# of the holders of its “smoker” and “non-smoker” life assurance policies. The following figures refer to medically examined lives (males and females combined). smokers actual expected deaths deaths 73 68.9

of the holders of its “smoker” and “non-smoker” life assurance policies. The following figures refer to medically examined lives (males and females combined). smokers actual expected deaths deaths 73 68.9 non-smokers actual expected deaths deaths 65 87.8 The “expected deaths” were obtained on the basis of the 1965-70 Select Basic Table; that is, the expected deaths in respect of a given sex, age and duration were obtained by multiplying the central exposed to risk by the appropriate central death rate (according to the standard table), and then aggregating over all smokers and non-smokers respectively. Assume that the force of mortality of smokers is α1 times the force of mortality of the standard table (for the same age and duration), and that the corresponding factor for non-smokers is α2. (i) Using a 5% significance level, test the null hypothesis that α1 = α2. (ii) Calculate the p-value of the test-statistic used in (i). (iii) Find estimated standard deviations of ˆα1 and ˆα2. (iv) Find an approximate 95% confidence interval for α1 − α2. . R.A.M. Case and A.J. Lea (Brit. J. prev. soc. Med. (1955) 9, 62-72) found the following facts in respect of a group of men who had been poisoned by mustard gas in the years 1917-18: actual deaths from cancer of the lung and pleura: 29 expected deaths: 14.0 The expected deaths were calculated on the basis of the cancer rates applicable to the general male population. Find an approximate 95% confidence interval for the ratio of the force of mortality from this form of cancer in respect of those exposed to mustard gas to that of the general male population. 2.1 (i) Explain with the aid of formulae what is meant by the term ‘Standardised Mortality Ratio’. (ii) For a certain year, the estimated mid-year population of women in England and Wales and the number of deaths from cancer were as follows: Exact Estimated Number of ages population deaths 0-35 12,039,500 939 35-50 4,741,100 3,925 50-65 4,192,400 15,549 65-80 3,471,900 29,337 over 80 1,148,600 17,544 For the same year, the corresponding data for women in a certain district were as follows: Exact Estimated Number of ages population deaths 0-35 83,902 5 35-50 29,970 31 50-65 30,102 98 65-80 19,220 149 over 80 5,799 82 Calculate the Standardised Mortality Ratio with respect to cancer for women in this district for the year in question, using the national rates as the standard. 2.2 The following data refers to permanent assurances, males, in 1988-89 at durations 2 years and over. age-group Smokers Non-smokers (nearest Actual 100A/E Actual 100A/E ages) deaths by AM80 deaths by AM80 ultimate ultimate < 30 9 118 24 84 31-45 57 86 57 52 46-60 286 90 182 50 61-75 167 91 116 53 > 76 75 98 30 61 Total 594 91 409 53 Source C.M.I. Bureau (i) Suppose that, at all ages, the force of mortality of non-smokers is αN times the force of mortality of AM80 ultimate, whilst that of smokers is αS times the force of mortality of AM80 ultimate. Find approximate 95% confidence intervals for αN and αS. (ii) Suppose that, at all ages, the force of mortality of the combined group (of smokers and non-smokers together) is αC times the force of mortality of AM80 ultimate. Find an approximate 95% confidence interval for αC . 2.3 Let θ ∼ Poisson(mE), where E is a constant and where m takes the (unknown) value ◦m. Show that ˆm = θ/E is the maximum likelihood estimator of ◦m, and also that it is an unbiased and efficient estimator of ◦ Answer all the following questions well

## Leave a Reply

Want to join the discussion?Feel free to contribute!