Question 1
This question is fine, covering a mix of bookwork and familiar questions about finding the maximum likelihood estimate for a given distribution, for example as in Lab Sheet Three.
Question 2
This question is fine and illustrates the duality between confidence intervals and hypothesis testing. Part (a) follows the approach in Section 3.2.2 of the on0line lecture notes. Part (b) is analogous to Example 4.19 of the on-line lecture notes but with the t-distribution pivot rather than the standard normal.
Question 3
This question is also fine and a typical example of a paired sample t-test. Further examples are in Lab Sheet Nine and Lab Sheet Ten. In the 2023-24 exam, you will have access to the University Formula Book and so the hint, giving various quantiles, wouldn’t be given.
Question 4
This is fine, covering ideas about bias, mean squared error and consistency as well as comparisons between different estimators. Lab Sheet Four provides similar examples.
Question 5
This is fine, covering the fundamental (bookwork) principles of hypothesis testing and the use of the Neyman-Pearson lemma. Further examples of using the Neyman-Pearson lemma can be found in Lab Sheet Seven.
Question 6
This is fine covering ideas about F-tests and the independent two sample t-test. Further examples of these techniques can be found in Lab Sheet Ten and Lab Sheet Eleven. In the 2023-24 exam, you will have access to the University Formula Book and so the hints in part (b) and part (d), giving various quantiles, wouldn’t be given.
Question 1
This would be fine in the in-person exam world and tests understanding about confidence intervals.
Question 2
This would be fine in the in-person exam world. Note that the critical region is where we reject \(H_{0}\) in favour of \(H_{1}\) and so the region constructed in part (b) is where we reject \(H_{0}\). Thus, the written answer for part (c) is incorrect because the test statistic is outside the critical region; we should accept \(H_{0}\).
Question 3
This would be fine in the in-person exam world and tests understanding of the constructing of method of moments and maximum likelihood estimates (part (b) has to be neither as the sample median would arise from either the method of moments or maximum likelihood construction for the given density).
Question 4
This would be fine in the in-person exam world, covering the key ideas of properties for estimators.
Question 5
This would be fine in the in-person exam world. Part (b) covers a \(\chi^{2}\) goodness of fit test and further examples of these can be found on Lab Sheet Eleven. In part (b) for an in-person exam you would have been given either a specific level at which to perform the test or be expected to interpret the observed value of \(X^{2}\) using \(\chi^{2}\)-tables: as \(12.592 < 12.9 < 14.449\) then the p-value is in the interval \((0.025, 0.05)\). Parts (c) and (d) essentially use ideas derived in question 3 of the questions for marking and feedback of Lab Sheet Six.
Question 6
This would be fine in the in-person exam world. In part (e), you need quantiles of a Gamma distribution in order to perform the test. In an in-person exam setting, you would have been given a list of possible quantile values from which to choose the explicit critical value for the given data and significance level.
Question 1
This would be fine in the in-person exam world. In part (a) you should be able to calculate the mean and variance from a given data set using a calculator. In part (b), the critical values for the 92% confidence intervals are not given in the corresponding tables in the University Formula Book. For an in-person exam, you would have been asked to find 100\((1-\alpha)\)% intervals for values of \(\alpha\) that are available on the tables for the desired values of \(t_{n-1, 1- \alpha/2}\), \(\chi^{2}_{n-1, 1-\alpha/2}\), and \(\chi^{2}_{n-1, 1-\alpha/2}\).
Question 2
This would be fine for an in-person exam. In part (c), if I wanted an exact test based on the binomial distribution then you would be given an appropriate list of quantiles. Question 2 of the questions for marking and feedback section of Lab Sheet Seven would be an example of the style of this (but for the Poisson distribution): part (a) does an exact calculation and part (b) using a central limit theorem argument.
Question 3
This would be fine for an in-person exam.
Question 4
This would be fine for an in-person exam, the only change would be the data summaries given in part (d). We didn’t directly discuss how to calculate \(s_{D}^{2}\) from summaries of the distributions of \(X\) and \(Y\) including the covariance. Instead, we did this by manipulating the actual data and working with (or calculating) \(\sum_{i=1}^{n} d_{i}\) and \(\sum_{i=1}^{n} d_{i}^{2}\). Thus, based on the 2023-24 delivery of the unit, the data summaries given would have been: \[\begin{eqnarray*} \sum_{i=1}^{25} d_{i} \ = \ -52 & \mbox{ and} & \sum_{i=1}^{25} d_{i}^{2} = 4751. \end{eqnarray*}\]
Question 5
This would be fine for an in-person exam. In part (d), I would expect you to be able to do all the calculations involved in this question on a calculator.
Question 6
This would be fine for an in-person exam. In part (b), I would have most likely stated the first and second moments of the uniform distribution in the question in a style similar to question 3 of the questions for marking and feedback of Lab Sheet Four.
Question 1
This question is fine and we covered all of the ideas in it.
Question 2
This question is fine and we covered all of the ideas in it.
Question 3
This question is fine and we covered all of the ideas in it. There is a typo in the solution to part (c) where the p-value should be \[\begin{eqnarray*} P(X \geq x \, | \, \pi = \pi_{0}) & = & \sum_{i=x}^{n} \binom{n}{i} \pi_{0}^{i}(1-\pi_{0})^{n-i}. \end{eqnarray*}\]
Question 4
This question is fine and we covered all of the ideas in it.
Question 5
This question is fine and we covered all of the ideas in it. Whilst, in principle, you can do everything in parts (d) and (e), the relationship in (d) is not something that we discussed in the 2023-24 delivery of the unit and so I would be unlikely to construct a similar question in this way. However, thinking about the reasonableness of assumptions (as per part (e)) and the implications of these (part (f)) is something that I would expect students at the upper end to be able to do.
Question 6
Parts (a), (b), (c), and (d) are appropriate to the content we covered in 2023-24. We didn’t cover the delta method so I would not ask a question like part (e). In principle, you could do part (f) (using properties of the normal distribution) and part (g) (using similar ideas to question 3b of the questions for marking and feedback of Lab Sheet Six) but they do rely on the answer in part (e).
Question 1
This question is fine and we covered all of the ideas in it.
Question 2
This question is fine and we covered all of the ideas in it. The quantiles of the t-distribution in the typed solution for part (d) use the same notation that we used in the 2023-24 delivery of the unit (and so you should ignore the hand-written correction to these).
Question 3
This question is fine and we covered all of the ideas in it.
Question 4
This question is fine and we covered all of the ideas in it.
Question 5
This question is fine and we covered all of the ideas in it.
Question 6
This question is fine and we covered all of the ideas in it.