Reference no: EM133047380

EG5060 Statistics And Probability For Safety, Reliability And Quality - University of Aberdeen

Question set A

Question 1. A machine is producing metal pieces that are cylindrical in shape. A sample of pieces is taken and the diameters (in cm) are 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and 1.03. Find a 99% confidence interval for the mean diameter of pieces from this machine, assuming an approximate normal distribution.

Question 2. A soft-drink dispensing machine is said to be out of control if the variance of the contents exceeds 1.15 litres. If a random sample of 25 drinks from this machine has a variance of 2.03 litres, perform a hypothesis test at the 0.05 level of significance to check if the machine is out of control. Assume that the contents are approximately normally distributed.

Question 3. The daily water levels of two reservoirs M and N normalised to the respective full condition are denoted by two variables X and Y having the following joint probability density function

f(x, y) = c(x + y²), 0 < x < 1; 0 < y < 1

i) Determine the value of c that makes the function f(x,y) a joint probability density function over the given range.
ii) What is the probability that the water level in reservoir M is between 0.5 and 1?
iii) If reservoir M is half-full on a given day, what is the probability that the water level will be more than half full in reservoir N?

Question 4. In a particular study on beams made from composite laminates, the natural frequency of beams under loads were given as (in hertz): 230.66, 233.05, 232.10, 229.48, 231.58. Using a probability plot, check if there is evidence to support that the frequencies are normally distributed.

Question 5. The distribution of ocean wave height, H, may be modelled with the Rayleigh probability density function as

fH(h) = h/α² e^-1/2(h/α)2, h ≥0

where α is the parameter of the distribution. Suppose that the following measurements of the wave height were observed (in m): 3.11, 3.27, 3.84, 4.67,
2.82, 3.53, 3.21, 1.92, 3.87

Evaluate the maximum likelihood estimate of the parameter α.

Question 6. The marks of a class of 9 students on a midterm report (x) and on the final examination (y) are as follows:

x	77	50	71	72	81	94	96	99	67
y	82	66	78	34	47	85	99	99	68

Given that
∑x_i² = 57557 ;∑x_i y_i = 53258; ∑y_i² = 51980

i) Assuming that a simple linear regression model is appropriate, obtain the least squares fit relating y and x. Calculate the coefficient of determination for this model and provide an interpretation.

ii) Calculate the sample correlation coefficient between x and y.
iii) Estimate the final examination mark of a student who received a mark of 85 on the midterm report.

Question 7. Suppose that X is a continuous random variable with probability distribution

f_x(x) = e^-x, x ≥ 0

Determine the probability distribution for
i) Y = X²
ii) Y = ln X

Question 8. The following is a set of 12 measurements of a water-quality parameter in ppm: 47, 53, 61, 57, 65, 44, 56, 63, 58, 49, 51, 54

Comment on the suitability of normal distribution for modelling this water- quality parameter by performing a Kolmogorov-Smirnov goodness-of-fit test at 1% significance level.

Sample Question set B

Question 1. Suppose that X is a continuous random variable with probability distribution

f_x(x) = x/18, 0 ≤ x ≤ 6

i) Find the probability distribution of the random variable Y = 2X + 10.
ii) Find the expected value of Y.

Question 2. In a research study on the thermal inertia properties of autoclaved aerated concrete, five samples were tested, and the interior temperature (°C) was reported as follows:
23.01, 22.22, 22.04, 22.62, and 22.59.
i) Construct a normal probability plot of the interior temperature and comment on the model suitability.
ii) Construct a 95% two-sided confidence interval for population standard deviation.

Question 3. It is known that a sample of 14.9, 16.8, 13.6, 17.3 and 15.4 comes from a population with the density function

fx(x) = x/θ²e^-(x/θ), 0 ≤ x ≤ ∞

in which θ>0 is the parameter of the distribution. Find the maximum likelihood estimator for θ.

Question 4. Determine the covariance and correlation for the joint probability density function f_XY(x, y) = 0.1 xy over the range 0 < x < 3 and 0 < y < x.

Question 5. The biochemical oxygen demand (BOD) test is conducted over a period of time in days at a particular location and the resulting data are as shown below:

x, Time (days)	1	2	4	6	8	10	12	14
y, BOD (mg/litre)	0.6	0.7	1.5	1.9	2.1	2.6	2.9	3.7

Given that
∑ x_i = 57; ∑ y_i = 16; ∑ x_i² =561 ;∑ x_iy_i =148.8; ∑ y_i² =39.98;

i) Assuming that a simple linear regression model is appropriate, obtain the least squares fit relating y and x.
ii) Draw a scatter plot of the data superimposed by the obtained regression model.

Question 6. In a test of a device to generate electricity from wave power at sea, 60 observations are made of the root mean square bending moment M of a component (in Newton metres). The data are summarised as follows. The sample mean is 5.08 and the sample variance is 3.29. Test the hypothesis that M has a normal distribution by performing a chi-square goodness-of-fit test at 5% significance level.
Class   Frequency
 M≤3         5
 3<M≤4    12
 4<M≤5    18
 5<M≤6    11
 6<M≤7     5
 7<M≤8     4
 8<M         5

Question 7. The life in hours of a 50-watt light bulb is known to be normally distributed with σ = 25 hours. A random sample of bulbs has a mean life (x¯) of 1014 hours. If the total width of the two-sided confidence interval on mean life is to be eight hours at 90% confidence, what sample size should be used?