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An analyst wishes to compare the results from investing in a certain category of hedge funds, f, with those from the stock market, x. She uses on appropriate index for each, which over 12 years each produced the following returns ( in percentages to one decimal place). Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 Market (x) -5.0 -15.4 -25.0 16.6 9.2 18.1 13.2 ) -32.8 25.0 10.9 -6.7 Funds () 2.1 -3.7 -1.6 17.3 11.6 9.7 14.4 13.7 -19.8 19.5 -1.2 0.3 [r=0.101, [x* =0.3612, [/ =0.622, [/2 =0 1710. _of =0.1989 It is assumed that observations from different years are independent of each other. Below is a scatter plot of market returns against fund returns for each year. 25%% 20% X 15% X 10%% X X X X -40% -30% -20% X -10% .5%% C 10% 20% 30% Market -10% -15% X -20% Funds -75% (i) Comment on the relationship between the two series. [1] The hedge fund industry often claims that hedge funds have low correlation with the stock market. (ii) (@) Calculate the correlation coefficient between the two series. (b) Test whether the correlation coefficient is significantly different from 0. [7] Calculate the parameters for a linear regression of the fund index on the market index. [2] (iv) Calculate a 95% confidence interval for the underlying slope coefficient for the linear model in part (iii). [4] (v) Comment on your answers to parts (i)(b) and (iv). [2] [Total 16]The stem and leaf plot below shows 40 observations of an exchange rate. 1.21 9 1.22 4569 1.23 2679 1.24 3467889 1.25 011222345677778 1.26 00346688 1.27 1.28 For these data, Ex =50.000. (i) Find the mean, median and mode. [3] (1i) State, with reasons, which measure of those considered in part (i) you would prefer to use to estimate the central point of the observations. LIJ [Total 4] An insurance company experiences claims at a constant rate of 150 per year. Find the approximate probability that the company receives more than 90 claims in a period of six months. [4] The random variable X has a distribution with probability density function given by 2x ; 00 where 0 is the parameter of the distribution. (i) Derive expressions in terms of 0 for the expected value and the variance of X. [3] Suppose that X, X2,... \\",, is a random sample, with mean N', from the distribution of X. (ii) Show that the estimator 0 - 3.X 2 is an unbiased estimator of 0. [2] [Total 51