Reference no: EM133047386

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

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?