Description

1 Introduction
1. No files are to be submitted via e-mail. Submissions are to be made via Moodle.
Question 1 (10 marks)
1. Calculate an estimate of the average number of days to recovery using the provided data. Calculate a 95% confidence interval for this estimate using the t-distribution, and summarise/describe your results appropriately. Show working as required. [4 marks]
3. It is of interest to determine if there are any differences, at a population level, in recovery times for patients in different countries. Test the hypothesis that the population average time taken to recover for the Israeli cohort is the same as in the NSW cohort. Write down explicitly the hypothesis you are testing, and then calculate a p-value using the approximate hypothesis test for differences in means with (different) unknown variances presented in Lecture 5. What does this p-value suggest about the difference in mean recovery time between the two cohorts of patients?
[3 marks]
Question 2 (10 marks)
The exponential distribution is a probability distribution for non-negative real numbers. It is often used to model waiting or survival times. The version that we will look at has a probability density function of the form
p(y |v) = exp −e−vy − v (1)
where y ∈ R+, i.e., y can take on the values of non-negative real numbers. In this form it has one parameter: a log-scale parameter v. If a random variable follows an exponential distribution with log-scale v we say that Y ∼ Exp(v). If Y ∼ Exp(v), then E [Y ] = ev and V [Y ] = e2v.
1. Produce a plot of the exponential probability density function (1) for the values y ∈ (0,10), for v = 1, v = 0.5 and v = 2. Ensure the graph is readable, the axis are labeled appropriately and a legend is included. [2 marks]
2. Imagine we are given a sample of n observations y = (y1,…,yn). Write down the joint probability of this sample of data, under the assumption that it came from an exponential distribution with log-scale parameter v (i.e., write down the likelihood of this data). Make sure to simplify your expression, and provide working. (hint: remember that these samples are independent and identically distributed.) [2 marks]
3. Take the negative logarithm of your likelihood expression and write down the negative loglikelihood of the data y under the exponential model with log-scale v. Simplify this expression. [1 mark]
4. Derive the maximum likelihood estimator vˆ for v. That is, find the value of v that minimises the negative log-likelihood. You must provide working. [2 marks]
5. Determine the approximate bias and variance of the maximum likelihood estimator vˆ of v for the exponential distribution. (hints: utilise techniques from Lecture 2, Slide 27 and the mean and variance of the sample mean) [3 marks]
Question 3 (8 marks)
It is frequent in nature that animals express certain asymmetries in their behaviour patterns. It has been suggested that this might be nature’s way of “breaking gridlocks” that might occur if we were to act purely rationally (for example, why does a beetle decide to move one way over another when put in a featureless bowl?). An interesting observational study, undertaken by a European researcher in 2003 examined the head tilting preferences of humans when kissing.
The data was collected by observing kissing couples of age ranging from 13 to 70 in public places (mostly airports and train stations) in the United States, Germany and Turkey. The observational data found that of 124 kissing pairs, 80 turned their heads to the right and 44 turned their heads to the left.
You must analyse this data to see if there is an inbuilt preference in humans for the direction of head tilt when kissing. Provide working, reasoning or explanations and R commands that you have used, as appropriate.
1. Calculate an estimate of the preference for humans turning their heads to the right when kissing using the above data, and provide an 95% confidence interval for this estimate using the techniques discussed in Lecture 4. Summarise/describe your results appropriately. [3 marks]
2. Test the hypothesis that there is no preference in humans for tilting their head to one particular side when kissing. Write down explicitly the hypothesis you are testing, and then calculate a p-value using the approach for testing a Bernoulli population discussed in Lecture 5. What does this p-value suggest? [2 marks]
3. Using R, calculate an exact p-value to test the above hypothesis. What does this p-value suggest?
Please provide the appropriate R command that you used to calculate your p-value. [1 mark]
4. It is entirely possible that any preference for head turning to the right/left could be simply a product of right/left-handedness. To test this we obtain handedness of a sample of different people. It was found that 83 people were right-handed and 17 were left handed. Using the hypothesis testing procedure for testing two Bernoulli populations from Lecture 5, test the hypothesis that the rate of right-handedness in the population is the same as the preference for turning heads to the right when kissing this data. Summarise your findings. What does the p-value suggest? [2 marks]