please address all the questions. Thank you

A poll was conducted with respect to a future referendum, where voters will answer either "yes" or "no" to a question on a political issue. A random sample of 1,106 eligible voters were asked in the poll and 603 of them answered "no". Letp denote the true population proportion of "no" voters. We wish to predict the result of the referendum, by testing the hypotheses Ho: p = 0.5 against H : p >0.5. (1) Perform a suitable test of these hypotheses at the 5% level of significance. stating your conclusion in terms of a predicted result for the referendum. [4] (ii) (a) Determine a 90% central confidence interval for p. (b) Explain the effect of using a larger sample on the confidence interval in part (ii)(a). [4] (ii) (a) Give the definition of the P-value of a hypothesis test. (b) Derive the -value of the test in part (i) stating your conclusion in terms of a predicted result for the referendum. Comment, with reference to your answers in parts (i) and (iii)(b), on the advantages of using a P-value approach for testing, compared with a fixed significance level. 16] [Total 14] An analyst wishes to model the number of Initial Public Offerings (IPOs) on the local stock exchange each calendar month. Let X, be a random variable denoting the number of IPOs in month i, where all My's are independent. The analyst wishes to model X, using a Poisson distribution but is aware that there is less activity during the summer so uses the following rates: i = July. August i = July, August Write down the probability function of X;. [2] The analyst wishes to estimate u and 2. using data from the last 12 months. (ii) Show that the log likelihood is given by ((x;2,u) = >x; InA -(10+20)2+ (* ,w/ + X Ang ) Inu + C where C is a constant, independent of a and A. [3] (iii) Derive the maximum likelihood estimators for a and ). You are not required to confirm that these estimators maximise the likelihood function. [5] [Total 10]For the Venn diagram below, use shading to identify the following regions: A (i) A'nB (ii) A'MB' (iii) (AnB)' (iv) (AUB)