Answer the questions below with explanations

In an extrasensory perception experiment carried out in a live television interview, the interviewee who claimed to have extrasensory powers was required to identify the pattern on each of 11] cards, which had been randomly assigned with one of ve different patterns. The cards were visible only to the audience who were asked to "transmit" the patterns to the interviewee. When the interviewee failed to identify any of the cards correctly, she claimed that this was clear proof of the existence of ESP, since there was a strong mind in the audience who was willing her to get the answers wrong. (i) State the hypotheses implied by the interviewee's conclusion and carry out a 5% test on this basis. Comment on your answer. [3] (ii) State precisely the hypotheses that the interviewer could have specied before the experiment to prevent the interviewee from \"cheating" in this way, and determine the number of cards that would have to be identied correctly to demonstrate the existence of ESP at the 5% level. [2] [Total 5] A random variable X is believed to have probability density function, f(x), where: f (x) = 31'(1 +x)4 x>0 In order to test the null hypothesis 1 =50 against the alternative hypothesis 1 = 60, a single value is observed. If this value is greater than 93.5, Ho is rejected. (i) Calculate the size of the test. [2] (ii) Calculate the power of the test. [2] [Total 4]1 Why is mortality by "amounts" often lighter than mortality by "lives"? 2 (i) Which form of selection is likely to apply in respect of lives who retire in ill-health before the normal retirement age? (ii) The following data refer to male pensioners who retired before the normal retirement age. The period of investigation was 1979-82, and the standard table was PA(90) (males), which may be taken as broadly suitable for "active" members and "normal" retirees. Exact ages Actual deaths 100A/E Under 55 308 388 55-60 1,247 145 60-65 4, 155 113 (a) Find approximate 95% confidence intervals for the ratio of the force of mortality of these early retirees to that of the given standard table (in each age-range separately). (b) State briefly what these results imply about the state of health of these early retirees. .3 Define the following terms briefly: (i) temporary initial selection; (ii) self-selection; (iii) reverse selection; (iv) adverse selection (or anti-selection.) 4 The following table refers to the experience of a large friendly society's industrial branch life assurance business on male lives in 1991-1993. ("Industrial branch" business refers to policies in which the premiums are collected at regular intervals from the policyholder's home.) Age Group Actual Number 100 A/E 100 A/E (age last of Claims birthday) (A) on basis (1) on basis (2) 1-20 11 220 263 21-30 25 96 145 31-40 19 89 156 41-50 192 18 73 51-60 602 38 60 61-70 2,233 59 102 71-80 1,384 69 92 81-90 3,107 65 81 91-104 195 73 70 Total 11,098 65 87 On basis (1), the expected deaths (E) were calculated by reference to English Life Table Number 12 - Males, whilst on basis (2) the mortality table was A1967-70 ultimate. (i) Find an approximate 95% confidence interval for the true Standardised Mortality Ratio (S.M.R.) of the group of policyholders with respect to each of the standard tables (a) E.L.T. Number 12 - Males, and (b) A1967-70 ultimate. (ii) At which of the following age-ranges does the experience of the policyholders (relative to the standard tables) appear to be least favourable