1. In order to investigate the effectiveness of training programmes preparing new employees for routine tasks, 24 new employees (with the same background knowledge and experience) were randomly divided into 4 groups. Each group followed a different training programme. The table below shows the time (in hours) for each new employee to successfully complete the same test suite of routine tasks.
| Group |
| A |
B |
C |
D |
| 9 |
11 |
11.5 |
13.25 |
| 9.5 |
10 |
12 |
11.5 |
| 9.75 |
10 |
9 |
12 |
| 10 |
11.75 |
11.5 |
13.5 |
| 13 |
10.5 |
13.25 |
11.5 |
| 9.5 |
15 |
13 |
11.5 |
Implement a one-factor ANOVA model and compute 95% confidence intervals for the expected time taken to complete the test suite in each group. Test whether the training programmes have any effect. Supply your calculated output, comments, and code.
2. Transistor Inc. wants to investigate the effectiveness of an additional chemical bath treatment on the yield of server-grade devices. They try three new treatments, as well as their usual process. The table below lists the percentage yield of server-grade devices, for five different production plants.
|
Treatment |
| Plant |
Usual |
A |
B |
C |
| 1 |
13.8 |
11.7 |
14 |
12.6 |
| 2 |
12.9 |
16.7 |
15.5 |
13.8 |
| 3 |
25.9 |
29.8 |
27.8 |
25 |
| 4 |
18 |
23.1 |
23 |
16.9 |
| 5 |
15.2 |
20.2 |
19.9 |
13.7 |
Does the data suggest that adding a chemical bath treatment makes a difference? Use two-factor ANOVA to investigate this. Supply your calculated output, comments, and code.
3. For a time-homogeneous two-state Markov chain (Xt, t = 0, 1, 2, . . . ) with statespace E = {0, 1} and one-step transition matrix
where p, q 2 (0, 1), determine its stationary distribution.
4. For the same chain in Q3, with initial distribution P(X0 = 0) = a = 1 P(X0 = 1), determine formulas for the t-step probabilities P(Xt = 0) and P(Xt = 1). What is the limiting distribution of the Markov chain?
5. For the same chain in Q3 and Q4, simulate N = 103 outcomes of the chain with a = 1, p = 0.1, and q = 0.05. Plot the empirical average state together with the expected state for t = 0, 1, . . . , 100. Supply your plot and code. [2]
6. Consider five web-pages with links between them described by the (row-normalised) matrix
A random web-surfer is modelled as a Markov chain with one-step transition matrix
P = 0.85L + 0.15R=5 ,
where R is a 5 x 5 matrix of ones. Find the long-term probability Πi that the random web-surfer is on page i, for i = 1, . . . , 5 and use this to rank the five web pages.