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A professional gambler has said: "Flipping a coin into the air is fair, since the coin rotates about a horizontal axis, and it is equally likely to be either way up when it first clips the ground. So a flicked coin is equally likely to land showing heads or tails. However, spinning a coin on a table is not fair, since the coin rotates about a vertical axis, and there is a systematic bias causing it to tilt towards the side where the embossed pattern is heavier. In fact, when a new coin is spun, it is more than twice as likely to land showing tails as it is to land showing heads." After hearing this, you carried out an experiment, spinning a new coin 25 times on a polished table, and found that it showed tails 18 times. Do the results of your experiment support the gambler's claims about the probabilities when a coin is spun? [3]A random sample, x,....*jo, from a normal population gives the following values: 9.5 18.2 4.69 3.76 14.2 17.13 15.69 13.9 15.7 7.42 Ex; = 120.19 Ex, =1,693.6331 (i) Test at the 5% level whether the mean of the whole population is 15 if the variance is: (a) unknown (b) 20. [5] (ii) Test at the 5% level whether the population variance is 20. [3] [Total 8]3 It has been decided to try to graduate a set of mortality data consisting of the deaths, {0,}, and the initial exposed risk, {E,}, by reference to a standard table, the mortality rates of which are denoted by {q? }. The formula to be used is where a is an unknown constant. Show that the maximum likelihood estimator of the true value of a is the solution of the equation (1 -aq; ) =0 4 You have been given the following set of data, where a refers to the time since a certain illness was diagnosed. The group of patients was also subject to censoring. age, ET 148.0 39 106.0 19 85.5 12 72.5 6 66.0 59.5 53.0 3 48.5 You have assumed that {0x} follow binomial distributions. You have decided to try to fit the function qx (Q) = 01(02)* (x = 0, 1, 2, . .. , 7) You have found that the M.L.E. of o is the vector (0.2534, 0.75335), and have constructed the following table of graduated mortality rates and standardised deviations: X 4x = q. (6) 2x = Exq, (1 - 9.) 0 0.25340 +0.28287 1 0.19090 -0.30529 2 0.14381 -0.09125 3 0.10834 -0.70086 0.08162 +0.27565 0.06149 +0.72394 0.04632 +0.35613 0.03490 -0.54183 Test the goodness-to-fit of your graduation