MATH235 Week 8 - Assessed problems (coursework)

Submission is due on Tuesday in Week 9.

CW8.1

At the British Grand Prix at Silverstone, 6 July 2014, the following fastest laps were recorded for each of the 20 participating drivers:

Rank	Driver	Time (secs)	Rank	Driver	Time (secs)
1	Hamilton	97.20	11	Perez	98.70
2	Vettel	97.50	12	Grosjean	98.90
3	Rosberg	98.10	13	Vergne	99.30
4	Bottas	98.30	14	Sutil	100.00
5	Button	98.30	15	Bianchi	100.00
6	Kvyat	98.40	16	Maldonado	100.30
7	Ricciardo	98.50	17	Chilton	100.40
8	Alonso	98.60	18	Kobayashi	101.50
9	Hulkenberg	98.60	19	Gutierrez	102.60
10	Magnussen	98.70	20	Ericsson	104.30

Data are available on Moodle as silv2014.rda.

(a)

It is suggested that the lap times be modelled as independent draws from a normal distribution with mean and variance $\sigma^{2}$ . In this question, interest lies in the standard deviation $\sigma$ . Discuss any assumptions of this model, and whether they are reasonable.

[marks: 2]
(b)

Four drivers (Rosberg, Maldonado, Gutierrez and Ericsson) retired from the race early (due to either car failures or driver accidents). Explain why it may be sensible to exclude these drivers from the analysis.

[marks: 1]
(c)

Use R to assess the normality of the lap times of the 16 drivers who completed the race (using a QQ-plot, for example). Hint: use timef<-silv2014$lap[-c(3,16,19,20)] to extract these times.

[marks: 2]
(d)

Write down the likelihood and log-likelihood functions for $\sigma$ , assuming $\mu$ is some unknown constant.

[marks: 2]
(e)

Find the maximum likelihood estimate of $\sigma$ .

[marks: 1]
(f)

Calculate the observed information at the MLE for this model.

[marks: 2]