Suppose you know (from other sources, e.g. your analytics team, that 2 s = 0.5, and your objective is to conduct a two-tailed test

Suppose you know (from other sources, e.g. your analytics team, that σ²_s = 0.5, and your objective is to conduct a two-tailed test the hypothesis that μ_s =2. The estimator we will consider is the sample average of S. 

1. Use the properties of the expectation and summation operator to show that the sample mean is an unbiased estimator for this population moment. Show every step in your work and list each property used as you go along. 

2. Suppose you know from elsewhere (e.g. your analytics team) that σ²_s = 4. Conduct a two-tailed test of the hypothesis that μ_s =2 at the 5% level. What do you conclude based on your dataset? Is the true CEO salary statistically distinguishable from 2 million dollars? Show every step in your work.