1 Introduction
Submission Instructions: Please follow these submission instructions:
1. No files are to be submitted via e-mail. Submissions are to be made via Moodle.
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.
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]
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.
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]
