617
|
0.784
|
0.846
|
0.907
|
0.963
|
1.0
|
|
u/Uc
|
1.000
|
0.997
|
0.988
|
0.959
|
0.908
|
0.847
|
0.818
|
0.771
|
0.690
|
0.000
|
where r is the radius at which the velocity is measured, u is the velocity, and Uc is the centerline velocity. For R = 12.35 cm, and Uc = 30.5 m/sec, estimate the average velocity, Uave, in the pipe as defined by:
Uave = 2/R20∫Rurdr = 2Uc0∫R(u/Uc) ηdη.
where η = r/R.
(a) Explain why the Trapezoidal Rule is the most appropriate rule to use for the integration and why it is likely to under-estimate the value of Uave. Give your answer to two decimal places.
(b) Check your answer by writing a Matlab program to perform the trapezoidal integration. The data is given in the data file pipeflow.dat. The easiest way to read a data file into a Matlab program is to use the 'load' command to fill an array with the data for u/Uc and η.