MATH235 Week 2 - Assessed problems (coursework)

Table 0.8 gives the consumer price index (CPI) for the UK, US, France and Germany from 2001 to 2010. CPI is a measure of the change in time of the price paid by the customer for a basket of fixed goods and services; here it has been standardised to be 100 in 2005. The data were provided by the World Bank International Monetary Fund and were obtained from http://datamarket.com/.

Assume that the data for the US are an IID sample from a Normal$({\mu }_1,{\sigma }_1^2)$ distribution and that the data for Germany are an IID sample from a Normal$({\mu }_2,{\sigma }_2^2)$ distribution. We wish to test whether or not the mean CPI for the US is the same as the mean CPI for Germany.

1. (a)

   Explain why a paired approach is appropriate for this test.

   [marks: 1]

2. (b)

   Calculate the differences $d_i=x_i-y_i$.

   [marks: 1]

3. (c)

   Using your answer to part (b) or otherwise, test at the 10% level whether or not there is evidence that the mean CPI for the US is the same as the mean CPI for Germany. You should state your $p$-value.

   [marks: 3]

Table 0.6 shows UK interest rates for January split into two decades, 1960–1969 and 1970–1979. Is there evidence that the mean interest rate in the 1960’s is lower than the mean interest rate in the 1970’s?

1. (a)

   Calculate the pooled sample variance for the data.

   [marks: 2]

2. (b)

   Carry out an appropriate $t$-test at the 5% level. What do you conclude about the difference between the mean interest rates in the two decades?

   [marks: 3]

Challenge
The geometric distribution gives probabilities for the number of Bernoulli trials required before the first success. The distribution has a single parameter $p\in [0,1]$ which can be interpreted as the probability of success for each trial. In the game of ludo a ‘six’ must be thrown on a six-sided dice before a player can start. A player recorded the number of throws that they made before obtaining a six in each of ten games. The same dice was used throughout all games, and the results are as follows:

$$1,0,6,7,0,1,3,11,3,1.$$

Assuming that these data come from a Geometric$(p)$ distribution, is there evidence that $p$ is greater than $\frac{1}{6}$, i.e. that the dice is loaded?

1. (a)

   Given that the expectation of a $\mathrm{Geometric}(p)$ random variable is $p^{-1}$, explain why $\widehat{p}={\overline{X}}^{-1}$ is a sensible estimator for $p$. Obtain an estimate of $p$ using the above sample.

   [marks: 2]

2. (b)

   By writing appropriate code in R, carry out a non-parametric bootstrap routine to obtain a 90% confidence interval for $p$. Hint you might want to adapt the code given in workshop question WS2.1.

   [marks: 2]

3. (c)

   Is there evidence at the 5% level that $p$ is greater than 1/6?

   [marks: 1]