Twenty values were assumed to be observed from a continuous distribution with mean 0.

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The following data are an IID sample from a Cauchy (0, 1) distribution: 1.77, -0.23, 2.76, 3.80, 3.47, 56.75, -1.34, 4.24, -2.44, 3.29, 3.71, -2.40, 4.53, -0.07, -1.05, -13.87, -2.53, -1.75, 0.27, 43.21. a. Graph the log likelihood function. b. Find the MLE for 0 using the Halley's method. Try the following starting points: -11, 0, 1.5, 4, 4.7, 7, 38, and 51. Consider also using the mean and median of the data as starting points. Discuss the results. 

Twenty values were assumed to be observed from a continuous distribution with mean 0. The values obtained are as follows: 

102 112 131 107 114 95 133 145 139 117

93 111 124 122 136 141 119 122 151 123 

Suppose now that, as in a simulation, we have the option of continually generating additional data values. How many additional observations do you think will we have to generate if we want to be 99% certain that our final estimate of 0 is correct within ±0.5 units? Discuss how you will proceed (include assumptions and/or reasons), present the necessary steps (as in an algorithm), and implement in R. 

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