Reference no: EM133047382

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:

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.