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Problem 7 [Total 30 marks] A random sample of size n = 25 from the gamma distribution with parameters 2 and 1 yields a sample mean of y - 11.9 and a sample standard deviation of s = 8.7. (a) Calculate three approximate central 95% confidence intervals for the mean of the distribution u- 21. Each interval formula should involve the standard normal 0.975-quantile, namely 1.96. Then indicate which of the three intervals you think is the 'best' and which is the 'worst'. Briefly explain why (in no more than 100 words). 120 marks (b) Test the null hypothesis 1 - 5 against the alternative hypothesis 1 > 5, using a significance level of 0.025 and a suitable test statistic that is a function of Y but not of $2, whose distribution you may approximate using a suitable normal distribution. Clearly define the test statistic and rejection region, report the observed test statistic, and make a conclusion based on the results. [10 marks)