A marketing consultant was commissioned to conduct a questionnaire survey of the clients of a financial company. The total number of respondents was 650, of whom 220 had investments above a specified threshold. (i) Each respondent who had investments above the threshold was asked about the percentage of these investments that was held in the form of a certain type of trust. The respondents answered by ticking appropriate boxes and the results led to the following frequency distribution. percentage frequency 50 49 (a) Present these data graphically using a carefully drawn histogram. (b) Calculate the mean percentage for the full set of 220 such respondents. assuming that the frequencies in each category are uniformly spread over the corresponding range. [5] (ii) Calculate a 95% confidence interval for the percentage of such investors who would have investments above the threshold. [4] The same respondents with investments referred to in part (i) were also asked to specify their satisfaction with the current return received from their full portfolio of investment. This was in the form of a four-point qualitative scale: very satisfied, quite satisfied, a little disappointed, very disappointed. The following two-way table of frequencies was obtained. percentage in type of trust 50 very satisfied 1 6 quite satisfied 8 a little disappointed 10 35 29 37 very disappointed 3 4 7082 36 1959 27 In order to investigate whether there is any relationship between the percentage in such trusts and satisfaction with current return, a x test is to be performed. (iii) Calculate the expected frequencies for the above table under an appropriate hypothesis (which should be stated) and comment on why it would be inappropriate to carry out a x test directly with these data. [3] (iv) Combining die "very satisfied" and "quite satisfied" categories together and the "a little disappointed" and "very disappointed" categories together results in the following reduced two-way table. satisfied disappointed 50 9 35 43 33 13 41 30 16 Perform the required x2 test at the 5% level using this reduced table and comment on your conclusion. [7] [Total 19] 2 3 4 (i) Explain the difference between the central and the initial exposed to risk, in the context of mortality investigations. An investigation studied the mortality of infants aged under 1 year. The following table gives details of 10 lives involved in the investigation. Infants with no date of death given were still alive on their first birthday. Life Date of birth 1 August 2008 1 September 2008 1 December 2008 1 January 2009 1 February 2009 Date of death 1 February 2009 [2] G 1 March 2009 1 December 2009 7 1 June 2009 8 9 1 July 2009 1 September 2009 10 1 November 2009 1 December 2009 (ii) Calculate the maximum likelihood estimate of the force of mortality, using a two-state model and assuming that the force is constant. [3] (iii) Hence estimate the infant mortality rate, q [1] (iv) Estimate the infant mortality rate, 40 using the initial exposed to risk. [1] (v) Explain the difference between the two estimates. [2] [Total 9] (i) Define a Markov jump process. [2] A study of a tropical disease used a three-state Markov process model with states: 1. Not suffering from the disease 2. Suffering from the disease 3. Dead The disease can be fatal, but most sufferers recover. Let, p be the probability that a x+1 person in state i at age x is in state j at age x+t. Let u be the transition intensity from state i to state j at age x+1. (ii) Show from first principles that: d 13 1113 12.23 [4] The study revealed that sufferers who contract the disease a second or subsequent time are more likely to die, and less likely to recover, than first-time sufferers. (iii) Draw a diagram showing the states and possible transitions of a model which allows for this effect yet retains the Markov property. [3] [Total 9]